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Numbers and Algebra
Basic Number Theory - Integers, Fractions, Decimals
Q.02
'[4]From (1/y) + (1/z) = (1/3) and (1/z) ≤ (1/y), we have (1/3) ≤ (2/y), hence y ≤ 6. Combining this with y ≥ 6, we get y = 6.\nSubstitute y = 6 into (1/z) = (1/3) - (1/6) = (1/6) to solve for z, yielding z = 6.'
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'The sum of natural numbers, squares, and cubes from 1 to n is represented as follows: (1) 1+2+3+...+n = \\frac{1}{2} n(n+1) (2) 1^{2}+2^{2}+3^{2}+...+n^{2} = \\frac{1}{6} n(n+1)(2n+1) (3) 1^{3}+2^{3}+3^{3}+...+n^{3} = \\left\\{\\frac{1}{2} n(n+1)\\right\\}^{2}'
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'Prove that the following inequalities hold for natural number n.'
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"What is the 'beginning' of mathematics? What should be considered as the beginning of mathematics?"
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'Find the discriminant D of the following quadratic equation and determine the type of its roots:'
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"In Greek mathematics, how was the proof of 'Even + Even = Even' established?"
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'Find the sum of terms 10 to 20 of an arithmetic sequence with the 8th term as 37 and the 24th term as 117.'
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'Find the general term of the given recurrence relation.'
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'(2) \ \\frac{1}{1-\\frac{1}{1-\\frac{1}{1+a}}} \'
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'Exercise 20 Grid Point Counting\n(1) Let k be a non-negative integer. The number of non-negative integer pairs \\( (x, y) \\) satisfying \ \\frac{x}{3} + \\frac{y}{2} \\leqq k \ is denoted as \ a_{k} \. Express \ a_{k} \ in terms of k.\n(2) Let n be a non-negative integer. The number of non-negative integer triples \\( (x, y, z) \\) satisfying \ \\frac{x}{3} + \\frac{y}{2} + z \\leqq n \ is denoted as \ b_{n} \. Express \ b_{n} \ in terms of n.\n[Yokohama National University]'
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'17 (1) multiplied by 10, product 29 (2) multiplied by 0, product 2 (3) multiplied by -4, product 4'
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'Find the sum of the given sequence of fractions by decomposing them into partial fractions to simplify the calculation. Use transformations like \\( \\frac{1}{k(k+1)} = \\frac{1}{k} - \\frac{1}{k+1} \\) for example.'
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'When real numbers x and y satisfy x^{2}+y^{2} ≤ 3, the maximum value of x-y-xy is □.'
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'When a>0, b>0, compare the sizes of (a+b)/2, √(ab), 2ab/(a+b), and √((a²+b²)/2).'
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'169 (1) \ \\frac{110}{3} \ (2) \ \\frac{37}{12} \ (3) \ \\frac{9}{8} \ (4) \ \\frac{14 \\sqrt{14}}{3} \'
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'Translate the given text into multiple languages.'
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'Since it is expressed as -2 = -2 + 0 \\cdot i, the complex conjugate of -2 is -2 -0 \\cdot i, which is -2. Therefore, the sum of -2 and -2 is -4, and the product of -2 and -2 is 4.'
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'Find the real and imaginary parts of the following complex numbers. (1) 2-√3 i (2) (-1+i)/2 (3) -1/3 (4) 4i'
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'45 \ \\frac{1}{\\tan \\frac{\\pi}{24}}-\\sqrt{2}-\\sqrt{3}-\\sqrt{6} \ is an integer. Find its value.'
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'Practice 42 \n When the equation of the given line is organized in terms of k, \n k(3x-2y-10) + x - 4y + 10 = 0 \n Find the conditions for which this equation holds true regardless of the value of k.'
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'Let {a_{n}} be a sequence with the initial term {a_{1}} up to the nth term {a_{n}} and the sum denoted as {S_{n}}. If {S_{n}+a_{n}=4 n+2}, then {a_{1}=} A {, a_{2}=} B. Expressing {a_{n+1}} in terms of {a_{n}} gives {a_{n+1}=C {a_{n}+} D}. Therefore, the general term of this sequence is {a_{n}=E}.'
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'Find the sum S of the arithmetic sequence from the first to the 100th term, with the first term being 1 and the common difference being -2.'
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'Gaussian symbol and summation of series, recurrence relation'
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'The sequence {an} satisfies a1=1, and for all natural numbers m, a2m=a2m-1+1, a2m+1=2a2m.'
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'Find the sum S of the arithmetic sequence 2, 17/6, 11/3, 9/2, ⋯⋯, 12.'
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'Drawing n chords on a circle, where any two chords intersect inside the circle and no three chords pass through the same point. The number of parts divided by these chords is denoted as D_{n}. In this case, D_{3}=口の□, D_{4}=1□, and D_{n}=ウ□. Furthermore, the number of parts that form polygons among D_{n} parts is denoted as d_{n}. When n is greater than or equal to 4, d_{n}=エ□.'
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'Prove by mathematical induction that for any natural number m, a_{3m} is a multiple of 5.'
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'Prove that for a sequence {an} (where {an} are greater than 0), if the relation (∑an)^2 = a1^3 + a2^3 + ... + an^3 holds, then an = n.'
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"Exercise 81 (1) |x| ≥ 1, so |t| ≥ 1. The slope of line OA is 1/t, the coordinates of the midpoint of line OA are (t/2, 1/2), so the equation of the perpendicular bisector of line OA is y-1/2=-t(x-t/2), i.e., y=-tx+(t^2+1)/2 (|t| ≥ 1). (2) y=-tx+(t^2+1)/2 gives t^2-2xt-2y+1=0. Let f(t)=t^2-2xt-2y+1, the required condition is {about real numbers t that make f(t)=0's discriminant D satisfy (1)}, so D/4=x^2+2y-1 ≥ 0, hence y ≥ -x^2/2+1/2. Real numbers t that satisfy (1) are all in -1<t<1, that is, satisfy {D ≥ 0 f(-1) > 0 f(1) > 0 -1 <x <1}, i.e., {y ≥ -x^2/2+1/2 y < x+1 y < -x+1 -1 <x <1}. Consider excluding the case of |t|<1 from all real number solutions that satisfy condition 1."
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'P ≥ 2√(a * 1/a) + 2√(b * 1/b) + 2√(c * 1/c) + 2√(abc * 1/abc) = 2 + 2 + 2 + 2 = 8 Therefore (a + 1/b)(b + 1/c)(c + 1/a) ≥ 8'
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'Find the general term of the sequence {a_n} determined by the following conditions.'
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'The first term is 96, the common ratio is -1/2, so the sum of the first 7 terms is 96{1-(-1/2)^7}/(1-(-1/2))=96/(3/2)(1+1/128)=64*129/128=129/2'
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'Which of the following sequences is a geometric progression?'
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'When throwing several dice at the same time, what is the minimum number of dice needed for the probability of the product of the numbers thrown to be even to be at least 0.994? Where log_{10} 2=0.3010, log_{10} 3=0.4771.'
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'23\\ n \\ 1,2,3 | 4,5,6,7,8 | 9,10,11,12,13,14,15 \\ mid 16, \\ cdots \\ cdots \\ n(1) \\ Find the first and last number in group \ n \. \\ n(2) \\ Find the sum of all numbers in group \ n \. \\ n(3) \\ In which group and at which position is 2014?'
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'Next, when 4^{10} is expressed in base 9, let the number of digits be denoted as n'
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'Math \ \\Pi \ 63 Therefore, \\( \\quad P(-1)=-a+b, P(1)=a+b \\) From (1), (2) we have \ -a+b=5, a+b=7 \ Solving simultaneously gives \ \\quad a=1, b=6 \ Therefore, the remainder we seek is \ \\quad x+6 \'
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'(2) \\( \\alpha=\eta=\\gamma=1 \\Leftrightarrow \\alpha-1=\eta-1=\\gamma-1=0 \\Leftrightarrow(\\alpha-1)^{2}+(\eta-1)^{2}+(\\gamma-1)^{2}=0 \\)'
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'Let integers a, b be not multiples of 3, and let f(x)=2 x^{3}+a^{2} x^{2}+2 b^{2} x+1. Find the remainders when f(1) and f(2) are divided by 3.'
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'(4) \\sqrt[4]{16}=\\sqrt[4]{2^{4}}= 2, \\quad \\sqrt[4]{625}=\\sqrt[4]{5^{4}}= 5 ,'
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'Find the sum of the numbers that satisfy the following conditions among two-digit natural numbers: (1) Numbers that leave a remainder of 3 when divided by 5. (2) Odd numbers or multiples of 3.'
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'(3) Just assuming that when is a multiple of is not sufficient for proof.\n[1] When \n, and since both and are multiples of , are both multiples of .\nTherefore, for , is a multiple of .\n[2] When , assuming is a multiple of , and are multiples of , so using integers , can be expressed. Consider .\n\nSince is an integer, is a multiple of . Therefore, when , is also a multiple of .\nFrom [1], [2], it can be concluded that for all natural numbers , is a multiple of .'
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'Find the general term of a geometric sequence where the 3rd term is 12 and the 6th term is -96. Assume the common ratio is a real number.'
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'(1) Since , by the inequality of arithmetic mean and geometric mean, we have .'
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'For a natural number n, when √(2n)+1/2>1, an is an integer greater than 1. For a natural number m, when an = m, m ≤ √(2n) + 1/2 < m + 1, i.e., m - 1/2 ≤ √(2n) < m + 1/2. Since m - 1/2 > 0, then according to the above, (m - 1/2)^2 ≤ 2n < (m + 1/2)^2, we get m(m-1)/2 + 1/8 ≤ n < m(m+1)/2 + 1/8.'
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'Mathematics B\n287\nFrom (1) we have a=6\nSubstitute this into (2) we get 6(36-d^{2})=162\nTherefore d^{2}=9\nTherefore d=±3\nThus, the 3 numbers we seek are 3,6,9 or 9,6,3\nIn other words\n3,6,9\nSince the order of the 3 numbers is not specified, the answer can be one way.\nAnother solution is to let the sequence of 3 numbers forming an arithmetic sequence be denoted as a, b, c. Based on the conditions\n2b=a+c\na+b+c=18\nabc=162\nSubstitute (1) into (2) we get 3b=18, hence b=6\nAt this point, from (1) and (3) we obtain a+c=12, ac=27\nTherefore, a, c are two solutions of the equation x^{2}-12x+27=0. Solving (x-3)(x-9)=0 gives x=3,9\nIn other words\n(a, c)=(3,9),(9,3)\nThus, the 3 numbers we seek are 3,6,9'
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'The total number of terms from the 1st group to the 11th group is 66. Therefore, the 77th term of the sequence {an} is the number of 11th group which is (77-66=11). Therefore, based on (1), the 77th term of the sequence {an} is 12*11^2=1452.'
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'There are 2 red books and n blue books. Randomly line up these n+2 books on the shelf. Let X be the number of blue books between the two red books.'
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"Mathematics II\nThat is \\[ \\left(p, q\\right)=\\left(0,0\\),\\left(-2,-2\\) \\]\nWhen \\( \\left(p, q\\right)=\\left(0,0\\) \\), from (2) we get \ \\quad a=0 \\nSince this does not satisfy the condition \ a>0 \, it is not appropriate.\nWhen \\( \\left(p, q\\right)=\\left(-2,-2\\) \\), from (2) we get \ \\quad a=4 \\nThis satisfies the condition \ a>0 \.\nTherefore, the integer we seek \ a \ is \ \\quad a=4 \ \2\. Similar to \1\, considering two equations\n\ x^{2}+a x+b=0 \\n(4), \ y^{2}+b y+a=0 \\nall have integer solutions.\nLet's denote the two solutions of (4) as \ p, q \, according to the relationship between solutions and coefficients\n\ p+q=-a, p q=b \\nBecause \ a>0, b>0 \, therefore, \ \\quad p+q<0, p q>0 \\nTherefore, \ p, q \ are both negative integers, i.e., integers less than or equal to -1.\nTherefore, let \\( f(x)=x^{2}+a x+b \\), then the graph of \\( y=f(x) \\) has intersection points only with the part of the \ x \ axis which is -1 or below.\nThus \\( \\quad f(-1)=1-a+b \\geqq 0 \\) i.e., \ \\quad a \\leqq b+1 \\nCombining \ a>b \, it follows that \ \\quad b<a \\leqq b+1 \\nTherefore, in this case, (4) becomes \\( x^{2}+a x+(a-1)=0 \\), which leads to\n\\[ \\left(x+1\\right)\\left(x+a-1\\right)=0 \\]\nThen, the integer solutions are \ x=-1,-a+1 \.\nNext, let's consider (5) that is \\( y^{2}+b y+(b+1)=0 \\) and find the values of \ b \ for which it has integer solutions.\nAssuming the two solutions of (5) are \ r, s\\left(r \\leqq s\ \\), according to the relationship between solutions and coefficients we have\n\ r+s=-b \\]\n\\[ r s=b+1 \\nBy eliminating b from (7), (8) we get \ r s+r+s=1 \, therefore \\( \\quad \\left(r+1\\right)\\left(s+1\\right)=2 \\)\nSince \ r, s \ are integers, so are \ r+1, s+1 \.\nFurthermore, from (7) and (8), like \ p, q \, \ r, s \ are also integers less than or equal to -1, so\n\ r+1 \\leqq 0, s+1 \\leqq 0 \\nThus, from 9) we get\n\\[ \\left(r+1, s+1\\right)=\\left(-2,-1\\) \\]\nThat means\n\\[ \\left(r, s\\)\\right=\\left(-3,-2\\) \\]\nIn this case, from (7) we get \ \\quad b=5 \\nand therefore \ \\quad a=5+1=6 \\nThus, the integers we seek \ \\left(a, b\ \\ are \\) \\left(a, b\\)=\\left(6,5\\)"
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'Translate the given text into multiple languages.'
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'(1) When n ≥ 2, the number of numbers from the 1st group to the (n-1)th group is ∑_{k=1}^{n-1}(2 k+1)=2 ⋅ \\frac{1}{2}(n-1) n+(n-1)=n^{2}-1, therefore, the first number of the nth group is the {n^{2}-1+1}=n^{2} (term) of a sequence of natural numbers, and this is also true for n=1. Hence, the first number of the nth group is n^{2}, and the last number of the nth group matches the number of terms in the sequence of natural numbers included up to the nth group ∑_{k=1}^{n}(2 k+1)=2 ⋅ \\frac{1}{2} n(n+1)+n=n^{2}+2 n'
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'Arithmetic progression general term and sum\nGeneral term If the first term is and the common difference is \n\\[\na_{n}=a+(n-1) d\n\\]\nArithmetic mean\nSequence is an arithmetic progression \nSum of an arithmetic progression Sum from the first to the th term \n(1) First term , th term (last term) \n\\[\nS_{n}=\\frac{1}{2} n(a+l)\n\\]\n(2) First term , common difference \n\\[\nS_{n}=\\frac{1}{2} n\\{2 a+(n-1) d\\}\n\\]\nSum of natural numbers, sum of positive odd numbers\n\\[\n\egin{array}{l}\n1+2+3+\\cdots \\cdots+n=\\frac{1}{2} n(n+1) \n1+3+5+\\cdots \\cdots+(2 n-1)=n^{2}\n\\end{array}\n\\]'
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'The sequence a, b, c forms an arithmetic sequence, so 2b=a+c. The sequence b, c, a forms a geometric sequence, so c^2=ab. Since the product of a, b, c is 125, then abc=125. Substituting (2) into (3) gives c^3=125. As c is a real number, c=5. Substituting into (1), (2) gives 2b=a+5, ab=25. Eliminating b gives a(a+5)=50, so a^2+5a-50=0. Therefore, a=5, -10. From ab=25, we get b=25/a. Example: \\triangleleft 36-d^2=27. The arithmetic mean form of a series 2b=a+c is used. Two numbers with sum p and product q are the two solutions of the quadratic equation x^2-px+q=0 (Math II). 4 (common ratio)=(2nd term)/(1st term). 4 a_n=2*(-3)^n is incorrect. 4(-1)^{可效}=-1. It is allowed to consider r^3=-8 by dividing (2) by (1). 4 a_n=ar^{n-1}. Arithmetic mean form of an arithmetic sequence. Arithmetic mean form of a geometric sequence. Substituting ab=c^2 into (3). The first equation. Substituting (2nd equation) multiplied by 2.'
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'The sequence {a_n} is a geometric sequence with first term {a_1} = \\frac{1}{4}-\\frac{1}{3}=-\\frac{1}{12} and common ratio -\\frac{1}{8}, so find the general term of the sequence {a_n}.'
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"In mathematics A, I learned about the concept of 'permutations and combinations'. When arranging numbers from 1 to n in a row, if the k-th number from the left is not k, it is called a perfect permutation. Also, the number of perfect permutations of n items is denoted as W(n), known as the Monge-Montel number, where W(1)=0, W(2)=1, W(n)=(n-1){W(n-1)+W(n-2)} (n ≥ 3) holds (for more details, refer to Chart Math I+A p. 264). Here, let's consider expressing W(n) in terms of n based on the recursion formula. Note that, for simplicity, we will consider the recursion formula rewritten as follows."
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'Assuming the l-th term of the sequence {a_n} is equal to the m-th term of the sequence {b_n}, and given the equation 15l-2=7・2^{m-1}, solve for the variables l and m.'
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'Common ratio: A mathematical term referring to ratios and proportions.'
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'Using natural number n, compare the size of n! and 3^n.'
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"In round (1), the number of seats for parties B and C was a total of 5 seats, but as in round (2), by forming party E through merger and assuming that the total votes remain the same as before the merger, with no change in the votes of other parties, the number of seats became 6. Therefore, it is possible for the number of seats to change as parties merge, but the following properties are known regarding D'Hondt proportional representation."
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'(2) Let the two integer solutions of the quadratic equation \ x^{2}-x-m=0 \ be \\( \\alpha, \eta(\\alpha \\leqq \eta) \\) . From the relationship between the solutions and the coefficients\n\ \\alpha+\eta=1 \\nRearranging we get\n\\[ \\text { (1), } \\alpha \eta=-m \\]\nSince \ m \ is a natural number, it follows that \ \\alpha \eta<0 \, hence \ \\alpha \ and \ \eta \ have opposite signs, \ \\alpha<0, \eta>0 \.\nFrom (1), we have \ \\alpha=1-\eta \\nSince \ \\alpha<0 \, we have \ 1-\eta<0 \\nTherefore, \ \eta>1 \'
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'(3) 1 + 2 + 2^2 + ... + 2^n = 2^{n+1} - 1, so an = 2^{n+1} - 1. 2008 = 4 * 502, therefore, from (2), the remainder when dividing 2^{2008} - 1 by 17 is 0. Thus, 2^{2008} = 17k + 1 (where k is an integer). Hence an = 2^{2011} - 1 = 2^{2008} * 8 - 1 = (17k + 1) * 8 - 1 = 17 * 8k + 7. Therefore, the remainder when dividing an by 17 is 7, and since 2012 = 4 * 503, then an = 2^{4 * 503} - 1, so from (2), a_{2012} = 2^{2014} - 1.'
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'\\(\\frac{1}{b-a}\\left(\\frac{1}{x+a}-\\frac{1}{x+b}\\right) = \\frac{1}{b-a} \\cdot \\frac{(x+b)-(x+a)}{(x+a)(x+b)} = \\frac{1}{b-a} \\cdot \\frac{b-a}{(x+a)(x+b)} = \\frac{1}{(x+a)(x+b)}\\)'
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'When the sequence starts with a₁=1, it satisfies the conditions. When a₁=2, a₂=1, so it still satisfies the conditions. Next, consider the case when a₁>2. Assuming that for all natural numbers n, aₙ>2, we have a₁>a₃>a₅>a₇>⋯ from (2). Therefore, there exists a natural number m such that a₂ᵐ⁺¹≤2. This contradicts the assumption. Hence, there exists a natural number n such that 1≤aₙ≤2. If there exists a natural number n such that aₙ=1, it satisfies the conditions. If there exists a natural number n such that aₙ=2, then aₙ₊₁=1, which also satisfies the conditions. Therefore, regardless of the initial value of a₁, the sequence {aₙ} always contains an element with the value of 1.'
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'Among the two-digit natural numbers, the numbers that are divisible by 55 with a remainder of 3 are 5·2+3, 5·3+3, ..., 5·19+3. This forms an arithmetic progression with the first term as 13, the last term as 98, and 18 terms in total, therefore, the sum is 1/2·18(13+98)=999'
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'Calculate the total amount of principal and interest after depositing 200,000 yen with an annual interest rate of 5% for 7 years.'
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'Assuming the votes for Party 1, Party 2, and Party 3 are 300,000, 300,000, and 100,000 respectively.'
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'(2) 20=2^{2} \\cdot 5,10=2 \\cdot 5, therefore, 20^{x}=10^{y+1}, thus 2^{2 x-y-1}=5^{y+1-x} (1). Assuming y+1-x \\neq 0, then from (1) we get 2^{\\frac{2 x-y-1}{y+1-x}}=5 \\cdots\\cdots\\cdot(2). When x, y are rational numbers, 2 x-y-1, y+1-x are also rational numbers, and \\frac{2 x-y-1}{y+1-x} is also a rational number. Furthermore, from (2) we have 2^{\\frac{2 x-y-1}{y+1-x}}>1, so \\frac{2 x-y-1}{y+1-x}>0, therefore \\frac{2 x-y-1}{y+1-x}=\\frac{m}{n}(m, n are positive integers), which can be expressed as 2^{\\frac{m}{n}}=5. Multiplying both sides by n gives 2^{m}=5^{n}, the left side is a multiple of 2, whereas the right side is not a multiple of 2, leading to a contradiction. Hence y+1-x=0. In this case, from (1) we get 2^{2 x-y-1}=1, thus 2 x-y-1=0 (4), (5). Solving the system of equations yields x=0, y=-1'
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'What is the formula to find the general term of an arithmetic sequence?'
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'Exercise 19 Gauss symbol and sum of sequences, recurrence formula'
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"Proportional allocation of seats method (1)..... D'Hondt method Introduction to the allocation of seats in Japan's national elections using the method of proportional representation. In proportional representation elections, the number of seats each party wins is determined using a calculation method known as the D'Hondt method based on the number of votes received by each party. Let's explain what this 'D'Hondt method' is and provide specific examples. *The 'D'Hondt method' is a method devised by Belgian mathematician Victor D'Hondt (1841-1902)."
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'Given , we have and . This implies or . Solving these equations we get or . Therefore, or . Hence, we conclude that .'
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'59 (1) 8^(1/8) < 2^(1/2) = 4^(1/4) (2) 2^30 < 3^20 < 10^10 (3) 6^(1/6) < √2 < 3^(1/3)'
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'Given an arithmetic sequence {a_{n}}, where the first term is a and the common difference is d, each term is represented as follows: \na, a+d, a+2d, a+3d, ..., a+(n-1)d. Find the nth term a_{n} of this sequence.'
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'Using a natural number n, compare the sizes of n² and 4^(n-2).'
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'Find the number of real solutions of f(x)=x^{3}+3 x^{2}-9 x-9.'
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'The coordinates of point R are from (-2+6)/2, (5-3)/2) to (2,1)'
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'Find the number of combinations b_n of integers x, y, z that satisfy the constraints x≥0, y≥0, z≥0, and (x/3)+(y/2)+z≤n.'
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'Find the sum of the numbers between 100 and 200 that satisfy the following conditions: (1) Numbers that leave a remainder of 2 when divided by 7. (2) Multiples of 4 or 6'
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"Explain the following properties of Pascal's Triangle:\n1. The numbers at the two ends of each row.\n2. Properties of each number except the ones at the ends.\n3. Array of numbers."
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'Translate the given text into multiple languages.'
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'Let real numbers p, q satisfy |p|≤1, |q|≤1, |p-q|≤1. Define the maximum of 0, p, q as M and the minimum as m. Prove the following inequalities hold.'
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'Two particles are located at vertex A of triangle ABC at time 0. These particles move independently, moving to a neighboring vertex with equal probability every 1 second. Let n be a natural number, and let the probability that these two particles are at the same point after n seconds be pn.'
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'From the given recurrence relation, for any natural number n, there exists a natural number a_{n} such that a_{n}<a_{n+1}. Therefore, when n \\geqq 2, a_{1}, ... a_{n-1} are not multiples of a_{n}, but a_{n} is a multiple of a_{n}. Next, for n \\geqq 2, using mathematical induction on m, it can be shown that for any natural number m, a_{n+m}-a_{m} is a multiple of a_{n}.'
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'Derive the real and imaginary parts from the following complex numbers.'
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'Show that the three inequalities a(1-b)>1/4, b(1-c)>1/4, c(1-a)>1/4 cannot hold simultaneously when a, b, c are all positive numbers less than 1.'
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'Comprehensive exercise 369 Since 2^{4n}-1 ≡ (-1)^n-1 (mod 17), when n is even, 2^{4n}-1 ≡ 0 (mod 17) and when n is odd, 2^{4n}-1 ≡ -2 ≡ 15 (mod 17) Therefore, the required remainder is 0 when n is even and 15 when n is odd (3) 2008=4 × 502, thus from (2) we have 2^{2008}-1 ≡ 0 (mod 17), meaning 2^{2008} ≡ 1 (mod 17) Hence, 2^{2011} ≡ 2^3 · 1 ≡ 8 (mod 17) 2^{2012} ≡ 2 · 8 ≡ 16 (mod 17) 2^{2013} ≡ 2 · 16 ≡ 32 ≡ 15 (mod 17) 2^{2014} ≡ 2 · 15 ≡ 30 ≡ 13 (mod 17) Therefore, a_{2010} ≡ 2^{2011}-1 ≡ 7 (mod 17) a_{2011} ≡ 2^{2012}-1 ≡ 15 (mod 17) a_{2012} ≡ 2^{2013}-1 ≡ 14 (mod 17) a_{2013} ≡ 2^{2014}-1 ≡ 12 (mod 17)'
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'Since \\(\\sum_{k=1}^{n}\\left(a_{k}-k\\right)^{2} \\geqq 0\\), the expression \1 \\cdot a_{1}+2 a_{2}+\\cdots \\cdots+n a_{n}\ is maximum when \\(\\sum_{k=1}^{n}\\left(a_{k}-k\\right)^{2}=0\\), which means \\(a_{k}=k (k=1,2, \\cdots \\cdots, n)\\). Therefore, the required sequence is \1,2,3, \\cdots \\cdots, n\.'
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'The quotient obtained by dividing the number of votes for each political party by 1, 2, 3, ... is as shown in the following table.'
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'Up to which term should the sum be taken from the initial term to maximize the sum? Also, find the sum at that point.'
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'Exercise (1) Prove the inequality |x+y+z| ≤ |x|+|y|+|z|.'
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'Calculate the following expressions using common logarithm tables and round the answers to two decimal places: (1) 2.37 × 3.79 (2) 7.67 ÷ 2.86'
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'Maximum value of 102 is 16, coordinates of point P are (5 / sqrt(26), 1 / sqrt(26)) or (-5 / sqrt(26), -1 / sqrt(26))'
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'62\n(1) (A) 3\n(B) \ -\\frac{5}{2} \\n(2) \ \\frac{13}{4} \'
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'Find the first term a and common difference d of an arithmetic sequence, where the sum of the first 5 terms is 125 and the sum of the first 10 terms is 500.'
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'Arrange the natural numbers as shown in the right diagram.'
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'Prove that when |x|<1 and |y|<1, |left|\\frac{x+y}{1+xy}|right|<1'
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'Let a and d be integers. Define the sequence {an} as an arithmetic sequence with first term a and common difference d. Let the sum of the first n terms of the sequence {an} be denoted by Sn.'
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'Since the first term is 2 and the common difference is 17/6-2=5/6, and if the 12th term is considered as the nth term, then 2+(n-1)・5/6=12, thus n=13. Therefore, the sum of the arithmetic series is calculated as S=1/2・13(2+12)=91'
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'Consider the sequence {Fn} determined by the following conditions.'
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'For a real number x, let [x] denote the largest integer that does not exceed x. Define the sequence {a_{k}} as a_{k}=2^[\\sqrt{k}] (k=1,2,3,......). For a positive integer n, find b_{n}=\\sum_{k=1}^{n^{2}} a_{k}.'
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'Prove that for all natural numbers n, 2^{n+1}+3^{2n-1} is a multiple of 7.'
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'Let x be a positive number. Prove that the inequality (x+1/x)(x+4/x) ≥ 9 holds true. Also, determine the conditions under which the equality holds.'
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'When k = 0, x = -1, 0, 4; when k = 12, x = -2, 2, 3'
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'Since and , from (1), it is conjectured that the general formula holds. We will now prove this using mathematical induction.'
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'The condition for a point (x, y) to be inside ΔOAB is expressed as x>0, y>0, x+y<1. Let 2x+y=X (2), x+2y=Y (3), then 3x=2X-Y and 2×(3)-(2) leads to 3y=-X+2Y, thus x=(2X-Y)/3, y=(-X+2Y)/3. Substituting these into (1) we get x>0 implies 2X-Y>0, y>0 implies -X+2Y>0, and x+y<1 implies (X+Y)/3<1, meaning X+Y<3. Therefore, Y<2X, Y>1/2X, and X+Y<3. Thus, the range of the moving point (X, Y) or (2x+y, x+2y) is the region represented by the system inequalities y<2x, y>1/2x, x+y<3 when variables are changed to x, y. Hence, the desired range is the slanted line portion in the right diagram, excluding the boundary lines.'
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'Find the sum of the geometric progression (1) from the first term of the geometric sequence to the nth term Sn.'
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'If the sequence {a_{n}+b_{n}} has initial term {a_{1}+b_{1}=2} and common ratio 2 as a geometric progression, find its general term.'
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'Let {an} be a geometric sequence with a non-zero common ratio and initial term of 1. Also, let {bn} be an arithmetic sequence satisfying b1=a3, b2=a4, b3=a2.'
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'Find the following sums:\n(1) Sum of the arithmetic sequence 2, 8, 14, ..., 98\n(2) Sum of the arithmetic sequence with initial term 100 and common difference -8 from the first to the 30th term\n(3) Sum of the arithmetic sequence with the 8th term as 37 and the 24th term as 117 from the 10th to the 20th term'
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'Therefore, the desired maximum and minimum values are as follows:'
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'For the arithmetic sequence {an} with first term 77 and common difference -3, answer the following questions: 1. Find the general term an. 2. Which term becomes negative for the first time. 3. At which term from the first term does the sum become maximum and what is that sum.'
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'Chapter 1 Sequences - 219\nUsing the TR difference sequence, find the general term of the following sequence \ \\left\\{a_{n}\\right\\} \.\n19\n(1) \ 20,18,14,8,0 \, \ \\qquad \ (2) \ 10,10,9,7,4 \,'
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'Find the first five terms of the sequence represented by the following formulas.'
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'Find the sum of integers from 1 to 100 that are neither multiples of 3 nor multiples of 5.'
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'Find the sum of integers from 1 to 100 that are neither multiples of 3 nor multiples of 5.'
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'TRAINING 26\nLet \ n \ be a natural number. Using mathematical induction, prove the following equation:\n\\[\n1 \\cdot 4+2 \\cdot 5+3 \\cdot 6+\\cdots \\cdots+n(n+3)=\\frac{1}{3} n(n+1)(n+5)\n\\]'
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'By mathematical calculation, the arithmetic mean is greater than or equal to the geometric mean, we have , so . The equality holds when and , which means .'
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'Simplify the fractions in (1) and (2). Calculate the expressions in (3) to (5).'
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'Let n be an integer greater than or equal to 2. Using the binomial theorem, prove the following:'
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'Given an arithmetic sequence {an} with initial term 1 and common difference 4, and an arithmetic sequence {bn} with initial term -9 and common difference 6. Find the general term of the sequence {cn} formed by the terms common to both sequences, arranged in ascending order.'
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'In example 1, the number of votes for party B is 7000, and for party C is 6000. In example 2, what will happen in the case where the number of votes for party E is 13000?'
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"What is the minimum score for students ranking within the top 64000 in last year's test? Choose from the following 0-5 options."
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"Let's list some typical examples of fraction calculations.\n(1) Simplification\n......Simplification is dividing the numerator and denominator of a fraction by their common factor. A fraction that cannot be simplified further is called an irreducible fraction.\nExample:\n\\(\\frac{x^{2}+7x+12}{x^{2}+8x+15}=\\frac{(x+3)(x+4)}{(x+3)(x+5)}=\\frac{x+4}{x+5}\\)\n\\\frac{12}{15}=\\frac{3 \\cdot 4}{3 \\cdot 5}=\\frac{4}{5}\"
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'Dividing each side by 3 gives the following result.'
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'Find the first term and common ratio of the geometric sequence. The common ratio is a real number.'
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'Let {b_{k}} be a geometric sequence with first term 1 and common ratio 3. For each natural number n, let the largest b_{k} satisfying b_{k}≤n be denoted as c_{n}. Calculate Σ_{k=1}^{30} c_{k}.'
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'The 3rd degree equation x^3 + ax^2 + bx + 1 = 0, where coefficients a and b are integers, has 2 complex solutions and 1 negative integer solution. The number of integer pairs (a, b) that satisfy this condition is □.'
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'Find the general term and sum of a geometric series. Let the first term be a and the common ratio be r.'
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'The following sequences are geometric progressions. Find the values of x and y. \n(1) 3, x, 1/12, ......\n(2) 9, x, 4, y, ......'
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'In an arithmetic sequence with a first term of -83 and a common difference of 4, up to which term will the sum from the first term be the smallest? Also, determine the sum at that point.'
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'In the order of sum and product, the values are (1) 4, -3 (2) 3/2, 3 (3) -4/3, -5/3'
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'(2)(1/2)^{n} <0.001 Taking the common logarithm of both sides, we get n log_{10} 2>-3 Therefore, n>3/ \\log_{10} 2=9.96... The smallest natural number n that satisfies this inequality is n=10'
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'There is an arithmetic sequence {an} with the first term 7 and a common difference of 3, as well as an arithmetic sequence {bn} with the first term 8 and a common difference of 5. Let {cn} be the sequence formed by arranging the common terms of these two sequences in ascending order. Find the general term of the sequence {cn}.'
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'Prove that the following inequalities hold when a>0, b>0.'
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'When depositing 200,000 yen at the beginning of each year with an annual compound interest rate of 1%, calculate the total principal and interest at the end of the 10th year (i.e., the total amount of principal and interest at the beginning of each year). Use 1.01 raised to the power of 10 equals 1.105 for calculation.'
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'Given an annual interest rate r, annually saving a yen in compound interest for n years, find the total amount of savings at the end of n years.'
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'Chapter 1 Sequences - 219\nUsing the method of finite differences, find the general term of the following sequence \ \\left\\{a_{n}\\right\\} \.\n19\n(1) \ 20,18,14,8,0 \, \ \\qquad \ (2) \ 10,10,9,7,4 \,'
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'When the three points A(1,1), B(2,4), C(a,0) are the vertices of triangle ABC and form a right triangle, find the value of the constant a.'
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'Find the following values. (1) \ \\sqrt[4]{16} \ (2) \ -\\sqrt[3]{64} \'
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'Find the remainder when the polynomial is divided by .'
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'When the inequalities 4x+y≤9, x+2y≥4, and 2x-3y≥-6 are simultaneously satisfied, find the maximum and minimum values of x^2+y^2.'
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'Find two numbers that satisfy the following conditions.'
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'The sequence {a_n} is defined by the first term and the recursive formula .'
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'Prove the inequality 2^{n}>4 n+1 when n is an integer greater than or equal to 5.'
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'Determine whether the following sequences are arithmetic sequences or geometric sequences.\n1. Sequence 4, 7, 10, 13\n2. Sequence 3, 6, 12, 24'
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"Let's review the sum of natural numbers and the sum of arithmetic sequences!"
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'Find the general term of the sequence defined by .'
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'A student A, who commutes by bicycle, went to school at a speed of 12 km/h one day, and on the way back, walked with his friend at a speed of 6 km/h while pushing the bike. Now, on this day, at what average speed did student A travel?'
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'Find the number of terms n and the common difference d of an arithmetic sequence where the first term is 2, the last term is 38, and the sum is 200.'
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'For the sequence {an} where the sum from the initial term to the nth term is given by Sn=-n^2+24n (n=1,2,3,...), find the range of natural numbers n for which an<0, and calculate ∑_(k=1)^40|ak|.'
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'Find the general term of a geometric sequence {an} with initial term a and common ratio r.'
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'Find the numbers of an arithmetic sequence with a first term of 3 and a common difference of 4, up to the 5th term.'
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'\\sum_{k=1}^{n} k^{2}=\\frac{1}{6} n(n+1)(2 n+1)'
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'(4) \ \\\\sqrt[3]{54} \\\\times 2 \\\\sqrt[3]{2} \\\\times \\\\sqrt[3]{16} \'
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'Find the nth term of the following sequences:\n1. Arithmetic sequence with first term 3 and common difference 2\n2. Geometric sequence with first term 2 and common ratio 3'
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"Find the number of terms 'n' and the common difference 'd' of an arithmetic sequence with initial term -10, final term 200, and sum 2945."
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'Find the sum of an arithmetic series with first term 25, last term -10, and 16 terms.'
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'For an arithmetic sequence with initial term -0.2 and final term 0.6, if there are n terms between the initial and final terms, the sum is 405.'
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'Using mathematical induction to prove the following equation'
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'(1) Find the general term a_{n} of a geometric sequence with first term 7 and common ratio 1/2. (2) Find the common ratio and the general term a_{n} of the following geometric sequences. (a) 3, -3, 3, -3, ... (b) -16/27, 4/9, -1/3, 1/4, ...'
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'\ 65 a=-3, \\quad b=10 \, solution: \ x=-2,2 \\pm i \'
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"Create the pattern shown in Figure 1 using Pascal's triangle, where even numbers are represented by ○ and odd numbers are represented by ●. By following the four rules based on the properties of Pascal's triangle, mark the positions with ○ and ●."
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'Prove the inequality (A>B) by creating the difference (A-B). Use the following methods:'
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'Find the values of x and y in the arithmetic sequence.'
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'Standard 47: Determination of two numbers given their sum and product'
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'The star Vega (Weaving Maiden Star) is a zero-magnitude star'
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'Determine which column from the left the second digit of 2020 is located in.'
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'Find the sum of the following numbers for integers from 1 to 200: (1) Multiples of 4 (2) Numbers that are not multiples of 4.'
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'Find the first term and common ratio of a geometric sequence. The common ratio is a real number. (1) The third term is 18, and the fifth term is 162. (2) The second term is 4, and the fifth term is -32'
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'In the same election as example 1, party B and party C merged to form a new party E, while maintaining the same total number of votes before and after the merger. Assuming that the votes of the other parties remain the same, the votes for party A are 10000, party D are 4000, and party E are 15300.'
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'Understand the formula for the sum of a geometric series and conquer Example 13!'
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'Let TR be \ \\log _{10} 2=0.3010 \. Find the value of a natural number \ n \ that satisfies the following conditions.'
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'Find the first term and common ratio of a geometric sequence. The common ratio is a real number. (1) The 3rd term is -18, the 6th term is 486 (2) The 6th term is 4, the 10th term is 16'
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'Using the 2nd order difference sequence, find the general term of the following sequence {a_{n}}. (1) 20, 18, 14, 8, 0, ...'
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'Find the sum S of an arithmetic sequence with first term 25, last term -10, and 16 terms.'
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'Divide the sequence of natural numbers such that each group contains 2n numbers as follows: 1,2|3,4,5,6| 7,8,9,10,11,12 | 13,14, …… (1) Find the first number in the nth group. (2) Find the sum of all numbers in the nth group.'
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'Find the general term of the sequence {an}: 5, 11, 23, 41, 65, 95, ...'
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'For two distinct real numbers a, b, if a, 2, b form a geometric sequence in that order, and 1/2, 1/b, 1/a form an arithmetic sequence in that order, then a=□, b=□.'
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'Basic Example 3 Determination of the 4th series (1)...An arithmetic progression {an} in which the 5th term is 3 and the 10th term is 18'
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'Explain what an arithmetic progression is and find the 10th term of an arithmetic progression with initial term 5 and common difference 2.'
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'(4) A point (x, y) in the coordinate plane is called a lattice point when both coordinates are integers. In this problem, "inside the region" refers to including the interior and boundary of that region.'
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'Let a be a positive constant. Determine the range of values for a such that 2x^{2}+y^{2}-1=0, x^{2}+y^2-4x-4y+8-a=0 have common points.'
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'Find the sum of the following geometric progressions.'
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'Find the pattern in the following sequences and express the general term in terms of n that follows the pattern.'
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'Find the general term of the sequence {an} defined by a1=3, an+1 = an/(2an + 4).'
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'Find the 10th term of an arithmetic sequence with a first term of 5 and a common difference of 3.'
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'Basic 58: Use long division to find the quotient and remainder of a division operation.'
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'Find the sum of integers from 1 to 100 that are multiples of 6 and those that are not multiples of 6'
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'There are many glass plates of the same quality. When 10 glass plates are stacked and light passes through them, the intensity of the light becomes 2/5 of the original. How many more glass plates should be stacked to reduce the intensity of the transmitted light to below 1/8 of the original? Given that log10 2 = 0.3010 and log10 5 = 0.6990.'
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'Find the 5th term of a geometric sequence with first term 5 and common ratio 2.'
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'Using second order different sequence, find the general term of the sequence {an}. (2) 10,10,9,7,4, ...'
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'Basic 5: Three numbers forming an arithmetic progression'
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'Prove using mathematical induction that for all natural numbers n, 4n^3 - n is a multiple of 3.'
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'Proof of Equation (3)...condition is a proportionality'
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'Find the general term and sum of an arithmetic sequence.'
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'Prove that the inequality |1+ab| > |a+b| holds when |a| < 1, |b| < 1.'
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'Translate the given text into multiple languages.'
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'For the sentences X and Y regarding the underlined part f of question 6, choose the correct combination of true or false.'
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'Q. (2) Answer the following questions by providing appropriate values as integers.'
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"When the side length of the black square is 9 cm, what range of integers should be arranged in the white square's grid? Please list all possible options."
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'For sentences X and Y regarding the underlined part b in question 2, choose the correct combination of true or false from below.'
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'To completely react 11.2mL of hydrogen, at least 5.6mL of oxygen is required. The volume of air containing 5.6mL of oxygen can be determined from the percentage of air in Table 1, which is 5.6 ÷ 0.21 = 26.66, rounded to 26.7mL.'
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'Person A leaves school between 0 and 60 minutes later, arrives at station K between 12 and 72 minutes later, and arrives at station M between 14 and 74 minutes later. In addition, the train leaves station K at times divisible by 8, and the train leaves station M at times divisible by 5, as shown in the diagram 1. However, it is not possible to determine the difference in waiting time from diagram 1, which is why diagram 2 is created by shifting the M station diagram 2 minutes to the right to align the arrival times at the station. From diagram 2, it is apparent that the waiting times are equal when arriving at the station in the bold line section. In this case, if Person A determines the time to leave school at station M, it can be deduced from 45-14=31 minutes later to 50-14=36 minutes later (A is 45-2=43 minutes later). If determined at station K, the elapsed time can be narrowed down from 43-12=31 minutes later to the time seen in the figure.'
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'There is a stack of 144 cards with numbers 1, 2, 3, ..., 143, 144 placed on top of each other as a stack with a box next to it.'
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"Answer the questions regarding Data 2, 'Graph of atmospheric pressure changes'.(1) Select appropriate words or symbols to fill in the [ ] and enclose them in a circle.\nAround a typhoon, the closer to its center, the lower the atmospheric pressure. Therefore, it is understood that the graph created from our school's observation data is [(I) [ (low) ]. Additionally, from the graph in Data 2, we can determine the time when the typhoon's center approached each observation point. Comparing Tokyo and Choshi in the graph, it becomes clear that Tokyo's graph showed [(III) [ (low) ] as the first to approach the typhoon's center, while Choshi's graph showed [(V) [ (high) ] as the first."
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"(5) The sedimentation rate of the Chiba section is 2 meters per thousand years, so the time required to stack from the layer of volcanic ash formed 773,000 years ago to the layer 1.6 meters above is 1000×1.6/2=800 (years). Therefore, from 773,000-800=772,200 years ago, the Earth's magnetic field shifted to its current orientation."
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'When comparing the numbers at the same index of columns A and B, which number has the largest difference? List all possible answers.'
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'(6) The train heading from Makuhari Station to Makuharihongo Station travels 600m in the first 60 seconds, and then 20 × 17.5 = 350m in the remaining 17.5 seconds. Therefore, the position where the trains pass each other is at a distance of 950m from Makuhari Station.'
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'Events happened in 1428, 1392, and 1489, so in chronological order, it should be I-II.'
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"The average values of temperature and other data announced by the Japan Meteorological Agency are calculated by averaging the numbers from the years where the last digit of the year is '1' and continuing for 30 years. Starting from May 19, 2021, the data for the years 1991 to 2020 have replaced the previous data from 1981 to 2010."
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'There are points A and B upstream and downstream of a river, and boats P and Q sail back and forth between them. Boat P departs from upstream point A, arrives at B, and immediately returns to A. Boat Q departs from downstream point B, arrives at A, and immediately returns to B.\nBoats P and Q depart simultaneously from points A and B, meet at point C, then meet again at point D. The distance between A and C is in a 3:2 ratio with the distance between C and B, and C and D are 120 meters apart.\nIn still water, the speeds of boats P and Q are constant, with the speed of boat Q being 1.5 times the speed of boat P. Boat P takes 48 minutes for a round trip between A and B. Additionally, assume that the speed of the river current is constant.\nAnswer the following questions:\n(1) How many minutes does boat Q take for a round trip between A and B?'
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'When the side length of the black square is 14 cm, the number of white squares is (14+1) x 4 = 60. Thus, 60 can be represented as the product of two integers: 60 = 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10.'
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'Please fill in the blanks appropriately. The vertical axis value of point (2) represents the population number of generation (A), and the vertical axis value of point (3) represents the population number of generation (B).'
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'(1) Since the cross-sectional area inside the plastic tube is 0.25 cm^2, the volume of nitrogen at 20°C is 0.25 x 14.0 = 3.5 (cm^3), and the volume of oxygen is 0.25 x 30.0 = 7.5 (cm^3).'
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"Finding the number of cases\n(1) First, find the 20th number of Mr. A. As shown in Figure 1, when the digit in the thousands place is 1, there are 4 possibilities for the hundreds place, 3 possibilities for the tens place, and 2 possibilities for the ones place, so for a four-digit number, we find that the 24th number from the left is 1976. From here, drawing a tree diagram from largest to smallest as shown in Figure 2, we can determine that the 20th number from the smallest is 1947. Also, Mr. A's card number is 2938."
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'(7) (1)~(3) To accumulate layers of the same thickness of 1m, it takes 500 years in Chiba, while it takes 5000 years in Italy. Therefore, the speed of layer accumulation in Chiba is 10 times faster, calculated as 1/500 ÷ 1/5000 = 10.'
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'Mathematics: 100 points (estimated score)\n1. Each 7 points x 3\n2. (1) 8 points\n (2) to (4) each 5 points x 3 < Each fully answered > □\n3. □ Each 7 points x 8'
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'Arrange white squares with a side length of 1 cm around a black square with a side length of 1 cm. The diagram below shows white squares arranged around black squares with side lengths of 1 cm, 2 cm, 3 cm, and so on from left to right. Inside the white square grids, the integer A is used A times, and two or more consecutive different integers are arranged starting from a certain integer. For example, as shown on the left side of figure 1, when the side length of the black square is 2 cm, using 3 of 3, 4 of 4, and 5 of 5 can be arranged precisely. However, as shown on the right side of figure 1, it is not possible to arrange precisely 4 of 4, 5 of 5, and 6 of 6. Also, as shown in figure 2, when the side length of the black square is 8 cm, integers from 1 to 8 and from 11 to 13 can be arranged precisely.'
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'2020 Shibuya Educational Institute Makuhari Junior High School Mathematics 1st Test\n1 (3) What card is left on the mountain at the end of the operation?'
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'At a certain time, 12 lights were on. How many possible times are there?'
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'In relation to part c under question 3, the ancient battlefield of the Battle of Yashima is located in present-day Kagawa Prefecture. Kagawa Prefecture is the birthplace of former Prime Minister Masayoshi Ohira. Choose the correct combination of statements A to D regarding events in the 1970s when Masayoshi Ohira served as Foreign Minister and Prime Minister from the options below and answer by number.'
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'For sentences X and Y regarding the underlined part d in question 5, choose the correct combination of true or false as the answer.'
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'Volcanic ash layers serve as clues to compare distant layers. Choose the appropriate option in the square brackets to explain the reasons, and encircle it with ○.'
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'(3) Definition of Unit\nThe standard unit of "mass" began to be "kilogram prototype" at the end of the 19th century. The reason is that the mass of "1000 cm^3 of water" varies depending on the conditions of the water. The "kilogram prototype" is a solid metal, so its mass does not vary with conditions. Think of a condition that would change the mass of "1000 cm^3 of water" and write it down.'
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'When A is divided by 15, the quotient and remainder are denoted as □, so A ÷ 15 = □ remainder □. Now, when P ÷ Q = R remainder S, then P = Q × R + S. Therefore, A = 15 × □ + □ = (15 + 1) × □ = 16 × □, which means A is a multiple of 16. Similarly, when A is divided by 17, the quotient and remainder are denoted as △, so A ÷ 17 = △ remainder △. Hence, A = 17 × △ + △ = (17 + 1) × △ = 18 × △, showing that A is a multiple of 18. Therefore, A is a common multiple of 16 and 18. Additionally, from the calculation on the right, we find that the least common multiple of 16 and 18 is 2 × 8 × 9 = 144, and thus A is a multiple of 144. Finally, since dividing yields A ≤ 16 × 14 = 224, the only number that satisfies the conditions is 144.'
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'React 11.2 mL of gas 3 with air. Provide the minimum volume of air required to ensure that gas 3 does not remain, rounded to the first decimal place.'
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'If the liquid is continued to be poured at the same rate after figure 3, find the time it takes for containers A and B to fill up respectively, and answer which container will be filled first.'
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'2020 Shiba Education Academy Makuhari Middle School 2nd (2)\n(2) How many minutes after departure did ship P and ship Q meet at point D?'
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'(2) The illuminance at a distance of 100 cm from the light bulb is 120 lux, and at a distance of 50 cm, it is 500 lux. Therefore, 120 divided by 500 equals 0.24, so the illuminance at a distance of 100 cm is approximately one-fourth of the illuminance at a distance of 50 cm.'
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'Hara Takashi organized the Rikken Seiyukai as its president, and formed the first full-fledged party cabinet in 1918, which corresponds to the 7th year of Taisho.'
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"In Japan, television broadcasting began in 1953, followed by the start of a high economic growth period in the late 1950s. During this time, household electrical appliances began to spread to homes nationwide, with black and white televisions, electric washing machines, and electric refrigerators being popularly known as the 'Three Sacred Treasures'. Air conditioning and automobiles, along with color television, were referred to as '3C', and became common in the latter half of the high economic growth period. The Public Election Law and Maintenance of Public Order Law were enacted in the late Taisho era in 1925, the same year that radio broadcasting began."
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'Regarding the underlined part B in question 3, choose the correct combination of true or false for the following sentences X and Y'
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'There are 33 types of candles A, B, C. When you light 3 candles, they will burn at a certain rate. After lighting A, light B after 10 minutes, then C after another 5 minutes. Candle C burned out first, followed by candles A and B burning out simultaneously. The graph below shows the time it takes for all candles to burn out after lighting candle A, as well as the relationship between the longest and shortest candle lengths. The length of a burned-out candle is considered to be 0 cm. Answer the following questions.'
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'For sentences X and Y related to the underlined part b in question 3, choose one from the following combinations and answer with the number which is correct.'
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'For the sentence X and Y regarding the underlined part c in question 3, select the correct combination of true or false, and then choose one number from the options below to answer. X Yokohama City was one of the first designated government ordinance cities in Japan, along with Nagoya City, Osaka City, Kyoto City, and Kobe City. Due to historical reasons, Y in Yokohama City, dyeing industries such as silk handkerchiefs and scarves have become local industries.\n\egin{tabular}{|llllllllll|}\n\\hline 1 & \ \\mathrm{X} \ & True & \ \\mathrm{Y} \ & True & 2 & \ \\mathrm{X} \ & True & \ \\mathrm{Y} \ & False \\\\\n3 & \ \\mathrm{X} \ & False & \ \\mathrm{Y} \ & True & 4 & \ \\mathrm{X} \ & False & \ \\mathrm{Y} \ & False \\\\\n\\hline\n\\end{tabular}'
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'Due to its original green color, the color change of red cabbage solution is considered as red. This color change indicates that the pH of solution A is below 2.5. An acidic solution that exactly neutralizes the 5 mL of solution B has a pH of 3.5, calculated as 7-(10.5-7)=3.5. Assuming solution A has a pH of 2.5, compared to the acidic solution with a pH of 3.5, the acidity level is 10 times stronger, indicating that the volume of solution A needed to neutralize solution B is 1/10 of B. If the pH of solution A is lower than 2.5, the volume of solution A needed for neutralization will be even less.'
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'Reading the graduations on this scale, P’ is 4mm, Q is 26mm. Therefore, the length of P’Q is found to be 26 - 4 = 22mm.'
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'Math problem (1) at Shibuya Gakuen Makuhari Middle School in 2021'
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'For the following sentences X and Y regarding the underlined part c in question 3, choose one correct combination of true or false from the table below and answer with the corresponding number.'
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"When the side length of the black square is 14 cm, what range of integers should be lined up in the white square's grid of squares? Please answer with all possible options."
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"Shinichi walks from his home to his friend's house along a straight road. Initially, he was running towards his friend's house but got tired, so he started walking from the midpoint between his home and his friend's house. As a result, he arrived 20 minutes later than if he had run all the way. On his way back, his mother comes to pick him up by car. Shinichi walks to his friend's house, while his mother drives from home, both leaving at the same time. They meet on the way back, where Shinichi gets into the car, and they are supposed to return home together. However, Shinichi left his friend's house 10 minutes later than planned. His mother, who left as scheduled, continues driving until she meets Shinichi, picks him up, but it takes longer than planned. Shinichi's walking speed is x, running speed is 2x, and car speed is 5x."
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'Provide appropriate values for the following blanks.'
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"In this manner, set A's tens place to 9 and B's tens place to 9. Thus, the remaining card difference is 2-1=8-7=1, so the largest difference is found in the pairs (6491, 4392) and (6497, 4398) (both differences are 2099). Next, consider the cases where A becomes 6491 or 6497. From (1) and (2), we know that there are 24 integers with a thousandth place of 1 or 4. Additionally, there are 6 integers with a thousandth place of 6 and a hundredth place of 1. Arranging integers with a thousandth place of 6 and a hundredth place of 4 in ascending order gives {6417, 6419, 6471, 6479, 6491, 6497}, so we find that 6491 is the 59th number and 6497 is the 60th number."
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'Measure the weight of 12 seeds and find the weight of water lost from the 12 seeds.'
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'If train A runs 0.2 km/h slower than the actual speed, it arrived at station K 18 minutes late from the scheduled time.'
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'(1) "Meters per second" is a unit that represents the distance traveled in one second, so it is a unit of speed.'
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"In question 1, the blanks A to O will be filled with either 'low' or 'high'. Choose the correct symbol combination representing the blank filled with 'high' from the options below and answer with the corresponding number."
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'Question 1 is worth 3 points each × 4, Questions 2 to 5 are worth 4 points each × 4, Question 6 is 6 points, Question 7 is 4 points, Question 8 is 10 points, Question 9 is worth 3 points each × 2'
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'Question 7 pertains to the underlined part f, Owari Province was a country located in the western part of present-day Aichi Prefecture. The prefectural capital of Aichi is Nagoya, but the figure on the right, Figure 5, depicts the situation of the rice disturbance that erupted in Nagoya. Looking at this Figure 5, choose the correct combination of the following statements A~D concerning the rice disturbance.'
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'Which of the following is a solid at room temperature? (1) Sodium hydroxide (2) Aluminum (3) Salad oil (4) Disinfectant alcohol (5) Carbon dioxide (6) Oxygen'
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'2021 Shibuya Education Academy Makuhari Middle School 2nd time question (3) (2) It takes 12 minutes to cover the distance at a speed of 1 km per minute, so the distance between M station and K station is 1 * 12 = 12 km.'
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"The 3rd Abe Cabinet was a coalition cabinet of the Liberal Democratic Party (LDP) and Komeito, with ministers chosen from Komeito as well. Prime Minister Shinzo Abe resigned on September 16, 2020, with a total tenure of 3188 days, surpassing Taro Katsura's 2886 days to become the longest-serving prime minister in history. Furthermore, since the formation of the second cabinet on December 26, 2012, the continuous tenure has been 2822 days, surpassing Eisaku Sato's 2798 days, also making it the longest in history. Therefore, the statement is correct."
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'Question 1 In relation to underlined part a, during the Jomon period, there was a custom of burying the deceased as shown in the image on the right. What is this type of burial called? Answer in kanji.'
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'When various groups were divided to conduct experiments 1 and 2, some groups found that the mixed solution did not enter the round bottom flask vigorously. Select all applicable reasons from the following options and provide the corresponding symbols.'
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'2021 Shibuya Education Academy Makuhari Middle School 2nd (2)'
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'2 (2) ÷ C = 15 with a remainder of 15, therefore, B = C × 15 + 15 = 15 × (C + 1) so B is a multiple of 15. Similarly, B ÷ D = 17 with a remainder of 17, B = D × 17 + 17 = 17 × (D + 1) so B is a multiple of 17. Thus, B is a common multiple of 15 and 17, with the least common multiple of 15 and 17 being 15 × 17 = 255, so B is a multiple of 255. Furthermore, C is greater than or equal to 16, and D is greater than or equal to 18, therefore B is at least 17 × (18 + 1) = 323. Thus, when 999 is divided by 255 with a remainder of 234, the largest 3-digit integer is calculated to be 255 × 3 = 765.'
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'The fractions that cannot be simplified are in ascending order from small to large {1/2021, 2/2021, 3/2021, ...}, and in descending order from large to small {2020/2021, 2019/2021, 2018/2021, ...}. Adding them up in pairs, the sum of each pair is 1/2021+2020/2021=2/2021+2019/2021=3/2021+2018/2021=1. Moreover, as there are 2020-88=1932 fractions that cannot be simplified, the number of pairs is 1932÷2=966. Therefore, their sum is 1×966=966.'
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'The 73 lugworms eaten by sandpipers A ate double the organic matter of 2.19 x 2 = 4.38 grams in two days, 15th and 16th.'
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'Select the correct option regarding the underlined part a in the following sentences X and Y.'
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"(3) The density of water (weight per unit volume) is highest at 4°C, and decreases at temperatures higher or lower than 4°C. In other words, the mass of '1000 cm^3 of water' varies with temperature, making it unsuitable as a standard for weight."
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"Explain why, regardless of the length of a black square's side, it is not possible to arrange only two consecutive integers in the white square's grid."
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'In the photo of Figure 6, the main scale and vernier scale are aligned at the 3.5 mark of the vernier scale. What is the diameter of the button in millimeters? Please answer to two decimal places.'
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'As shown in the diagram, there is a stack of 144 cards numbered 1, 2, 3, ..., 143, 144, placed on top of each other, and next to it there is a box.'
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'For the sentence X・Y regarding the underlined part j in question 10, which combination of correct and incorrect is the correct one?'
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'With 1 km = 1000 m and 1 hour = 60 minutes = 3600 seconds, 72 km/h is equal to 72 × 1000 ÷ 3600 = 20 (m/s).'
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'What is the maximum time interval from when you hear the sound of A to when you hear the sound of B (rounded to one decimal place)?'
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'White and black stones are arranged in a single row from left to right without having the same color stones appear consecutively more than 3 in a row. The diagram on the right was drawn to consider the ways in which 4 stones can be arranged using white and black stones combined. (1) How many ways are there to arrange the stones using a total of 6 white and black stones?'
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'Question 5 a Byodo-in Phoenix Hall is a hall built by Fujiwara no Yorimichi in 1053, in the latter half of the 11th century. b In 784, towards the end of the 8th century, Emperor Kanmu relocated the capital from the strong Buddhist influence of Heian-kyo to Nagaoka-kyo in Kyoto.'
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'Answer the questions about the heat released during the combustion of methane, propane, and butane.\n(1) Perform the following calculations:\n a) The amount of heat released when 0.7g of methane combusts\n b) The weight of 1L of propane and the heat it releases\n c) The weight of 1L of butane and the heat it releases'
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'Choose the appropriate content in the following [], and mark it with ○.'
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'Calculate the amount of heat required to raise the temperature of ice from -20°C to 0°C.'
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"The United Nations, established in October 1945 with 51 founding member countries after the end of World War II, has its headquarters in New York City in the eastern United States. As of the end of 2021, 193 countries are members of the organization. The expenses required for the UN's activities are mainly covered by contributions from member countries. These contributions are allocated every three years based on factors such as each country's economic power, and are decided by the General Assembly. Japan's share of contributions has ranked second after the United States for many years, but in recent years, it has dropped to third place, after the United States and China."
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'These are equal to 6 times K or N-number or 3N. Identify one of them.'
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'Which of the following is the same as the gas or precipitate obtained through operations a and b? Please answer with symbols.'
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'By counting the sub-scale between PQ, you can determine the length between PQ. How long is PQ in millimeters? Please answer up to two decimal places.'
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'In 607 AD, Ono no Imoko was dispatched to Sui (China) as a envoy during the time of Empress Suiko. In 804 AD, Kukai crossed over to Tang as a scholar monk on a mission ship, learned Esoteric Buddhism, returned to Japan, and became the founder of Shingon sect in Japan by building Kongobuji on Mount Koya (Wakayama Prefecture).'
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'I happened in 1936, II in 1925, III in 1914, IV in 1918. The Taisho era lasted until December 1926, after which the Showa era began, so it should be II-III-I-III.'
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'What is the distance between the smallest graduation lines on the scale in millimeters? Please answer up to two decimal places.'
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'(5) As the temperature rises by 1 degree Celsius, kerosene increases by 0.14% of the standard, and nitrogen gas increases by 0.36%. Therefore, 0.36 ÷ 0.14 = 2.57..., which is approximately 2.6 times.'
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'In the second exam of 2020, you scored 203 points. Does this mean you have reached the passing score?'
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'Using the 2021 Shibuya Education Academy Makuhari middle school (2nd round) (26) question, calculate the answer up to the third decimal place.'
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'Question 6. Choose the correct combination of true or false for the following sentences X and Y regarding the underlined part e.'
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'It can be inferred that many school facilities were built around 1978 based on the fact that most of these facilities are over 40 years old. After the end of World War II in the late 1940s, there was a baby boom, and by the early 1970s when that generation became parents, a second baby boom occurred. It is predicted that when the children born during this time go to school, there will be a shortage of school facilities such as classrooms and school buildings, leading to many local governments carrying out new constructions or renovations.'
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'When assigning teams to a tournament bracket, the arrangement where the match-ups in the first round are all the same and the potential match-ups in the second round are also all the same is considered to be the same. For example, the assignments in Figures 2, 3, 4, and 5 are considered the same, while the assignments in Figures 2 and 6 are considered different.'
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'In the food chain of the intertidal zone, consider migratory birds utilizing organic matter. Observing a sample of Eastern Curlew (hereafter, Curlew) that visits the intertidal zone, the amount of organic matter consumed by them was determined. Calculate the appropriate numerical values (to two decimal places) for the following sentences.'
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'2020 Shibuya Educational Academy Makuhari Middle School 第1次 (24) (3) For figure 2, answer the suitable numbers for the following parentheses.'
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'(2) There are 6 possible combinations of 4 seats. In each case, there are 24 ways for 4 people to sit (4 × 3 × 2 × 1), so the total is 24 × 6 = 144 ways.'
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'Starting from 1/2022, where the denominator decreases by 1 and the numerator increases by 1, a total of 2022 fractions are lined up. Look for fractions that can be simplified, such as 4/6=2/3. Answer the following questions: (1) Which position from the left is the first one that can be simplified? (2) Which position from the left is the third one that can be simplified? (3) Which position from the left is the 25th one that can be simplified?'
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'For the sentences X and Y regarding the underlined part b in question 2, choose the correct combination of true or false from the options below and answer with the corresponding number.'
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'Problem regarding the speed and transmission of sound'
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'When A leaves S Middle School between 2:00 PM and 3:00 PM, the waiting time for the train at any station remains the same. At what time does A leave S Middle School from 2:00 PM to what time?'
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'From (5) and (4), the length of D can be calculated as 4 + 0.55 = 4.55(mm).'
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"Question 1: For the sentence X and Y regarding the underlined part 'a', choose one correct combination of true or false from the options below."
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'For question 2, choose the correct combination of true or false statements regarding the underlined part b in relation to the following explanations X・Y.\nX A revision of the Public Offices Election Law, including an increase of 6 seats, was passed to eliminate the disparity in votes in the House of Councillors elections.\nY A revision to the Imperial Household Law was made to allow the abdication of the Emperor for one time.\n1. X - True, Y - True\n2. X - True, Y - False\n3. X - False, Y - True\n4. X - False, Y - False'
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'[Math] 100 points (estimated score) 8 points each for 1 and 2 × 6, 7 points each for 3 and 4 × 4 boxed 5 × 8 points each for 3'
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'Question 56 points question 6 to question 8 each 4 points (3 questions)'
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'In the Pleiades star cluster, we can see two types of red-colored stars. Bright red stars and dark red stars. How can bright red stars be perceived differently from dark red stars? Fill in the blank to complete the sentence.'
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'Question 9. Choose one correct combination from the following options regarding the explanation about the underlined part h (X) and Y.'
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'When dividing the seating arrangements for 3 seats into 5 cases, each case has 6 ways for 3 people to sit (due to permutations). Therefore, the total number of combinations is calculated as 22x6=132.'
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'Ship Q arrived at A after 36 ÷ 2 = 18 minutes from departure. By that time, Ship P had advanced 6 minutes from B, so the distance between both ships when Q arrived at A was 36 - 1 × 6 = 30. Therefore, the two ships met at D after Ship Q turned back from A, which is 30 ÷ (4+1) = 6 minutes later. This is 24 minutes after departure.'
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'Find the ratio of water depth when the same amount of water is poured into A and B.'
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'From <Experiment 2>, calculate the weight of 12 grains of seeds for each group and the weight of water lost when the 12 grains of seeds turn into popcorn, rounded to the first decimal place.'
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'You have four lights that can illuminate red, blue, yellow, and green, arranged in a row. Each time you press the switch, the colors of these four lights change according to a certain rule. Starting from the initial state, how many different rules can there be that result in the four lights illuminating different colors?'
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'Choose one correct combination of X and Y statements regarding the time when the work in question 3, part c, was completed.'
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'In condition 4, 11.2mL of hydrogen and 22.4mL of oxygen (33.6 - 11.2 = 22.4) are mixed, so 5.6mL of oxygen is used in the reaction, leaving 16.8mL.'
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'Question 9 1) Both prefectural governors and local council members have a term of 4 years. 2) The right to stand for election for senators and prefectural governors is granted to those aged 30 and above, while the right for representatives, mayors, and local council members is granted to those aged 25 and above. 3) In 2015, an amendment to the Public Offices Election Act lowered the age for suffrage from 20 to 18 for national parliamentarians, mayors, and local council members. 4) Prefectural governors have the authority to dissolve the prefectural assembly, but do not have the authority to dissolve city (ward) and town/village councils.'
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'For a non-zero integer c, calculate 8 △ c. Find the largest possible value of 8 △ c.'
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'Question 12 (Example) Part stipulating that the number required to convene an extraordinary session is at least one-fourth of the total members of any of the houses of the legislature.'
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'Choose one option from the list below and answer with a number.'
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'For the statement X and Y regarding the policy in the underlined part of question 5, choose one correct option as the correct and incorrect combination from the table below.'
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'If A leaves S Middle School between 2 p.m. and 3 p.m., there is less waiting time for the train at the station to go to K Station than to go to M Station. How many minutes in total does A spend from leaving S Middle School at 2 p.m. until 3 p.m.?'
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"(4) 1 day is 24 hours, 1 hour is 60 minutes, 1 minute is 60 seconds, so 1 day is 24×60×60=86400 seconds. The number '86400' is derived from this."
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'Read the sentence in (), use explanatory text and numerical values in the table to find the integer that fits in ().'
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"Since it took 12 minutes for boat P to travel from point A to point B, the downstream speed of boat P is 1800 divided by 12 equal to 150 meters per minute. Furthermore, the ratio of boat P's downstream speed to the river flow speed is 3:1, thus the river flow speed is 150 multiplied by 1/3 equal to 50 meters per minute. Converting this to speed per hour, it becomes 50 multiplied by 60 divided by 1000 equal to 3 kilometers."
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'In 2019, at the Shibuya Education Academy Makuhari Middle School, the following experiment was conducted: 5 mL of aqueous solutions of substances A to F commonly used in daily life were taken in test tubes and their colors were observed. A was toilet cleaner, B was water with shirataki (konjac) noodles, C was carbonated water, D was laundry bleach, E was rice vinegar, F was mirin. A was green in color, while B to F were almost colorless and transparent, with D to F slightly yellow. It was also discovered that the green color of A does not change with acidity or alkalinity. Then, equal amounts of purple cabbage solution were added to the A to F aqueous solutions, resulting in the following color changes. Question 1. (2): Which of A to F are alkaline aqueous solutions? Select all and provide the symbols.'
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'When the total number of black stones is 1000, how many rounds at most can the first white stone be surrounded by black stones?'
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'(5) The definition of a meter is based on the speed of light, which is 299,792,458 meters per second. The speed of light was first measured in 1676 through observations of the moons of Jupiter.'
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'You want to set the readable length with an interval of 0.02mm. You need to change the length of the sub-scale to 49mm. How many parts should you divide 49mm into?'
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'Generations 1 to 3 increase to 10→99→690. Then, continuing the same process from point 3 onwards, as generations progress, the number of individuals fluctuates and eventually approaches a certain fixed number (the intersection of the individual number graph and line L is 570). Furthermore, the amplitude of these fluctuations gradually decreases. Therefore, () and (セ) are options to choose.'
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'When divided by 9, 60 gives a quotient and a remainder of 6 each, which are equal. Also, when divided by 11, 60 gives a quotient and a remainder of 5 each, which are equal.'
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'[Math] 100 points (estimated score)\n1 (1) 2 points each × 3\n(2),\n(3) 7 points each × 2<(3) is a complete answer > '
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"Question 9 has the word 'national tax' just after the underlined part. In addition, it is the travelers who bear the tax of 1000 yen, but the tax is collected by the airline or shipping company by adding it to the ticket price 'as a general rule', so it is the airline or shipping company that pays taxes to the country, which is an indirect tax where the burden and payer of the tax are different. Income tax in 1 is a direct tax as national tax, resident tax in 2 is a direct tax as local tax, liquor tax in 3 is an indirect tax as national tax, and local consumption tax in 4 is an indirect tax as local tax, so 3 is chosen."
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'Problem of finding teams that have the potential to become runners-up'
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'Problem to find the number of ways of allocating when A enters I\nConsider a 1st round of 4 matches as I to IV, and consider the case where A enters I. In this case, there are 7 ways to choose the other team that enters I. Furthermore, the two teams entering I will be chosen from the remaining 6 teams, so there are 6×5÷2×1=15 possible combinations for I. Also, the two teams entering {II, III, IV} will be chosen from the remaining 4 teams, so there are 4×3÷2×1=6 possible combinations for II, but swapping II and IV results in the same combination, making the combinations for II and IV to be 6÷2=3. Therefore, the total number of different allocations is 7×15×3=315.'
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'For candles A, B, C, find the length before lighting them.'
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'Choose the correct combination of X and Y sentences regarding events from 1960 to 1965 related to the underlined part k, and answer with the correct number from below.'
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'When student A returns home from S middle school, they can choose to use either K station on the Seaside Railway or M station on the Wakaba Railway. K station is located south of S middle school and takes 12 minutes to walk from S middle school. In addition, M station is located north of S middle school and takes 14 minutes to walk from S middle school. At K station, trains depart every 8 minutes after 2 p.m., and at M station, trains depart every 5 minutes after 2 p.m. Answer the following questions.'
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'Compare the number of lights turned on at a certain time with the number of lights turned on one minute later. At what time does the number of lights turned on increase the most after one minute. List all possible times.'
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'Question 1 is 3 points each for a total of 4 questions; Question 2 is 9 points; Question 3 and 4 are 5 points each for 2 questions; Question 5 is 11 points; Question 6 and 7 are 3 points each for a total of 4 questions.'
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'For the sentence X and Y regarding the underlined part f in question 8, choose the correct combination of true or false from the options below.'
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'Arrange white and black stones in a row without having more than three stones of the same color in a row. The diagram on the right shows the possible arrangements using a total of 4 white and black stones.'
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'(5) The train from Makuhari station to Makuharihongo station travels 600m in 60 seconds, and the train from Makuharihongo station to Makuhari station travels from Figure 7 for 60 seconds, at a speed of 20 × 40 ÷ 2 + 20 × (60-40) = 400 + 400 = 800m. Therefore, at this time, the distance between the two trains is 2100 - (600+800) = 700m. As both trains are moving towards each other at 20 m/s, they will meet after 700 ÷ (20+20) = 17.5 seconds. Hence, the time taken for the two trains to meet after departing is 60 + 17.5 = 77.5 seconds.'
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'To a solution of B with pH 10.5, 5mL of A solution was added. How many milliliters are needed for exactly neutralization? Encircle the appropriate answer in the following square brackets.'
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'At 20°C, what is the volume of gas in 20 cm^3? Provide your answer to one decimal place for nitrogen and oxygen respectively.'
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'Solve the following problem regarding a sequence and a pattern.'
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"In the Chinese historical book 'Records of the Three Kingdoms,' it is written that in the early 3rd century, in 239 AD, Queen Himiko of Yamatai-koku sent envoys to Wei (China) and was bestowed with the title of 'Queen Who Is Loyal to Wei' and a bronze mirror by the emperor."
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'By doubling the amount of water in A midway, find the depth of the water at 32.5 minutes and the time at that moment.'
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'At room temperature, a small amount of the following substances were added to water and stirred with a glass rod. Select all substances that do not dissolve in water and write down their symbols.'
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'Which container, A or B, doubled in amount per minute? Please provide the time as well.'
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'2020 Shibuya Gakuen Makuhari Junior High School Mathematics 1st Exam\n1 (1) Use operations P and Q to manipulate cards from 1 to 144. What is the card to be placed in the box at the 42nd position?'
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'Among the gases produced in the underlined parts (1) to (6), there is only one type of gas that is different. Choose from (1) to (6) and also provide the name of the gas produced.'
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'From (3) more than (2), the distance between AD is known to be 4 × 6 = 24. Also, the distance between AC is 36 × 3/(3+2) = 21.6, so the distance between CD is 24-21.6 = 2.4. This corresponds to 120 m, so the distance of 1 is 120 ÷ 2.4 = 50 m, and the distance between AB is 50 × 36 = 1800 m, which is calculated to be 1800 ÷ 1000 = 1.8 km.'
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"Another example of a 'unit of assembly' is [meters per second], which is obtained by dividing distance by time. What physical quantity is measured in meters per second?"
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'In Basic Example 40, we are dealing with a sequence starting from the 0th term.'
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'Define a square S_n and a circle C_n (n=1,2,⋯⋯) as follows. C_n is inscribed in S_n, and S_{n+1} is inscribed in C_n. If the side length of S_1 is a, find the total circumference.'
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'Prove that the inequality √(ab) < (b-a)/(log b-log a) < (a+b)/2 holds when 0 < a < b.'
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'Let α, β be the two solutions of the equation x^2-2px-1=0, with |α|>1.'
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'If the inequality -4 ≤ x ≤ a holds, and the maximum value of y=√(9-4x)+b is 6, while the minimum value is 4. In this case, what are the values of a and b?'
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"Explain what is meant by the term 'initial term', provide its definition."
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'Solve the inequality when the function takes the value found in (1).'
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'Let the line with the smaller slope be denoted as \\ell, drawn from point (2,1) to the parabola y=\\frac{2}{3}x^{2}-1.'
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'Let a, b be natural numbers. Prove that if ab is a multiple of 3, then either a or b is a multiple of 3.'
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'(1) \ \\frac{4}{5}<x<4 \ (2) \ x \\leqq-2, \\quad 1 \\leqq x \ (3) \ 1<x<4 \'
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'When a=4, b=6, the largest integer x that does not satisfy inequality (1) is x= ◻.'
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'There are 2 ways to choose 3 numbers whose sum is a multiple of 3 from 0, 1, 2, 3, 4: [1] {0,1,2}, {0,2,4} [2] {1,2,3}, {2,3,4}. [1] Since the hundreds digit is not 0, there are 4 different 3-digit numbers for each group.'
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'Explain how to calculate the expected value of scores and find the expected value.'
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'Math I\nD = a^{2}-4 \\cdot 1 \\cdot\\left(-a^{2}+a-1\\right)=5 a^{2}-4 a+4 \\\n=5\\left(a-\\frac{2}{5}\\right)^{2}+\\frac{16}{5}>0\n\nTherefore, D>0 always holds.\n-3<-\x0crac{a}{2}<3 implies -6<a<6\nf(-3)=-a^{2}-2 a+8 f(-3)>0 implies\n a^{2}+2 a-8<0\nSolving which gives -4<a<2\nf(3)=-a^{2}+4 a+8 f(3)>0 implies\n a^{2}-4 a-8<0\nThe roots of a^{2}-4 a-8=0 are a=2 \\pm 2 \\sqrt{3}\nHence 2-2 \\sqrt{3}<x<2+2 \\sqrt{3}\n\n(a+4)(a-2)<0\nwhen a= -(-2)\\pm \\sqrt{(-2)^{2}-1 \\cdot(-8)}\n\nFind the common range of (1), (2), (3)\n2-\\sqrt{3}<a<2'
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'Prove that the product of three consecutive integers m-1, m, m+1, (m-1)m(m+1) is a multiple of 6. Similarly, (n-1)n(n+1) is a multiple of 6. Therefore, prove that m^3n - mn^3 is a multiple of 6.'
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'When using the numbers 1, 1, 1, 2, 2, 2, 3, 3 to form an 8-digit integer, how many integers can be formed?'
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'Find the number of two-digit natural numbers that satisfy the inequality 6x + 8(6 - x) > 7.'
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'Find all the integer solutions to the following system of equations.'
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'Basic Problem 39 Determining the Elements of a Set'
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'A and B work part-time together for 4 days a week. Show that there is at least one day each week when A and B work together.'
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'Find the smallest fraction that, when multiplied by 34/5, 51/10, and 85/8, results in a natural number product.'
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'Let a be a natural number. If a+5 is a multiple of 4 and a+3 is a multiple of 6, prove that a+9 is a multiple of 12.'
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'Let p, q, r be three consecutive odd numbers (p<q<r). Prove that pqr + pq + qr + rp + p + q + r + 1 is divisible by 48.'
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'Find the non-negative integer value of k for which the equation in x, k x^{2}-2(k+3) x+k+10=0, has real solutions.'
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'Prove that m^3 n - m n^3 is a multiple of 6 when m and n are integers.'
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'Find the number of integer tuples (a, b, c, d) that satisfy the following conditions:'
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'In the case where x < A in part (a), just as in the case when x≥A, it is seeking the range of values of x that satisfy (2). If we define the range of x values as (4), choose two appropriate contents for * from the following 0-ろ.'
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'Determining coefficients from maximum and minimum (2)'
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'From the 7 numbers 1, 2, 3, 4, 5, 6, 7, how many different 5-digit even numbers can be created without repeating any digit?'
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'Find the largest three-digit natural number that leaves a remainder of 1 when divided by 12 and a remainder of 4 when divided by 7.'
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'Choose 3 different numbers from the 7 numbers 0, 1, 2, 3, 4, 5, 6 to form a 3-digit number. How many integers can be created that satisfy the following conditions? (1) is a 3-digit number (2) is a multiple of 3 (3) is a multiple of 9'
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'Find the range of values of the constant for which the quadratic equation has two distinct real roots in the range . Let be the function, and the graph of is a parabola that opens downwards with its axis as the line . The condition for the equation to have two distinct real roots in the range is when the graph of intersects the -axis at two different points within the range . Thus, if we let the discriminant of be , the following conditions must simultaneously hold true. [1] [2] The axis lies within the range [3] [4] '
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'For the subset A, B of the universal set U={x | 1 ≦ x ≦ 10, x is an integer}, given that A ∩ B = {3,6,8}, complement of A ∩ complement of B = {4,5,7}, and A ∩ complement of B = {1,10}. Find the sets A, B, and A ∪ B.'
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"(1) How many natural numbers are there that become three digits when represented in decimal and also in quinary?\n(2) Prove that there are no natural numbers which become four digits in both decimal and quinary representations.\n[Similar to Tokyo Women's University]"
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'When traveling from point A to point B, 5 km apart, starting by walking at a speed of 5 km per hour, and later switching to running at a speed of 10 km per hour, what distance should one run at 10 km per hour or more in order to arrive at point B in 42 minutes or less?'
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'In example 27, when trying to find the integer part and the decimal part, I was told that the answer was incorrect. What is the mistake?'
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'Throwing three indistinguishable dice of the same size, how many ways are there for the sum of the numbers to be a multiple of 7?'
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'There are three men: Matsuo, Takeo, and Baio, and three women: Yukimi, Tsukimi, and Hanami, a total of 6 people holding hands to form a ring. How many ways can the ring be formed in the following way:\n1. Matsuo and Yukimi holding hands.\n2. Men and women holding hands alternately.\n3. Three men and three women hold hands in a row.'
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'Using the numbers 0, 1, 2, 3, 4, create integers greater than or equal to 1 and arrange them in ascending order.'
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'An expression in which only the signs change when swapping any two characters is called an alternating expression.'
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'Find the largest three-digit natural number such that when divided by 11, the remainder is 9, and when divided by 5, the remainder is 2.'
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'(1) (A) \ \\frac{7}{9} \ (B) \ \\frac{41}{11} \ (C) \ \\frac{45}{37} \ (2) 5'
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'State the inverse and contrapositive of the following proposition regarding integers a, b, c, and discuss their truth values. If 240 ulcorner a^{2}+b^{2}+c^{2}► is odd, then at least one of a, b, c is odd. Inverse: If at least one of a, b, c is odd, then a^{2}+b^{2}+c^{2} is odd. Inverse is false (Counterexample: a=1, b=1, c=0) Contrapositive: If a, b, c are all even, then a^{2}+b^{2}+c^{2} is even. Contrapositive is true (Proof) If a, b, c are all even, then integers k, l, m can be used to represent a=2k, b=2l, c=2m, and thus a^{2}+b^{2}+c^{2}=(2k)^{2}+(2l)^{2}+(2m)^{2}=2(2k^{2}+2l^{2}+2m^{2})'
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'Find the largest three-digit natural number that leaves a remainder of 9 when divided by 11 and a remainder of 2 when divided by 5.'
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'Principle of multiplication (applies to three or more items as well). If there are a ways in which event A can occur, and for each of those cases event B can occur in b ways, then there are a x b ways in which both A and B can occur.'
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'Let D be the set of all integers divisible by 3. (4) Let C = {x+y | x ∈ A, y ∈ B}. Show that C is equal to the set of all integers divisible by 3.'
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'Permutation, Circular Permutation, Permutation with Repetition'
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'Example 97(1): Determine the range of existence of solutions to a quadratic equation with x > 2.'
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'Math A \n㓢解 When using the numbers 0 to 5, a 4-digit or less integer can be constructed with 4 digits.\n\\[6^{4}=1296 \\text { (count) }\\]\n\nExcluding the case of 0000, the total number of positive integers we are looking for is \ 6^{4}-1=1295 \ (counts)'
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'Number of elements in a set, basic concepts 1) Number of elements in a set Number theorem Let A, B be finite sets (sets with a finite number of elements). Also, n(P) denotes the number of elements in the finite set P. (1) Number of elements in the union set 1 n(A ∪ B)=n(A)+n(B)-n(A ∩ B) 2 If A ∩ B=∅, then n(A ∪ B)=n(A)+n(B) (2) Number of elements in the complement set n(A^)=n(U)-n(A) where U is the universal set In this book, the above (1) and (2) are called the number theorem. For sets, refer to Mathematics I pages 68, 69.'
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'For the positive divisors of 6400: Find the sum of all those that are multiples of 5.'
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'Find the remainder when 13 raised to the power of 30 is divided by 17.'
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'Let n be a positive integer. Prove the following: (1) n² + 1 is a multiple of 5 if and only if the remainder when n is divided by 5 is 2 or 3.'
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'When two dice are rolled simultaneously, let the smaller number be X and the larger number be Y (or the number if they are equal). If a constant a is an integer from 1 to 6, find the following probabilities.'
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'Find the value of x^2 + 4xy + 3y^2 + z^2 when x=199, y=-98, z=102.'
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'In a certain high school, 140 students were surveyed about their proficiency in Japanese, Mathematics, and English, with a scale of (3)10. The results showed that 86 students were proficient in Japanese, 40 students were proficient in Mathematics. Furthermore, 18 students were proficient in both Japanese and Mathematics, 15 students were proficient in both Japanese and English, 101 students were proficient in either Japanese or English, and 55 students were proficient in either Mathematics or English. Additionally, there were 20 students who were not proficient in any of the three subjects. At this point, the number of students proficient in all three subjects is represented by A, and the number of students proficient in only one subject is represented by B.'
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'65 (1) when x= ± 1, the maximum value is 5, and there is no minimum value. (2) when x= 3/4, the maximum value is 71/64, and there is no minimum value.'
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'(1) Choose 3 numbers from 1, 2, 3 allowing duplicates. Find all combinations where the sum of the selected numbers is a multiple of 3.\n(2) Prepare 3 cards with the number 1, 3 cards with the number 2, and 3 cards with the number 3, totaling 9 cards. When randomly selecting 3 cards from these, calculate the probability that the sum of the numbers on the cards is a multiple of 3.'
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'A computer consists of switches that represent two states: on and off. By considering on as 1 and off as 0, binary becomes the basis of the structure. A bit is the smallest unit that represents the amount of information. With n bits, it is possible to represent 2^n different information.'
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'Answer the following question about subsets of real numbers.'
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'Calculate the total number of circular permutations of 7 people, and then consider the total number of ways to arrange them so that women are not adjacent.'
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"Using De Morgan's laws A∪B = A∩B, A∩B = A∪B, let's find the number of elements in the intersection of sets A and B where the integer is not divisible by 53 or 8. The complement set's number can be obtained by subtracting from the total number of elements in the universal set."
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'When there are 4 10-yen coins, 6 100-yen coins, and 2 500-yen coins, how many different total amounts can be made? Note that it is not possible to make a payment if all the coin quantities are 0.'
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'Given sets A = {8, 12}, B = {4n | 1 ≤ n ≤ 6, n is an integer}, express set B by listing its elements.'
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'For two integers a and b, if there exists an integer k such that a=bk, then b is called a divisor of a, and a is a multiple of b. Since a=bk, it can also be written as a=(-b)⋅(-k), therefore if b is a divisor of a, then -b is also a divisor of a.'
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'Let n be an integer. Prove that if n^2 is a multiple of 5, then n is also a multiple of 5.'
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'(1) (1) \\n is one of the integer solutions of (1). \\Therefore From (1)-(2) we get \\[ 8(x-2)-3(y-5)=0 \\] Which means . Since 8 and 3 are coprime, is a multiple of 3. Therefore, for an integer , it can be expressed as . Substituting this into (3) we get . Therefore, all the integer solutions of (1) are given by \\[ x=3k+2, y=8k+5 \\quad (k \\text{ is an integer})$'
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'When going from point A to point B, which are 5 km apart, one starts walking at a speed of 5 km per hour and then switches to running at a speed of 10 km per hour. In order to reach point B in 42 minutes or less, how many kilometers should one run at a speed of 10 km per hour?'
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'Let a, b be integers. When a is divided by 7, the remainder is 3, and when b is divided by 7, the remainder is 6. Find the remainder when the following numbers are divided by 7.'
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'How many natural numbers below 1000\n(1) are divisible by 2 or 7?\n(2) are not divisible by 2?\n(3) are not divisible by either 2 or 7?'
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'Set and necessary and sufficient condition problem 4) Set and necessary and sufficient conditions The set of all those satisfying conditions p, q is denoted by P, Q respectively, the following holds.'
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'Find the sum of all 4-digit integers that can be formed using 1, 2, 3, 4, 5, 6, 7.'
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'Another solution: There are three cases where the product of three numbers is a multiple of 4. 1. When one number is 4 and the other two are odd, there are 3 possibilities for the dice with 4, 3 dice sizes (1 x 3 x 3) x 3 = 27 (possibilities). 2. When two numbers are even and one is odd, there are 3 possibilities for the odd dice size, three dice sizes (3 x 3 x 3) x 3 = 81 (possibilities). 3. When all three numbers are even, 3 x 3 x 3 = 27 (possibilities). Therefore, according to the sum rule, the total number of cases is 27 + 81 + 27 = 135 (possibilities).'
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'Please explain how to specify k when classifying an integer n as a non-negative integer or a natural number n.'
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'Multiply a two-digit natural number B by 9 and add 72, if the hundreds digit is 6 and the ones digit is 5, find the value of B.'
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'The weight of box A is 95 g, and the weight of box B is 100 g. There are 20 balls weighing 12 g each. When these balls are divided into boxes A and B, box A is heavier. So, when one ball is moved from box A to box B, box B becomes heavier. How many balls were initially put in box A?'
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'(4) The number of cases where Y=a is equal to the number of cases where Y ≤ a minus the number of cases where Y ≤ a-1. The number of cases where Y ≤ a is the number of permutations of taking 2 out of a total of a numbers from 1 to a with duplicates allowed, which is a^2. When 2 ≤ a ≤ 6, the number of cases where Y ≤ a-1 is the number of permutations of taking 2 out of the numbers from 1 to a-2 with duplicates allowed, which is (a-1)^2. Therefore, the number of cases where Y=a is a^2-(a-1)^2. When a=1, there is only 1 case where Y=1, which also satisfies this formula. Hence, the required probability is (a^2-(a-1)^2)/36 = a/18 - 1/36'
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'Find all sequences of three natural numbers (a, b, c) that satisfy the conditions a < b < c and 1/a + 1/b + 1/c < 1/3, and determine the sequences where c is the smallest.'
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'The average for 11 years of scallop data is 296,332 t, with an additional catch of 235,952 t in 2017, what is the average for 12 years?'
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'There are two types of cards, one with 3 written on it and the other with 7 written on it, totaling 30 or more cards. Also, the sum of all the numbers on the cards is 110. In this case, find out how many 3 cards and how many 7 cards there are.'
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'Question 103 (1) a= \\pm 6, \\pm 12, \\pm 24, \\pm 48, \\pm 96'
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'How many non-negative integer solutions (x, y, z) satisfy x + y + z = 9?'
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'Consider the real numbers as the universal set, and let A={x | −1 ≤ x < 5}, B={x | −3 < x ≤ 4}, C={x | k−6 < x < k+1} (where k is a constant).'
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'There are 6 cards with the numbers 1 to 6 written on them. Put them into 3 boxes A, B, C.'
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'Taking a=1, c=3 as constants, vary only the value of b. Here, D=b^{2}-12.'
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'100 diagram\n(1) y ≤ 0\n(2) y ≤ 1/2\n(3) 0 ≤ y ≤ 6\n(4) 1 ≤ y < 4'
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'Using the 5 numbers 0, 1, 2, 3, 4, create a 5-digit integer with all digits being different, and arrange these numbers in ascending order. The same number will not be used more than once.'
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'When two dice are odd numbers and the third die is a 2 or a 6, there are 3 possibilities of getting a 2 or 6: (3 × 3 × 2) × 3 = 54 (possibilities). Therefore, the total number of cases is 216 - (27 + 54) = 216 - 81 = 135 (cases).'
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'Let a, b be constants, represent x²-5x+6≤0 as (1), and x²+ax+b<0 as (2). There are no values of x that satisfy both (1) and (2), but when x lies in the range 2≤x<5, it satisfies either (1) or (2). In this case, a=2, b=100 million.'
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'Find all pairs of integers x and y that satisfy the following equation: (1) (x + 2)(y - 1) = -6'
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'Using the PR contract style, answer the following questions. 1) Find the remainder when 4124 (1) 13^2017 is divided by 5. 2) Prove that for all positive integers n, 3^3n-2 + 5^3n-1 is a multiple of 7.'
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'This is a route to F₁, represented by 6 → and 3 ↑, so there are a total of 9!/(6!3!)=84 ways.'
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'Find the number of permutations that can be made by taking any 4 characters from the word mathematics.'
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'Today is Sunday, and 10 days later is Wednesday. What day of the week is it 100 days and 1 million days later?'
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'Please explain why there is no solution for the inequality |3x-6|<0.'
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'Two brothers have a total of 52 pencils. Now, after the elder brother gives exactly one-third of his pencils to the younger brother, he still has more than the younger brother. Furthermore, after giving 3 more pencils, the younger brother ends up having more. Find out how many pencils the elder brother originally had.'
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'Out of 42 students, 35 students use bicycles and 30 students use trains. Therefore, the maximum number of students who do not use either bicycles or trains is A, and the minimum number of students who use both bicycles and trains is B. The minimum number of students who use only bicycles is C, and the maximum number is D.'
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'Let U = {x | x is a positive integer less than or equal to 15} be the universal set. For subsets A, B, and C of U, A = {x | x is a multiple of 3, x ∈ U}, C = {2,3,5,7,9,11,13,15}, and C = (A ∪ B) ∩ (¬(A ∩ B)) holds true.'
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'Arrange the numbers 3, 4, 5, 6 to form a four-digit number m. Let n be the four-digit number formed by reversing the digits of m. Show that m + n is a multiple of 99.'
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'Utilization of integer solutions for linear Diophantine equations'
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'\ 116 \\quad R=\\frac{14 \\sqrt{3}}{3}, r=\\sqrt{3} \'
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'Find the largest 4-digit natural number that leaves a remainder of 2 when divided by 106 and a remainder of 5 when divided by 6.'
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'Find the natural number solutions of a linear Diophantine equation'
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'When you choose 6 out of the 7 letters in HGAKUEN to form a string and arrange it in dictionary order, what is the position of GAKUEN in the list? Assume that each letter cannot be repeated.'
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'The set {a, b, c, d, e} has 5 elements, and deciding whether each element belongs to the subset or not determines one subset. Therefore, the number of subsets is 2^5=32 (count).'
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'Regarding the positive factors of 6400: Determine the number that are even.'
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'Assume a positive integer represented in decimal is converted to octal and becomes a 3-digit number abc(8), and when converted to septenary becomes a 3-digit number cba_(7). What is this number in decimal form?'
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'Answer the following dates:\n100 days later is Tuesday, 1 million days later is Monday'
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'(1) Find an integer a that is a multiple of 6 and a divisor of 96.'
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'Throwing three indistinguishable dice of the same size, how many combinations are there where the sum of the numbers is a multiple of 8?'
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'In computers, characters are represented by assigning a numerical value called "character code" to each character. Using the table below, please provide the binary representation of the character \'A\'.'
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'PRACTICE 21\n(1) Select 6 letters from the 7 letters of HGAKUEN to form a string and arrange it in dictionary order, what is the position of GAKUEN? Note that the same letters should not be repeated.\n[Kitahai Gakuen University]\n(2) Using different 5 letters A, B, , , D, E, each used once, when listed in order by the dictionary method, what is the 63rd permutation?'
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'There are A numbers in total that can be formed by selecting three different numbers from 0, 1, 2, 3, 4 including the basic example question 140. Among these, there are B numbers that are multiples of 3.'
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'Using the properties of square roots of real numbers, find the following values.'
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'Out of 101 students, 43 like bananas, 39 like strawberries, and 51 do not like either bananas or strawberries.'
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'When throwing 3 dice, how many ways are there for the product of the three numbers to be a multiple of 4?'
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'Chapter 1 The Number of Cases - 217\nWhen 3x ≥ x+y+z=10\nand x ≥ \\frac{10}{3}\nSince x is a natural number, x ≥ 4\nMoreover, y ≥ z ≥ 1 and x ≤ 8\nTherefore, \\quad 4 ≤ x ≤ 8\nWhen x=4, \\quad y+z=6\nTherefore, there are 2 cases: (y, z) = (4,2), (3,3)\nWhen x=5, \\quad y+z=5\nTherefore, there are 2 cases: (y, z) = (4,1), (3,2)\nWhen x=6, \\quad y+z=4\nTherefore, there are 2 cases: (y, z) = (3,1), (2,2)\nWhen x=7, \\quad y+z=3\nTherefore, there is 1 case: (y, z) = (2,1)\nWhen x=8, \\quad y+z=2\nTherefore, there is 1 case: (y, z) = (1,1)\nThus, the representation of 10 as the sum of three natural numbers is 2+2+2+1+1=8 (cases)'
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'Prove the condition for the function f(n) to be an integer'
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'When throwing two dice, how many ways are there for the sum of the two dice to be 5 or 6?'
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'Out of 100 people, 50 have visited city A, 13 have visited city B, and 30 have visited city C. The number of people who visited both cities A and B is x, the number of people who visited cities A and C is 9, and the number of people who visited cities B and C is 10. The number of people who visited cities A, B, and C is 3, and the number of people who have not visited any of the cities is 28. Find the value of x.'
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'Solution to EXERCISES\n26 (1) {1,1,1},{1,2,3},{2,2,2},{3,3,3}\n(2) 5/14'
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'When a>0, x>\\frac{1}{a}. When a=0, there is no solution. When a<0, x<\\frac{1}{a}. When a>-1, x>2. When a=-1, there is no solution. When a<-1, x<2.'
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'(2)\n22x + 37y = 2\n(1st)\nx = -5, y = 3 is one of the integer solutions.\n22 * (-5) + 37 * 3 = 1\nMultiplying both sides by 2,\n22 * (-10) + 37 * 6 = 2\n(2nd)\nSubtracting (1) from (2) gives\n22(x + 10) + 37(y - 6) = 0\nwhich simplifies to\n22(x + 10) = -37(y - 6)\nSince 22 and 37 are coprime, x + 10 is a multiple of 37.\nThus, for some integer k,\nx + 10 = 37k\nSubstituting in,\ny - 6 = -22k\nTherefore, the solution is\nx = 37k - 10, y = -22k + 6 (where k is an integer)'
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'Select 3 different numbers from the 7 numbers 0, 1, 2, 3, 4, 5, 6 to create a 3-digit integer.'
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"Let the number of elements in the set be 102, denoted as A and B. When n(A)+n(B)=10 and n(A∪B)=7, find n(A'∩B+B∩A'). Here, n(X) represents the number of elements in set X."
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'Find all integer solutions to the following equations.'
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'Among three consecutive natural numbers, the square of the smallest number is equal to the sum of the other two numbers. Find these three numbers.'
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'Represent the result of the following calculations in the specified numeral system.'
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'How many integers between 100 and 200 satisfy the following conditions?'
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'Conditions for an expression containing n to become a natural number'
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"(2) Let's assume there are two rooms A and B. Even if there is an empty room, the number of ways to place 9 people in rooms A and B is\n\\[2^{9}=512 \\text { (ways) }\\]\n\nExcluding the case where one room is empty,\n\\[512-2=510 \\text { (ways) }\\]\n\nFinally, if we do not distinguish between A and B, the number of ways is\n\\[510 \\div 2=255 \\text { (ways) }\\]"
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'How many integers between 100 and 200 satisfy the following conditions:\n(1) Integers that are not divisible by 4\n(2) Integers that are divisible by 4 but not by 5\n(3) Integers that are not divisible by either 4 or 5'
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"The proposition 'Let n be an integer. If n^2 is a multiple of 3, then n is a multiple of 3' is true. Use this to prove that √3 is an irrational number."
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'Let x be a real number, and let a be a positive constant. Find the range of values for a such that 2 < x < 3 is a sufficient condition for a < x < 2a.'
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'How many different ways can you create 8 terms from the 3 types of characters x, y, and z in question (2)?'
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'Consider a grid similar to the one on the right. Find the number of ways to place the natural numbers 1 to 4 in such a way that no row (horizontally) or column (vertically) contains the same number.'
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'Find the ways to arrange 10 people in a line such that specific 3 people are next to each other.'
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'What are the maximum and minimum number of elements in a set?'
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'When the set A is {2, 4, 6, 8}, please enumerate the elements included in set A.'
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'There are 3 red balls each with a number from 1 to 3, 2 blue balls each with a number from 1 to 2, and 2 black balls each with a number from 1 to 2. Now we need to line up these 7 balls in a row.'
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'Create positive integers using 5 types of numbers: 0, 1, 2, 3, 4, and arrange them in ascending order.'
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'58\n(1) \ \\left[ \\frac{1}{\\sqrt{3}} \\right] = 0, \\left[ -\\frac{1}{2} \\right] = -1 \,\n\ \\frac{[-1]}{2} = -\\frac{1}{2} \\n(2) Omitted'
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'When selecting three different numbers from integers 1 to 8, how many different ways are there to make the selection?'
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'35 (1) \ x=1 \ (2) \ x=-\\frac{1}{4}, \\frac{5}{2} \'
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'When 4 people play rock-paper-scissors once, how many total ways are there to throw hands?'
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'For the values of a, b in the range -2 ≤ a ≤ 1 and 0 < b < 3, find the range of possible values for 1/2 a - 3 b.'
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'Integer solution to an indeterminate equation (1)'
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'(1) Find the set A. (2) Find the intersection of the complement of B and the complement of C.'
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'Find all integer solutions to the following equations. Also, find the pair of x and y where x is the smallest positive integer satisfying condition (2).'
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'[1] For a subset A of the universal set U, find the set of all elements in U that are not in A.'
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'(1) (a) \\n(b) \\n(c) \\n(ii) Since , then , so \\n\\n(2) (a) \\n(b) \\n(c) '
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'Arrange the numbers 1, 2, 3, 4, 5 to create a 5-digit integer. In this case, there are a total of 13 different integers that can be formed. Among them, the integers ending with 2 can be formed in 6 ways, while odd integers can be formed in 7 ways.'
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'When 4 men and 5 women line up in a row, how many ways are there for the following arrangements?\n(1) All 4 men are adjacent\n(2) Men are not adjacent to each other'
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'When the values of a, b are in the range -2 ≤ a ≤ 1, 0 < b < 3, find the range of possible values for 1/2a - 3b.'
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'Given that sets A, B are subsets of the universal set U with n(U)=50, n(A)=30, n(B)=15, n(A \\cap B)=10, find the number of elements in the following sets.'
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'Consider a grid similar to the one on the right. Find the total number of ways, denoted by K, to fill the squares in each row and each column with the natural numbers 1 through 4 such that no number is repeated.'
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'What are some methods to classify integers based on the remainder when divided by 3? Please explain the advantages of each method.'
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'Find the range of values for the constant when the quadratic equation with real number coefficients satisfies the following conditions: (1) has one positive and one negative root. (2) has two distinct negative roots.'
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'52 (1) positive (2) negative (3) positive (4) 0 (5) 0 (6) positive'
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'Prove that p, q, r are three consecutive odd numbers such that p=2n-1, q=2n+1, r=2n+3 (n is an integer), then pqr + pq + qr + rp + p + q + r + 1 is divisible by 48.'
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"When dealing with problems involving the quantity of integers, it is important to convert words like 'at least' in the problem statement into set conditions, and the key is to correspond the problem content with set symbols. For example, 'A and B', 'A or B', 'A and at least one of B' correspond to A ∩ B, A ∪ B; 'A is not' corresponds to ¬A. Let the set of all integers from 1 to 100 be U, the set of multiples of 3 be A, and the set of multiples of 8 be B. Visualizing this with Venn diagrams can make it easier to understand."
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'Basic Example 28 (1)\nDepart from point O and go to point A, write down the number of routes to each point within the rectangle with segment OA as one diagonal. Furthermore, do the same for the route from point A to point P.\nAs a result, there are 10 routes from O to A, and 150 routes from O to A to P.'
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'(2) By the power rule\n\\[ \egin{array}{r} \\mathrm{AD} \\cdot \\mathrm{AB}=\\mathrm{AE} \\cdot \\mathrm{AC} \\\\ \\text { Therefore } \\quad 2 a(2 a+3 b) \\\\ =3 a(3 a+b) \\end{array} \\]'
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'Prove that when the four given numbers 3, 4, 5, 6 are rearranged to form a four-digit number m, and the digits of m are rearranged in reverse order to form a four-digit number n, then m + n is a multiple of 99.'
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'Basic Example 38 set represented by inequalities'
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'How many natural numbers below 1000\n(1) can be divided by 2 or 7?\n(2) cannot be divided by 2?\n(3) cannot be divided by 2 or 7?\nLet the set of all natural numbers below 1000 be denoted as U, the set of numbers divisible by 2 be denoted as A, and the set of numbers divisible by 7 be denoted as B.'
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'Two brothers have a total of 52 pencils. Now, if the older brother gives exactly one-third of his pencils to the younger brother, he still has more. If he gives 3 more pencils, the younger brother will have more. Find out how many pencils the older brother initially had.'
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'Find all pairs of integers x and y that satisfy the following equations: (1) (x-1)(y+1)=4, (2) xy-3x-2y+3=0.'
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'When the value is large, it is not easy to find the approximate value of the square root of x. In such cases, you can find a natural number n such that n^{2} ≤ x < (n+1)^{2}, and then consider the square roots of each side.'
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'Write down all the subsets of set A = {0, 1, 2}.'
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'Solve the following system of linear equations to find one solution with integer values of x and y:\n\n(1) 5x - 3y = 1\n(2) 2x + 3y = 1'
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'When listing all permutations using five different characters A, B, C, D, E in dictionary order, what is the 63rd permutation?'
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'When rolling two dice, how many ways are there to get a sum of 4 or 7?'
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'Range of existence of solutions for a quadratic equation'
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'Investigate the truth of the proposition. Assume that a and b are integers. If a^2+b^2 is even, then a + b is even.'
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'From the 14 natural numbers from 1 to 14, when selecting 3 different numbers to form a group, find the number of groups for the following conditions: (1) groups consisting only of odd numbers (2) groups that include the number 1 (3) groups that include at least one multiple of 3'
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'Let the maximum value of f(x) in the range 0 to 4 be p, and the maximum value of f(x) in the range 2 to 6 be q. If true, then p=q. Choose the option that applies from (0) to (3).'
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'Every integer n can be divided into m ways of representation by taking the remainder with a natural number m. For example, there are 5 ways (0,1,2,3,4) for the remainder when divided by 5, and n can be represented as n=5k, n=5k+1, n=5k+2, n=5k+3, n=5k+4.'
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'To find the largest integer x that does not satisfy inequality (1), x=□.'
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'Find the number of permutations when selecting a class representative, chairman, and secretary from 10 people.'
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'In how many ways can 10 be expressed as the sum of 3 natural numbers? How many ways can it be expressed as the sum of 4 natural numbers? The order of addition does not matter.'
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'Prove that there are no natural numbers that are four digits when expressed in both decimal and base-5.'
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'203 Basic List Question 125 Conditions for Obtuse (Acute) Triangle'
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'(5) The catch of scallops in 2017 was 235,952 tons. At that time, the average catch of scallops from 2006 to 2017 for 12 years was C tons. Round to the nearest tenth. Choose one value from the following (0) to (3) that fits C.'
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"A company X conducted a survey to determine which is easier to write, their own pencil A or a pencil B from another company Y. Two-thirds of all respondents answered 'A is easier to write'. Later, company Y improved pencil B and conducted the survey again, in which 14 out of 30 people answered 'A is easier to write'. Can it be concluded that the ease of writing with A has decreased compared to B? Using the concept of hypothesis testing, analyze each of the following cases. Suppose an experiment was conducted where a fair die was rolled 30 times, recording the number of occurrences of 1 to 4, and this experiment was repeated 200 times. The results are as shown in the table."
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'Let A be the set of integers divisible by 36, and B be the set of integers divisible by 15. Define C = {x+y | x ∈ A, y ∈ B}, prove that C is the set of integers divisible by 3.'
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'Find the total number of ways for girls to line up when they are not next to each other.'
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'Distribute 252 chocolates to n children, and 360 candies to n children, such that all chocolates and candies are distributed without any leftovers. Find the maximum value of n and the corresponding values of a and b. Note that all characters represent natural numbers.'
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'Let P = {x | x + 2 < 1}, Q = {x | x < 3}. From |x + 2| < 1, we have -1 < x + 2 < 1, hence -3 < x < -1, thus P = {x | -3 < x < -1}. On the other hand, Q = {x | -3 < x < 3}, therefore P is a proper subset of Q.'
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'In a row of 7 squares, colored with red, blue, and green without adjacent squares being the same color. How many ways are there to color the squares so that the colors are symmetrical from left to right?'
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'Person A and person B both work part-time, with each working for 4 days a week. This implies that there is at least one day each week when person A and person B work together.'
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'When n=2019, find the remainder when 4 n^{3}+3 n^{2}+2 n+1 is divided by 7.'
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'Find the sum of all 4-digit numbers that can be formed using only the digits 1, 2, 3, 4.'
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'A 10-liter barrel is filled with 10 liters of oil. Using a 5-liter bucket and a 3-liter bucket, consider the steps to divide this oil into 6 liters and 4 liters. Assuming the barrels and buckets have no scale markings, only the following operations (a)~(c) can be performed. Provide the steps with the minimum number of times operation (a) is performed.'
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'Arrange the 10 characters N, A, G, A, R, A, G, A, W, and A in a single row from left to right.'
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'If set A is {1, 3, 4} and set B is {2, 5, 6}, what is the union of set A and B?'
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'Examine the truth of the following propositions. Where a and b are integers.'
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'Let a be a natural number. If a+5 is a multiple of 4 and a+3 is a multiple of 9, prove that a+21 is a multiple of 36.'
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'When -1 ≤ a ≤ 1, f(x) is minimum at x=a. Hence, f(a)=-a^{2}-a+6 ≥ 0, which implies a^{2}+a-6 ≤ 0. By rearranging the left side, we get (a+3)(a-2) ≤ 0. Solving this gives -3 ≤ a ≤ 2. The common range between -1 ≤ a ≤ 1 and this is -1 ≤ a ≤ 1'
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"The cases where the sum of two cards' integers is a multiple of 3 are:"
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'358 Mathematics IProvide the answer to the given mathematical problem. Select one answer that fits the criteria, choose one from 0 to 5.'
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'Consider a triangle with side lengths of 3cm, 4cm, and 5cm, and divide each side into intervals of 1cm. In this case, there are a total of 12 division points (including the vertices of the triangle). Find the total number of triangles that can be formed by 3 points out of these 12 points.'
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'Find a pair of integers x, y that satisfy the following equations: (1) 19x+26y=1 (2) 19x+26y=-2'
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'Prove that at least one of the three different real numbers a, b, c is equal to 1 if the equation a + b + c = abc holds true.'
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'Let be an integer, and . Prove that when is even, is divisible by 24, and when is odd, is not divisible by 4.'
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'Find the values of constants a and b such that the range of the function y=ax-a+3(0≤x≤2) is 1≤y≤b.'
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'When selecting 3 different numbers from integers 1 to 8, how many different ways are there to select? (2) 16.'
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'(1) Let U={1,2,3,4,5,6,7,8} be the universal set. For subsets A={2,5,6} and B={1,3,5}, find the sets A ∩ ¬B and ¬A ∪ B.'
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'Find the fractions b/a (where a and b are coprime natural numbers) that meet the following conditions when given the fractions 34/5, 51/10, and 85/8. In order for b/a to be a natural number, each fraction must satisfy the condition that multiplying it by a/b results in a natural number.'
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'(1) The integers that are divisible by at least one of 3 and 8 are the set of integers that are divisible by either 3 or 8.'
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'(2) -24 = 13 \\cdot(-2) + 2 rearranging gives 2 = -24 - 13 \\cdot(-2)'
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'There are 7 squares lined up in a row, and we want to paint them with red, blue, and green colors such that adjacent squares are not of the same color. In this case, how many ways are there to paint the squares so that the colors are symmetric?'
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'(2) The number of the set A of integers that are not divisible by 3 is denoted by n(A).'
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"Arrange the given 10 letters N, A, G, A, R, A, G, A, W, A from left to right in a single row. (1) How many total ways are there to arrange these 10 letters? (2) How many total ways are there for the arrangement where the consecutive 6 letters 'NAGARA' appear? (3) How many total ways are there for the arrangement where the 3 letters N, R, W appear in order? Note that configurations where N, R, W are not consecutive should also be included."
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'Since there is overlap in the progression from A to C to F, counting 20 x 1 = 20 (ways) results in counting duplicates. Therefore, the required number of cases is 84 + 84 - 20 = 148 (ways)'
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'Let a, b, c be three mutually coprime natural numbers. Prove that when a^2 + b^2 = c^2, one of a and b is even and the other is odd.'
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'Translate the given text into multiple languages.'
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'There are 5 ways to form a sum of 8 with three natural numbers'
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'In a certain class, a survey was conducted on 12 types of books to determine whether they had been read. The results showed that half of the total number of people read book A, one third read book B, one fourteenth read both books, and 10 people read neither. How many people are in this class.'
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'Question 78 (1) a=3 (2) b=\\frac{4+\\sqrt{2}}{2}, c=\\frac{4-\\sqrt{2}}{2}'
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'There are 3 red balls with numbers 1 to 3 written on them, 2 blue balls with numbers 1 to 2 written on them, and 2 black balls with numbers 1 to 2 written on them. These 7 balls need to be arranged in a row.'
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'(2) 131 (1) Express the decimal number 28 in binary and ternary. (2) Express the decimal number 0.248 in quinary.'
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'Prove that when A = {1, 3, 9} and C = {2, 6}, A ∩ C = ∅.'
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'Describe real numbers, including the definition of absolute value.'
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'Chapter 2 Real Numbers, Linear Inequalities: 4 Real Numbers'
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'Find the maximum natural number n that satisfies the inequality 1-(n-1)/3>n/4.'
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"Take a look at these five cards. Then tell me whether your birthday is on each card or not. Just with that information, I can immediately guess your birthday. For example, if you answer, 'My birthday is on cards A, B, and E,' then add the numbers in the top left corner of cards A, B, and E (16+8+1=25), and I can instantly guess 'Your birthday is on the 25th'. You can guess any number from 1 to 31 the same way, so make these cards and give it a try."
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'111 (1) (A) 0.5625 (B) 0.92 (2) (A) 0.11_(2) (B) 0 . 2_(3)'
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'Find all the integer values of x that satisfy the system of inequalities {2 x-1<3(x+1), x-4 ≤ -2 x+3}.'
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'Problem to find integer solutions to an indeterminate equation (1).'
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'Find the number of natural numbers below 400 that satisfy the following conditions.'
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'Remainder when sum, difference, and product of two integers are divided by m'
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'Given 2<x<5, -1<y<3, find the range of possible values for the following expressions.'
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'Let a be a natural number. Prove that if a+4 is a multiple of 5 and a+6 is a multiple of 8, then a+14 is a multiple of 40.'
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'53 (1) x=32 k+13, y=-37 k-15 (where k is an integer) (2) x=91 k+2, y=138 k+3 (where k is an integer) (3) x=68 k-84, y=-97 k+120 (where k is an integer)'
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'Explain the definition of recurring decimal and provide two examples.'
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'When rolling a six-sided die twice, how many ways are there for the product of the outcomes to be a multiple of 12?'
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'(1) (T) 2, (イ) 1, (ら) 1, (工) 3\n(2) (J) 3, (力) 2√7'
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'The solution to 96 is (1) does not exist (2) all real numbers (3) does not exist (4) all real numbers'
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'When throwing one large, medium, and small die simultaneously, calculate the following number of cases: (1) when the product of the three dice is a multiple of 5 (2) when the product of the three dice is a multiple of 4'
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'Calculate the remainder when 100 multiplied by 24 raised to the power of 32 is divided by 5.'
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'Consider a permutation of 5 numbers 1, 2, 3, 4, 5 arranged in a single row from left to right. How many permutations (*) are there in total?'
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'Let a, b, c be natural numbers that do not have any common factors other than 1. When a, b, and c satisfy a^2 + b^2 = c^2, prove the following: (1) One of a, b is even and the other is odd.'
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"When the statement 'a set containing only 1 is a subset of set A' is represented using symbols, which of the following is the most appropriate?"
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'Multiply 150 by a two-digit natural number n in order to make it a square of a certain natural number. Find the maximum value of n that satisfies this condition.'
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'Express the following numbers in decimal notation.'
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'Let n be an integer. Find the remainder when n^2 is divided by 3. Prove that if integers a, b, c satisfy a^2 + b^2 = c^2, then at least one of a, b must be a multiple of 3.'
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'Let n be an integer. Using the following information, prove (1) and (2):\nThe product of two consecutive integers is a multiple of 2\nThe product of three consecutive integers is a multiple of 6\n(1) When n is odd, the remainder of n^{2} + 2 divided by 8 is 3\n(2) n^{3} - 3n^{2} + 2n is a multiple of 6'
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'Out of 50 students in a certain class, 30 students use the train for commuting, 40 students use the bus, and 26 students use both. In this class, the number of students who do not use either the train or the bus is A, and the number of students who use the train but not the bus is B.'
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'Find the intersection and union of sets A, B, C.'
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'Let x, y be real numbers. Choose the appropriate □ in 1-3) below.\n(1) For x y=1, x=1 and y=1 is the condition for □.\n(2) For x>0 and y>0, x y>0 is the condition for □.\n(3) In △ABC, when AB=BC=CA, what is the condition for ∠A=∠B=∠C?'
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'(1) \\\\( -\\frac{3}{8} \\\\\\\\n(2) \\\\( -\\frac{11}{16} \\\\\\\\n'
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'When the unit price is 15 yen, the maximum sales amount is 1125 yen'
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'Find all integer solutions to the equation 55x-16y=1.'
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'Represent the following addition and subtraction in the given base.'
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'On the number line, the distance from the origin to the point representing a real number a is called the absolute value of the real number a, denoted by |a|. Regarding the absolute value of a real number a, the following holds true. When a is a positive number or zero, |a|=a, but what is |a| when a is a negative number?'
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'Find all the integer solutions to the following equations.'
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'Please determine the truth value of the following proposition.'
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'A school needs to create a pamphlet for the school festival. The printing cost is 4000 yen for up to 100 copies, but beyond 100 copies, it costs 27 yen per copy. How many copies at least need to be printed in order to keep the cost per copy below 30 yen? Do not consider consumption tax.'
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'118 (1) 100010_{(2)} (2) 10001_{(2)} (3) 2132_{(5)} (4) 103_{(4)}'
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'When there are 3 integers x that satisfy the inequalities x^2+2x-8>0 and x^2-(a+3)x+3a<0 simultaneously, find the range of the constant 60a.'
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'When 4 people played a game with a perfect score of 50 points, the scores were a, 43, b, c (points). Where a, b, and c are integers and 0<c<b<43<a. If the average of these scores is 43 points, the variance is 6.5, and the range is 7 points, answer the following questions: (1) Define x=a-43, y=b-43, z=c-43, and find the values of x+y+z and x^2+y^2+z^2. (2) Find the values of a, b, c.'
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'Problem regarding the use of integer solutions in solving linear diophantine equations.'
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'Choose all the correct options from the following (1) to (4).'
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'The three sets of equations for integer values of k are: (1) x=18k-1, y=-17k+1 (2) x=19k-1, y=37k-2 (3) x=7k+1, y=-12k+1'
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'(1) How many ways are there to throw a die 3 times and have the sum of the faces be 7?\n(2) A and B play a game where the first to win 3 times wins. How many possible scenarios are there to determine the winner within 5 rounds without a tie?'
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'Find the maximum natural number n that satisfies the inequality 1-(n-1)/3>n/4.'
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'Let the set of all positive integers less than or equal to 10 be the universal set U, and let subsets A and B be A={1,3,6,8,10}, B={2,3,6,8,9}. Find the following sets: (a) A ∩ B, (b) A ∪ B, (c) Complement of A, (d) A ∩ complement of B. For the example problem (2) above, find the complements of set A and the intersection of the complement of A and B.'
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'Express the natural number N in base-5 and base-7, both as two-digit numbers where the digits are reversed. Represent N in base-10.'
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'Find the total number of non-negative integer solutions for the equation x+y+z=10. Also, find how many positive integer solutions exist.'
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'When all elements of set A and set B are the same, we say that A and B are equal, and denote it as A=B. For example, let R be the set of all single-digit positive even numbers, and let Q={2,4,6,8}.'
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'(3) Let n be an integer. Prove that n³ - n is a multiple of 6.'
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'Problem to find natural number solutions for a fractional equation.'
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'The correct calculation is (1). The calculation in (2) is incorrect.'
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'Prove the following statements:\n(1) If a, b are multiples of 6, then a-b and 3a+8b are also multiples of 6.\n(2) If a, b are multiples of -2, then a^{2}-b^{2} is a multiple of 4.\n(3) If 5a-b and a are multiples of 9, then b is a multiple of 9.'
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'Using the given relationships, calculate the number of pairs of integers in a specific state.'
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'When integers a and b satisfy the equation 2a + 3b = 42, find the maximum value of ab.'
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'Find all pairs of natural numbers x and y that satisfy the equation 3xy = 4x + 2y. Also, consider x ≤ y.'
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'107 (1) x = 11k, y = 12k (k is an integer) (2) x = 8k-1, y = -23k + 3 (k is an integer)'
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'If a natural number N, which has 10 digits in base 4, is represented in base 2, how many digits will it have?'
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'For a real number x, define two conditions p and q as follows.p: -1 ≤ x ≤ 3q: |x-a|>3Let the negation of conditions p and q be denoted by ¬p, ¬q.1. The range of values of a for which the proposition p → q is true is a ∈ [ , ] form.2. When a= , x= is a counterexample to the proposition p → q.'
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'Let a and b be integers. When a is divided by 11, the remainder is 7, and when b is divided by 11, the remainder is 4. Find the remainder when the following numbers are divided by 11: (1) a + b (2) b - a (3) ab (4) a^2 - b^2.'
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'When N is between 2^18 and 2^19 (exclusive), it has 19 digits; when N is between 2^19 and 2^20 (exclusive), it has 20 digits.'
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'Using the trigonometric table, find the following:(1) Calculate the values of sin 15°, cos 73°, tan 25°;(2) Find the acute angles α, β, and γ that satisfy sin α=0.4226, cos β=0.7314, tan γ=8.1443;(3) Estimate the value of x and the approximate size of angle θ in the diagram. Round x to the nearest hundredth.'
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'If you are allowed to use the numbers 2, 4, and 6 repeatedly, how many 5-digit integers can you create?'
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"When buying a 100 yen item, the price is determined based on the quantity purchased. Similarly, when a car travels at 60 km per hour, the distance traveled is determined by the travel time. Let's learn about the relationship where one quantity determines another."
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'When x is greater than or equal to 1, y is greater than or equal to -1, and 2x+y=5, find the maximum and minimum values of xy.'
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'Problem of finding natural number solutions to indeterminate equations.'
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'Let A be the set of all rational numbers, then A contains the element 0.'
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'There are 4 red balls, 2 white balls, and 1 blue ball.\n(1) How many ways are there to arrange all 7 balls in a circle?\n(2) When stringing all 7 balls to make a necklace, how many different necklaces can be made.'
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'For example, the subsets of the set {2,6} are {∅,{2},{6},{2,6}}.'
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'How to decrease the coefficients to find one of the integer solutions'
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'If you bought several items of product A priced at 210 yen each and several items of product B priced at 170 yen each, and the total amount paid was 4400 yen. Find the number of items purchased for both products A and B.'
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'In set theory, it is common to start with a set U, and then consider subsets of U. In this case, U is called the universe set. For a subset A of U, the set of all elements of U that do not belong to A is called the complement of A with respect to U, denoted by Ā.'
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'Prove that when events A and B are mutually exclusive, the property n(A ∪ B) = n(A) + n(B).'
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'Let n be an integer. Prove that n^5-n is a multiple of 30.'
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'If a positive integer N is represented in base-5 and base-7 as two-digit numbers with their digits reversed, determine the decimal representation of N.'
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'24(2) Inverse: n is odd ⇒ n^2+1 is even, Converse: n is even ⇒ n^2+1 is odd, Contrapositive: n^2+1 is odd ⇒ n is even'
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'Find all integer solutions to the equation 97x + 68y = 12.'
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'Prove that when a is a positive number, and k is a positive number, then √(k^2 * a) = k √a.'
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'There are total 20 ways the 5 adults and 3 children can sit around a circular table without children sitting next to each other, considering rotations as identical.'
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'Find all pairs of integers x and y that satisfy the following equations:\n(1) ((x+1)(y-2)=7\n(2) xy-3x-2y+2=0'
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'Using the fact that the product of three consecutive integers is a multiple of 6, prove that 2n^3 + 3n^2 + n is a multiple of 6.'
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'(2) If m and n are odd, then both m and n are odd.'
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'How many integers satisfy the inequality -√10<x-5<√10?'
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'Using the method of finding the intersection and union of three sets, specifically find the intersection and union of sets V, W, X. For example: V = {1, 4}, W = {4, 5}, X = {4, 6}'
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'Given that when 91 is divided by the main number, the remainders for a and b are integers. If a divided by 8 has a remainder of 3, and b divided by 8 has a remainder of 6. Find the remainders when the following numbers are divided by 8: (1) a+b (2) a-b (3) ab (4) a squared'
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'Using the numbers 1, 2, 3, 4, 5, how many two-digit numbers can be created? The same number can be repeated.'
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'Find one pair of integers that satisfy the equation .'
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'In 71 (4), the maximum value is 17/4 when x=5/2 and there is no minimum value.'
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'When -2<x<5, -7<y<4, find the possible range of values for the following expressions.'
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'Explain the truth value of the proposition and check the truth value of the following proposition P.'
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'The given data represents the price of oranges per 1 kg in 8 stores. It is known that the value of a is a natural number 156. When the value of a is unknown, how many possible values can the median of the prices in the 8 stores have? Since there are 8 stores, the average of the 4th and 5th prices from the lowest would be the median. Arranging the prices other than a in ascending order gives 499, 500, ~~530, ~~550, 550, 555, 560. When a ≤ 530, the median price is (530+550)/2 = 540 (yen). When a ≥ 550, the median price is (550+550)/2 = 550 (yen). When 530 < a < 550, the median price is the average of a yen and 550 yen, which is (a+550)/2, and the natural number values of a satisfying 530 < a < 550 are a=531, 532, ... 549. Therefore, there could be 21 possible values for the median price.'
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'Find the integer solutions for the given equation 7l + 9m + 12n = 35.'
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"Explain the negation of propositions involving 'all' and 'some', and derive the negation of the following proposition."
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'110 (1) (A) 85 (B) 63 (C) 125 (2) (A) 110110_{(2)} (B) 13000_{(5)} (C) 14003_{(7)}'
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'When 2 men and 3 women line up in a row, how many ways are there for the following arrangements: (1) Both ends are women. (2) 2 men are adjacent. (3) Men are not adjacent.'
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"Arrange 10 letters, N, A, G, A, R, A, G, A, W, A horizontally from left to right.\n(1) How many ways are there to arrange the letters so that the continuous 6 letters spell out 'NAGARA'?\n(2) How many ways are there to arrange the letters N, R, W in that order? Include cases where N, R, W are not consecutive."
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'When throwing a large, medium, and small die simultaneously, how many ways are there for the product of the numbers to be even?'
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'(2) List all the subsets of the set A={0,1,2,3}.'
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'Determine the number of real solutions of the following quadratic equation ax^2+bx+c=0 using the discriminant D. Where D=b^2-4ac.'
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'Among the natural numbers below 500, find the number of elements in the following sets:\n(1) Set of numbers divisible by 3\n(2) Set of numbers divisible by 3, 5, or 7\n(3) Set of numbers divisible by 3 but not by 5\n(4) Set of numbers not divisible by 3, 5, or 7\n(5) Set of numbers divisible by 3 but not by 5 or 7'
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'56 (1) Converse: x ≠ -1 ⇒ x² ≠ -x, False Contrapositive: x = -1 ⇒ x² = -x, True Contrary: x² = -x ⇒ x = -1, False (2) Converse: x or y is rational ⇒ x+y is rational, False Contrapositive: x, y are both irrational ⇒ x+y is irrational, False Contrary: x+y is irrational ⇒ x, y are both irrational, False'
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'Determine the values of constants a and b to satisfy the following conditions.'
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'The range of y is -2<y≤0. When x=0, the maximum value is 0, and there is no minimum value.'
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"(√3)², (-√(3/2))², √((-7)²), -√((-9)²)'s values are required to be determined."
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'(2) Find the maximum and minimum values of x^2+y^2 when x≥0, y≥0, and x+y=2.'
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'Find all integer solutions to the following equations.'
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'The highest digit is greater than or equal to 1, and each digit of an n-ary number is between 0 and n-1.'
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'For example, let P={4,8} and Q={2,4,6,8}. Then, P is a subset of Q, and P⊂Q.'
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'If a natural number N with 5 digits in octal is represented in binary, how many digits will it have?'
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"How can you deepen your understanding when you don't understand basic examples?"
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'Find all pairs of natural numbers that satisfy the equation . Also, .'
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'Parts A, B, C produced in a certain factory use 7, 9, 12 screws respectively. After shipping, all the remaining screws from these parts were removed, and a total of 35 screws were found. Let the number of remaining parts A, B, C be denoted as l, m, n. Find all possible combinations of l, m, n.'
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'Find the integer x that satisfies \ \\sqrt{x} = \\sqrt{17 + \\sqrt{253}} - \\sqrt{17 - \\sqrt{253}} \.'
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'Maximum value of \\\frac{9}{4}\ is obtained at x = -\\frac{1}{2}'
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'Find all integer solutions of the equation 138x + 91y = 3.'
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'Find and write down the number of shortest paths'
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'The total number of different n circular permutations is (n-1)!'
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'Let f(x) = x^{2} + ax + b, and m be the minimum value at 0 ≤ x ≤ 1, express m in terms of a and b.'
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'Define the intersection, union, and complement of sets, and explain with an example.'
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'When there are 2 boys and 4 girls, find the number of ways they can be arranged with 2 boys at both ends.'
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'Let the set of all natural numbers less than 10 be the universal set U, with subsets A and B given by A={1,2,3,4,5}, B={1,3,5,7,9}. Find the following sets:\n(A) A ∩ B\n(B) A ∪ B\n(C) Complement of A\n(D) Complement of A ∩ B\nConsider the set of all real numbers as the universal set, with subsets A and B defined as A={x | -1 ≤ x ≤ 2, x is a real number}, B={x | 0<x<3, x is a real number}. Find the sets A ∩ B and A ∪ B.'
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'Find all integer solutions to the following equations.'
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'Using the contrapositive, prove the following propositions:\n(1) If n^2 + 4n + 3 is a multiple of 4, then n is an odd number.\n(2) If mn is even, then at least one of m or n is even.'
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'State the negation of the following conditions. Where x, y, m, n are real numbers.'
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'How many ways are there to find the shortest route?'
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"Let's solve the problem using the concepts of sets and their elements. Consider the set A.\n\nA={1, 2, 3, 4, 5}\n\nPlease write down all the subsets of set A."
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'Since the hundreds digit can be 3, 4, or 5, there are 3 possibilities. For each case, the tens and units digits can be arranged from the remaining 4 numbers in 4P2 ways. Therefore, by the multiplication principle, there are a total of 3 × 4P2 = 3 × 4 × 3 = 36.'
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'In a mathematical problem, there is a three-digit number abc in base n, where c≠0 and a>c. Determine the value of n such that the difference between abc in base n and cba in base n is 63 in decimal and 15, and find the decimal representation of abc. According to the conditions: 1 ≤ c < a ≤ n-1, 0 ≤ b ≤ n-1. Furthermore, an^2 + bn + c - (cn^2 + bn + a) = 15, hence (a-c)n^2 - (a-c) = 15, which leads to (a-c)(n^2-1) = 15. Since n is a natural number greater than or equal to 2, n^2 ≥ 4, thus n^2-1 ≥ 3. From (2), we get n^2-1 = 3,5,15, i.e. n^2 = 4,6,16. Since n^2 is a perfect square, only n=4,16 are suitable. Therefore, n=2,4. When n=2, according to (1), there are no integers a, c satisfying 1 ≤ c < a ≤ 1. When n=4, a-c=1 and 1 ≤ c < a ≤ 3, so the pairs of (a, c) are (2,1),(3,2). b can be 0,1,2,3. Consequently, when n=4, abc_(n)=201_(4) equals 33 in decimal, and 302_(4) equals 50. Since each increase in b results in a +4 in decimal for abc_(4), the decimal values of (4) are 33,37,41,45,50,54,58,62.'
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'When selecting three different numbers from 0, 1, 2, 3, 4 to form a 3-digit number, how many numbers can satisfy the following conditions: (1) an integer (2) even number'
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'Let n be an integer. Prove that n^5 - n is a multiple of 30.'
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'Find the number of pairs (x, y) of integers that satisfy 3x + 5y = 7 and 100 ≤ x + y ≤ 200.'
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'Choose all correct from the following 1~4. (1) −sqrt{0.25}= & ±0.5. (2) −sqrt{0.25}=0.5. (3) The square root of −49/−64 is ±7/±8. (4) The square root of −49/−64 is only ±7/±8.'
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'When the difference between two natural numbers a and b (a<b) is 3, and their least common multiple is 126, what is the value of a?'
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'18 (1) 27720 ways\n(2) 34650 ways\n(3) 5775 ways\n19840'
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'(1) Prove that when n is an odd number, the remainder when n² + 2 is divided by 8 is 3.'
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'There are 5 balls with the numbers 1, 3, 5, 7, 9 written on them in bag A, and 4 balls with the numbers 2, 4, 6, 8 written on them in bag B. A and B each take out one ball from their own bag, and the person with the higher number wins. The winner gets the points based on the number they drew, and the loser gets 0 points. In this case, which side, A or B, has the advantage?'
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'Find all the integer solutions of 37x + 32y = 1.'
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"Therefore, 'Birthday is on cards A, B, E' means the birthday can be represented as 'sum of 16, 8, and 1'. Since 16, 8, and 1 are written on the top left of the cards, you can quickly guess the birthday by just adding them up. Let's consider the case of 13. 13 is on cards B, C, E. 'Birthday is on cards B, C, E' means the birthday can be represented as 'sum of 4, 8, and 1', so from 8+4+1=13, we know it's the 13th. I see. So that's how birthdays are guessed. By the way, some people may have already noticed that the principle of creating this table is closely related to binary."
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'There are 4 students who want to buy 1 pencil for 50 yen and 1 notebook for 70 yen to distribute. In order for each student to receive the same number of pencils and notebooks, the total cost of the purchase is 1640 yen. Determine the number of pencils and notebooks purchased. Let the number of pencils per student be x and the number of notebooks be y, where x ≥ 1, y ≥ 1. Based on the conditions, we have 50 × 4x + 70 × 4y = 1640.'
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'When selecting 3 different fruits from 4 fruits (A), (B), (C), (D), there are 4 ways to choose.'
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'Out of the 6 numbers 0, 1, 2, 3, 4, 5, how many 4-digit integers can be formed with different 4 numbers, like 312?'
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'For A = {1,3,6,8,10}, B = {2,3,6,8,9}, find the following sets:\n(A) A ∩ B\n(B) A ∪ B\n(C) Complement of A\n(D) Complement of (A ∩ B)'
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'Find the largest three-digit natural number n such that n+2016 is a multiple of 5 and n+2017 is a multiple of 12.'
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'A certain product sells 100 units per day when the price is 10 yen each. For every 1 yen increase in price, the daily sales decrease by 5 units, and for every 1 yen decrease in price, the daily sales increase by 5 units. What price should the product be set at to maximize the daily sales revenue? Find the maximum value of the sales revenue and the price at that time. Taxes are not considered.'
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'Circular Permutations\nArranging objects in a circular manner is called a circular permutation. In circular permutations, rotating the arrangement to make it the same is considered the same arrangement. Specifically, considering a circular arrangement of 4 people, A, B, C, D. First, create a permutation where 4 people are in a line. There are a total of 4! = 24 ways, and listing them out as follows.'
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'How many ways are there to form a 3-digit odd number by selecting 3 different numbers from the 5 numbers 1, 2, 3, 4, 5?'
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'Problem to find integer solutions to a linear Diophantine equation (2) (basic).'
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'Find the maximum value of the function f(x)=(1-x)|x+2| when 43-2 5/2 ≤ x ≤ 2.'
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'Find all pairs of integers x and y that satisfy the following equations.'
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'All real numbers except -1 (2) All real numbers (3) No solution (4) x = 2/3'
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'Find the number of natural numbers below 300 that satisfy the following conditions: (1) multiples of 5 (2) multiples of 8 (3) numbers that are not multiples of 5 (4) multiples of 5 and 8 (5) multiples of 5 or 8'
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'Basic Example Problem\nCalculating Expected Value of 45\n(1) Roll a die, take the number rolled as points. Find the expected value of the points.\n(2) Toss two 10-yen coins simultaneously, when receiving the coin with the head facing up, determine the expected value of the amount received.'
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'Find all integer solutions to the following equations'
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'Circular permutations and permutations with repeated elements.'
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"Please explain the meanings of the symbols 'α ⟂ β' and '≡ 466'."
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'Divide 10 students into several groups. (1) How many ways are there to divide them into groups of 2, 3, and 5 people? (2) How many ways are there to divide them into groups of 3, 3, and 4 people? (3) How many ways are there to divide them into groups of 2, 2, 3, and 3 people?'
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'Define what a subset is and check for subsets using the following example.'
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'Explain the definition of terminating decimal and provide two examples.'
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'Find the maximum value of the function f(x) = (1-x)|x+2| when -5/2 ≤ x ≤ 2.'
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'Given AB = 2, BC = x, CA = 4-x in triangle ABC. Find the range of values \u200b\u200bof x.'
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'Let a be a natural number. Prove that if a + 2 is a multiple of 7 and a + 3 is a multiple of 3, then a + 9 is a multiple of 21.'
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'Consider the following mathematical equations:\n(1) Find all pairs of integers x, y that satisfy x^{2}+3y^{2}=36.\n(2) Find all pairs of natural numbers x, y that satisfy x^{2}+xy+y^{2}=19.'
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'Want to buy 1 pencil for 50 yen and 1 notebook for 70 yen for 4 students. To ensure each student receives the same number of pencils and notebooks, the total amount paid was 1640 yen. Find the number of pencils and notebooks purchased.'
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'When the average of a data set containing 7 values 1, 5, 8, 12, 17, 25, and a is 12, find the value of a.'
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'Express the decimal number 2.875 in binary form.'
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'Since the units place can be 1, 3, or 5, there are 3 possibilities for choosing. Regardless of which case, for the hundreds place and the tens place, there are 4 remaining numbers to choose 2 and arrange them, which results in 4P2 possibilities. Therefore, by the product rule, 3 × 4P2 = 3 × 4 × 3 = 36 (items)'
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'In how many ways can you answer the 6 questions using \, \triangle, \\times\?'
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'From the numbers 1, 2, 3, 4, 5, how many different three-digit numbers can be formed that satisfy the following conditions:\n1. The number is greater than or equal to 300\n2. The number is odd'
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'How many natural numbers less than 9 satisfy the condition that the parabola y=x^2+ax+3 intersects the x-axis at two distinct points?'
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'39 (1) 6,432 ; 48,54\n(2) 12; 378 ; 42,108 ; 54,84'
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'Regarding employment figures, it is important to note that the sum of male employment and female employment equals the total employment. For example, if the percentage of male employment is 60%, then the percentage of female employment is 40%.'
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'Find all pairs of natural numbers x, y that satisfy the equation 3xy = 4x + 2y. Subject to the condition x≧y.'
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'When rolling two dice simultaneously, how many ways are there to get a sum that is a multiple of 6?'
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'10 (1) 1440 possibilities\n(2) 720 possibilities\n(3) 1440 possibilities'
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'Represent a natural number N in base 7 and base 5, both as 3-digit numbers where the digits are in reverse order. Express N in base 10.'
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"Explain the basic relationships between sets A and B and De Morgan's laws."
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'When one man and one woman, as well as three male and three female students, sit around a circular table at equal intervals, find the total number of arrangements for the following:'
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'Find the number of occurrences of the following conditions among natural numbers below 400.'
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'Find one pair of integers x, y that satisfy the equation 31x+17y=1.'
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'Find the number of combinations of natural numbers x, y, and z that satisfy the equation x+3y+z=10.'
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'Combining 1 apple for 160 yen and 1 orange for 130 yen, you want to buy a total of 20 fruits and put them in a 200 yen basket, while keeping the total cost below 3000 yen. If you want to buy as many apples as possible, how many apples can you buy? Note that there is no consumption tax considered.'
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'For example, when an integer is divided by 2, the remainder is 0 or 1, so every integer can be expressed in one of the forms 2k or 2k+1 using an integer k. Also, when an integer is divided by 3, the remainder is 0, 1, or 2, so every integer can be expressed in one of the forms 3k, 3k+1, or 3k+2 using an integer k.'
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'Find all pairs of natural numbers that satisfy .'
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'Using the formula for variance s² = x̄² - (x̄)², calculate the variance of the data consisting of 6 values: 10, 7, 8, 0, 4, 2. Round to 2 decimal places.'
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'Prove the following property for a positive number a: (√a)^2 = a'
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'Prove the following when n is an integer:\n1. n^{2}+3 n+6 is even.\n2. n(n+1)(5 n+1) is a multiple of 3.'
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'Let TR(a) be a natural number. Prove that if a+4 is a multiple of 5, and a+6 is a multiple of 8, then a+14 is a multiple of 40.'
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'Represent the following decimal numbers in the notation inside the square brackets.'
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'There are 1 red jade, 2 blue jades, 2 yellow jades, and 2 white jades.\n(1) In how many ways can these 7 jades be arranged in a circular form.\n(2) When threading all 7 jades to make a bracelet, how many different bracelets can be made.'
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'A rectangular wall has a vertical side of 3m 24cm and a horizontal side of 1m 80cm. We want to cover it seamlessly with square papers of the same size. To maximize the paper size, what should be the length of each side in centimeters? Also, determine the number of papers needed.'
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'Find the value of x that satisfies the following condition: 2x + 3 = 7.'
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"49 (1) (A) {3,6,8} (B) {1,2,3,6,8,9,10} (C) {2,4,5,7,9} (D) {1,10} (2) A'={x|x<-1,2<x, x is a real number}, A'∩B={x|2<x<3, x is a real number}"
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'Let the universal set be U, its subsets be A, B. If A is a proper subset of B, then which of the following sets do A ∩ B, A ∪ B, A ∩ complement of B correspond to (1~(3))?'
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'The set of all elements that belong to both sets A and B is called the intersection of A and B, denoted by A∩B. Also, the set of all elements that belong to at least one of the sets A and B is called the union of A and B, denoted by A∪B.'
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'55 (1) x is not a positive number (x is 0 or below) (2) x=0 and y ≠ 0 (3) x<0 or x≥1 (4) x, y are not both irrational numbers (x, y are both rational numbers) (5) At least one of m, n is 0 or below (m≤0 or n≤0)'
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'There are how many ways for 5 adults and 3 children to sit around a circular table without adjacent children sitting together? Count rotations as the same arrangement.'
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'Choose the multiples of 2, 3, 4, 5, and 9 from the following numbers.'
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'(2) Prove that when a is odd, b is a multiple of 4.'
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'Find one pair of integers (x, y) that satisfy the equation 53x + 29y = 1.'
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'Find all pairs of natural numbers x and y that satisfy the equation 5x + 3y = 23.'
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'Prove the following equation for three sets with subsets A, B, C in the universal set U'
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'How many ways are there to choose 3 different numbers from 1, 2, 3, 4, 5 to create a 3-digit number? (1) Numbers greater than or equal to 300'
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'When creating a 6-digit integer using the following 6 numbers, how many integers can be created? (1) 2,2,3,3,3,3 (2) 4,4,5,5,6,6'
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'For the set P = {1, 2, 4, 5, 10, 20}, please determine if the following elements belong to P.'
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'Find the number of terms in a geometric sequence where the first term is 2, the common ratio is 3, and the sum is 242.'
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'Find the general term of the following arithmetic sequences.'
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'In this factory, the total amount of products P and Q produced in a day, x+y (kg), reaches its maximum when (x, y) = (a certain number, another number), and at that time, x+y = yet another number. Please provide the values for the certain number and the yet another number.'
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'Express the order of the three numbers using inequality symbols.'
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'The coordinates of point Q are (\\frac{(-2) \\cdot 8+3 \\cdot 7}{3-2}, \\frac{(-2) \\cdot 1+3 \\cdot 6}{3-2}). The coordinates of point R are (\\frac{2 \\cdot 7+3 \\cdot(-3)}{3+2}, \\frac{2 \\cdot 6+3 \\cdot 1}{3+2}). Therefore, calculate the coordinates of the centroid.'
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'Find the general term of the sequence {an} determined by the following conditions.'
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'Arrange natural numbers starting from 1 in groups of 1, 2, 4, and so on, where each group contains 2^(n-1) numbers. For example, 1|2, 3|4, 5, 6, 7|8, ...'
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'Find the conditions under which throwing a die n times ensures that paying 1000 yen will not result in a loss.'
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'For a natural number n (n≥2) and positive numbers α₁, a₂, ..., aₙ, it holds that:\n\\\frac{a_{1}+a_{2}+⋯+a_{n}}{n} ≥ \\sqrt[n]{a_{1} a_{2} ⋯ a_{n}}\\n(The equality holds if and only if a₁=a₂=⋯=aₙ).\nFor example, when n=3, \\\frac{a_{1}+a_{2}+a_{3}}{3} ≥ \\sqrt[3]{a_{1} α_{2} a_{3}}\\].\nAnd when n=4, \\[\\frac{a_{1}+a_{2}+a_{3}+a_{4}}{4} ≥ \\sqrt[4]{a_{1} α_{2} α_{3} α_{4}}\.'
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'Find the first and second terms of the following sequences: (1) (2) (3) (4) '
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'Given a common ratio of -3, find the first term of a geometric sequence such that the sum of the terms from the first to the sixth term is 728.'
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'Find the sum of the 6th to 10th terms of the geometric sequence 27, 9, 3.'
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'Find the 20th term of the sequence: \a_{1}=1, \\quad b_{k}=k, n=20\'
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'Find the sum of all irreducible fractions with a denominator of 3 between 1 and 100.'
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'In the given geometric progression, where the common ratio is a real number, find the following:'
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'Find the maximum and minimum values of the function f(θ) = 8√3cos^{2}θ + 6sinθcosθ + 2√3sin^{2}θ (0≤θ≤π).'
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'Using the following definition, please solve the problem: Definition: When a is 2 and m is 3, find the value of a^{-m}.'
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'(2) \ \\alpha=\eta \ [Refer to Example 42]\nFind the general term of the sequence \ \\left\\{a_{n+1}-\\alpha a_{n}\\right\\} \, and apply (6) to obtain 1.'
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'In an arithmetic sequence with first term 200 and common difference -6, let the sum from the first term to the nth term be Sn. The maximum value of Sn is A[□], and the value of 6 at that point is n=[□]. The value of Sn when it first becomes negative is U[□], and the corresponding value of n is n=[□].'
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'When the terminal side of θ is in the third quadrant, and cos θ = -4/5, find the values of sin θ and tan θ.'
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'Chapter 1 Sequences\n349\nn = 7k + 5, \nn = 13l + 11\nTherefore, 7k + 5 = 13l + 11\nHence, 7k - 13l = 6\nk = l = -1 is one of the integer solutions of (1), thus, (1) can be transformed into\n7(k + 1) - 13(l + 1) = 0.\nTherefore, 7(k + 1) = 13(l + 1)\nSince 7 and 13 are coprime, there exists an integer m such that k + 1 = 13m, which means k = 13m - 1. In this case, n = 7k + 5 = 7(13m - 1) + 5 = 91m - 2\nIf we set 200 ≤ 91m - 2 ≤ 500, then 202/91 ≤ m ≤ 502/91, and the integer values of m that satisfy this condition are 3, 4, and 5.\nTherefore, there are 3 natural numbers that satisfy the condition, and the sum of these 3 natural numbers is (91 * 3 - 2) + (91 * 4 - 2) + (91 * 5 - 2) = 1086'
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'Balls marked with numbers 1, 2, 3 are placed in bags containing 1, 4, 5 respectively, with a total of 10 bags. Consider this as the population of 70, answer the following questions:\n(1) Represent the population distribution with the variable X denoting the numbers on the balls.\n(2) Calculate the population mean m and the population standard deviation σ.'
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'Prove that for all natural numbers n, 3^(3sn)-2^n is a multiple of 25.'
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'Find the general term of a geometric sequence. Let the first term be a and the common ratio be r.'
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'Translate the given text into multiple languages.'
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'Find the n-th term and sum of first n terms of the following sequence (2) 19 (2) 1,4,10,22,46'
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'(1) \\( a_{n}=2 n\\left(\\frac{1}{3}\\right)^{n-1} \\)\n(2) \\( a_{n}=\\frac{n(3 n+1)}{2} \\)'
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'Find the first term and common difference of an arithmetic sequence in which the sum of the first 10 terms is 100, and the sum of the first 20 terms is 350. Also, find the sum of the terms from the 21st to the 30th in this sequence.'
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'Find the sum of the following numbers from 1 to 100:\n(1) Numbers that leave a remainder of 2 when divided by 5\n(2) Numbers that are not divisible by 3\n(3) Numbers that are multiples of 3 or 5'
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'Find the general term of the following geometric sequence {an}.\n(1) Initial term -4, common ratio 2\n(2) Initial term 5, common ratio -3'
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'The sequence {a_{n}} is defined as a_{1}=2 and the recurrence formula a_{n+1}=2-{a_{n}}/{2a_{n}-1}. (1) Find a_{2}, a_{3}, a_{4}, and speculate the expression for the general term a_{n} in terms of n.'
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"A pair of rabbits (one male, one female) born in a certain month, starting from the second month of birth, give birth to a pair of babies every month, and the newly born rabbits do the same. When increasing in this way, starting from a pair of rabbits born just this month, how many pairs of rabbits will there be after n months?\n\nThis is a question discussed by the 13th-century mathematician Fibonacci in his book. Representing a pair of rabbits as ○, ○, and mouth, let's check how many pairs of rabbits there are at the end of each month.\n\nArranging the number of rabbits at the end of each month (how many pairs)\n1,1,2,3,5,8,13,21, \\cdots \\cdots\nThis sequence {aₙ} is called the Fibonacci sequence and is expressed in a recurrence relation as\na₁=1, a₂=1, aₙ₊₂=aₙ₊₁+aₙ"
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'Suppose the sum of natural numbers greater than 10 and less than 3 that are divisible by 3 is 3657. Find the value of m.'
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'When the 3rd term is 72 and the 6th term is 243, find the first term and the common ratio.'
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'Please prove that the sum of natural numbers 1+2+3+...+n=∑_{k=1}^{n} k=\\frac{1}{2} n(n+1).'
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'Real numbers comparison In any two real numbers a, b, only one of the relations a>b, a=b, a<b holds true.'
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'Prove by mathematical induction that the inequality 2^n > 10n^2 holds when n is a natural number greater than or equal to 10.'
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'(1) Basic Properties of Inequality: 1. a>b, b>c ⇒ a>c 2. a>b ⇒ a+c>b+c, a-c>b-c 3. a>b, c>0 ⇒ ac>bc, a/c>b/c - a>b, c<0 ⇒ ac<bc, a/c<b/c Moreover a>0, b>0 ⇒ a+b>0 a>0, b>0 ⇒ ab>0'
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'When the sequence 7, a,-3 forms an arithmetic progression, find the value of a.'
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'The relationship between the sum average and the geometric mean of 5 is as follows: (a + b) / 2 is the arithmetic mean of a and b, where a > 0, b > 0, when a > 0, b > 0, √(ab) is the geometric mean of a and b. When a > 0, b > 0, the equality (a + b) / 2 ≥ √(ab) holds true only when a = b'
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'Calculate the sum of the terms from the first term to the nth term of the sequence.'
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'Consider all integers from 1 to 300. (1) Find the sum of all numbers that are divisible by 3 but not by 9.'
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'Prove the following statement: 33 Proof outline (1) The equality holds when a=b=c=0 (2) The equality holds when a=b=c>0'
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'Prove the following statement:\n29 omitted\n30 Proof omitted(1) The equality holds when a=\\frac{3}{2}(2) The equality holds when ad=bc'
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'Given a>0 and p as a real number, the range of values of p for which the curve y=f(x) and the line y=p have 3 common points is from 天<p<□.'
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'Prove by mathematical induction that for natural numbers n satisfying n ≥ 5, 2^n > n^2 holds.'
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'Four integers a, b, c, d (a < d) form an arithmetic progression in this order, and the sum of the four numbers is 32. Also, the equation bc = ad + 8 holds. Find the value of a.'
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'Prove that the following inequalities hold when x>0. Also, determine when the equalities hold.'
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'When k=5, x=1/2, 1/3, when k=20, x=2, 4/3, when k=-8, x=-3, 1.'
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"Let's consider an example where a certain amount borrowed from a bank or other similar institution is repaid in fixed installments over a specified period of time."
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'(2) The desired sum is\n\\[\n\egin{aligned}\n& \\left(1+2+2^{2}+\\cdots \\cdots+2^{50}\\right)\\left(1+3+3^{2}+\\cdots \\cdots+3^{70}\\right) \\\\\n= & \\frac{2^{51}-1}{2-1} \\times \\frac{3^{71}-1}{3-1}=\\frac{\\left(2^{51}-1\\right)\\left(3^{71}-1\\right)}{2}\n\\end{aligned}\n\\]'
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'In this factory, the total amount of products P and Q produced in one day, x+y (kg), is maximized when (x, y) = (テト, ナニ), and at that time, x+y = ヌネ. Fill in the appropriate numbers for Tet, Nani, and Nune.'
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'In the arithmetic progression {an} with a first term of 77 and a common difference of -5, (1) from which term does it become negative? (2) From the first term to which term does the sum become maximum? Also, find the maximum value of the sum.'
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'Considering 0 ≤ x ≤ 1, since y=x^{3}+3>0, the required area S is'
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'Given that the sequence {qn} is a geometric sequence with first term 3 and common ratio -2, find the general term of sequence {bn}.'
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'Find the sum as follows:\n(1) The sum of the arithmetic sequence \\frac{1}{3}, \\frac{5}{3}, 3, \\cdots \\cdots, 27\n(2) The sum of the arithmetic sequence with initial term -6 and common difference -8 from the initial term to the n-th term\n(3) The sum of the 19th to 51st terms of an arithmetic sequence where the 5th term is 2 and the 36th term is -60'
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'Prove that the sum ∑(k=1 to n) ka_k, where k is a continuous sequence of three integers product of the form n(n+1)(n+2), is a multiple of 3.'
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'Not true for a < 0, b < 0, but true for all other cases.'
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'There are three different numbers 6, x, 2x-6 that form a geometric sequence in that order. Find the value of x.'
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'Find the minimum value of x + \\frac{9}{x} when x > 0.'
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'Let n be a natural number. If the sum and product of two numbers x, y are integers, then x^n+y^n is also an integer. Prove this statement by mathematical induction.'
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'Find the value of k such that the sequence satisfies a_{0} < a_{1} < a_{2} < a_{3}, a_{3} > a_{4} > a_{5} > ... > a_{12}'
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'Consider the condition where for a certain positive number s, profit is maximized only at (x, y) = (0, s). Pay attention to the relative slopes of lines (1) and (5).'
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'Find the general term of the sequence 1,17,35,57,87,133,211.'
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'In a geometric progression {a_n} where the first term a_1 is a blank, the common ratio r is a positive, a_2 = 2-√2, a_4 = 10-7√2.'
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'Basic Example 41 Solution: Conditions for having imaginary solutions'
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'(1) Find the number that fits in the harmonic sequence 30, 20, □, ⋯.\n(2) Find the general term of the harmonic sequence {a_n} where the 5th term is 1/3 and the 9th term is 1/5.'
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'Translate the given question to the following languages.'
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'Given an arithmetic sequence {a_{n}} with first term x and common difference y, and a geometric sequence {b_{n}} with an integer first term z and a positive integer common ratio t. If c_{n}=2a_{n} + 3b_{n} (n=1,2,3, ...), and c_{1}=3, c_{2}=5, c_{3}=16. Find the values of x, y, z, and t. Also, find the general term of the sequence {c_{n}}.'
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'Prove that three different lines x-y=1 (1), 2x+3y=1 (2), ax+by=1 (3) intersect at one point, then the three points (1,-1), (2,3), (a,b) are on the same line.'
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'Prove the following statements: 27. Proof omitted(3) The equality holds only when x=y=0 28. Proof omitted(1) The equality holds only when a=0(2) The equality holds only when a=b'
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'Assume that the profit per kilogram of product P and Q are a million yen and 30,000 yen respectively. Now consider the profit per day. Let a be a positive number. (i) When a=1, the x and y that maximize the profit are (ノハ, ヒフ). (ii) When a is what value, the profit is maximized only by producing product Q and the maximum profit at that time is ムミミ yen. Provide the values of ノハ, ヒフ, ムミミ. Also, choose one option from 0~6 that fits the scenario in (ii). (0) 0<a<3/5 (1) 3/5<a<9/4 (2) 9/4<a<15/2 (3) a>15/2 (4) a=3/5 (5) a=9/4 (6) a=15/2'
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'Prove the following equation holds: (3) 2nC0 + 2nC2 + 2nC4 + ⋯⋯ + 2nC2n = 2nC1 + 2nC3 + 2nC5 + ⋯⋯ + 2nC2n-1 = 2^{2n-1}'
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'Choose one that applies to (3) from the following 0 to 3.'
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'The 4 points 1, 5, x, y are equally spaced from left to right. Find the values of x and y.'
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'Show that if a, b, x, y are positive numbers and a + b = 1, then √(ax + by) ≥ a√x + b√y. Also, determine when the equality holds.'
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'Express the magnitudes of 123 numbers using inequality signs: \ \\sqrt[3]{\\frac{4}{9}} < \\sqrt[4]{\\frac{8}{27}} < \\sqrt[3]{\\frac{9}{16}} \'
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'Find the quotient and remainder for the following expressions:\n12. (1) (a) Quotient 2x+y, Remainder 3y^{2}\n(b) Quotient 4y-x, Remainder 3x^{2}\n(2) Quotient 2x-3y+4, Remainder 0'
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'Find the minimum value of x+\\frac{9}{x+2} when x>0.'
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'When the equation represents a circle, find the range of values for the constant .'
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'For a sequence {a_{n}}, where each term is nonzero and the sequence formed by the reciprocals of the terms {\\frac{1}{a_{n}}} is an arithmetic progression, the original sequence {a_{n}} is called a harmonic sequence. In other words {\\frac{1}{a_{n+1}} - \\frac{1}{a_{n}} = d \\quad (constant)}'
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'Since the difference between the two solutions is 4, please find these two solutions.'
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'When using a slide rule to calculate and find the value of x that satisfies equation (a), as well as calculate values for (b) and (c), choose one appropriate method of using the slide rule from 0 to 3.'
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'Find the sum S of the series 1 / (1 * 4 * 7), 1 / (4 * 7 * 10), 1 / (7 * 10 * 13), ..., 1 / ((3n-2) * (3n+1) * (3n+4)).'
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'(1) Find the sum of terms from the first term to the nth term of the geometric sequence 3, 9a, 27a^2. (2) Find the sum of terms from the 11th term to the 15th term of the geometric sequence 512, -256, 128.'
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'In the arithmetic sequence {a_{n}} with the first term 77 and common difference -5\n(1) Which term becomes negative?\n(2) Up to which term from the first term is the sum the greatest? Also, find the maximum value of the sum.'
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'When an arithmetic sequence with the first term of 1 {an} and a geometric sequence with the first term of 1 {bn} satisfy a3=b3, a4=b4, and a5≠b5, find the values of a2, b2.'
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'When the sequence {a_n} satisfies {a_1 = 1, a_2 = 7, a_{n+2} = 2 (a_{n+1} + a_n)}, find the remainder when {a_n} is divided by 3.'
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'(1) Since \ x^{2} + 1 > 0 \, find the area being sought'
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'Prove that the following inequalities hold true when |a|<1, |b|<1, |c|<1: (1) ab+1>a+b (2) abc+2>a+b+c'
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'552\\n102\\n(1) \ b \\\n(2) \ a, d \\\n(3) \ C \'
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'There are three numbers in an arithmetic sequence with a sum of 15 and a product of 45. Find these three numbers.'
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'Find the sum of the first to the nth term of the given sequence.'
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'Let a be an integer. Find the value of a and all the solutions of the equation (3a-4)x^2-2ax+a=0 when it has integer solutions.'
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'There are three points A(3), B(-3), C(5) on the number line. Let point D be the point that divides the line segment AB into the ratio 2:1, and point E be the point that divides the line segment AC externally into the ratio 3:1. Find the coordinates of the point that divides the line segment DE into the ratio 3:4.'
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'216 \\x0crac{27}{4}\\n217 \3 \\sqrt{3} + \\frac{4}{3} \\pi\'
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'For a polynomial P(x), when divided by (x+1)^2, the remainder is -x+4, and when divided by (x-1)^2, the remainder is 2x+5.'
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'Find the general term of the sequence: 1/6, 1/9, 1/14, 1/21, 1/30, ...'
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'Find the value of m for the quadratic equation that has two integer solutions and .'
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'Find the general term of the following arithmetic sequence .'
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'For the quadratic equation , find the range of constants that satisfy the following conditions: (1) Having one solution greater than 1 and one solution smaller than 1, and (2) All solutions are greater than or equal to 2.'
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'Find all integer values of k for which the two solutions of the quadratic equation x^2-(k+4)x+2k+10=0 are both integers.'
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'The sequence {an} is a geometric sequence with a first term of 1 and a common ratio of 5. Find the smallest n that satisfies a1+a2+...+an≥10^100. It is given that log10 2=0.3010.'
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'Prove by mathematical induction that for all natural numbers n, the expression x^n + 1/x^n can be represented as an nth degree polynomial in t = x + 1/x.'
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'Find the general term of the sequence {a_{n}} defined by the following conditions.'
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'Choose the most appropriate one for O, K from ( ) to 3. However, you can also choose the same one.'
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'\ { }_{n} \\mathbf{C}_{r}={ }_{n} \\mathbf{C}_{n-r} \'
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'When positive real numbers x and y satisfy 9x^{2}+16y^{2}=144, find the maximum value of xy.'
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'Find the general term of the following sequences.'
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'Find the general term of an arithmetic sequence.'
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'Find all values of a for which one equation has real roots and the other equation has imaginary roots.'
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'Basic Example 18 Find the sum of the sequence containing n in the kth term\nFind the sum of the following sequence:\n\\[\n1 \\cdot(n+1), 2 \\cdot n, 3 \\cdot(n-1), \\cdots \\cdots,(n-1) \\cdot 3, n \\cdot 2\n\\]'
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'For an arithmetic sequence {an} with first term 51 and common difference -4, (1) determine the term at which negative number appears. (2) Find the term up to which the sum of terms from the first term is at maximum. Also, calculate this maximum value.'
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'Choose the most appropriate way of using a slide rule to calculate and satisfy the equation.'
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'Chapter 1 Sequences\nIf the sequence \ \\left\\{a_{n}\\right\\} \ satisfies \\( , a_{1}=1, a_{2}=7, a_{n+2}=2\\left(a_{n+1}+a_{n}\\right) \\), find the remainder when dividing \ a_{n} \ by 3.'
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'Find the number of lattice points contained in the region represented by the following system of inequalities, where \ n \ is a natural number.'
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'Find the sum of natural numbers between 10 and 100 that are divisible by 3.'
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'In a geometric sequence with a positive real number r as the common ratio, an=ar^(n-1) (n=1,2, ...) the sum of the first 5 terms is 16 and the sum of the 6th to 10th terms is 144. Find the sum of the 11th to 20th terms.'
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"(4) Let the total number of bicycles be 20. If the maximum capacity of point A is set to 8 bicycles, how many days does it take for point A's maximum capacity to be exceeded?"
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'Translate the given text into multiple languages.'
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'Investigate how the number of different real solutions of the equation x^3-3x-2-a=0 changes depending on the value of the constant a.'
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'In a geometric progression with a positive common ratio, let the sum from the first term to the nth term be Sn. If S2n=2 and S4n=164, find the value of Sn.'
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'In group sequences, it is important to pay attention to the regularity.'
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"Choose the most appropriate option that matches 'O' and 'K' from the following 0-3. You can choose the same option more than once."
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'Let a, b be nonzero real numbers. The following equations hold when a > 0, b > 0, but what about other cases? Investigate each of the following scenarios.'
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'A sequence consisting of the reciprocals of the terms forms an arithmetic sequence, called a harmonic sequence. (1) Find the number that fits in the blank in the harmonic sequence 30, 20, □, ...... (2) Find the general term of a harmonic sequence {a_n} where the 5th term is 1/3 and the 9th term is 1/5.'
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'Find the sum of the following numbers from 1 to 100:'
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'For a sequence (n=1,2,3,…), where each term an is a natural number and if m<n, then am<an holds for all natural numbers m and n. Prove that for all natural numbers n, n≤an holds true.'
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'Answer the following question about the position of numbers:'
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'The sum of terms from the first term to the nth term of the sequence a_n is represented as S_n = 3/4 n(n+3)(n=1,2,3 ...).'
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'The sequence {an} satisfies {a1}=1, {an+1}=(1+2/n) {an} (n ≥ 1). Find the general term of this sequence.'
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'Let n be a natural number. Prove by mathematical induction that 11^n - 1 is a multiple of 10.'
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'Given the first term is -128, the 6th term is 4, find the common ratio.'
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'Question 1 Sequence {a_{n}} is {a_{1}=1, a_{n+1}=4 a_{n}-5^{n} ......} (1) (n=1,2,3), (2) pattern of geometric sequence Find the general term of sequence {a_{n}} using either of the following strategies.'
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'(2) \ a_{n}=\\frac{1}{2^{n}}+\\frac{1}{2^{3 n-2}} \'
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'Initially, the amounts of salt in the saltwater in containers A and B are 5 g and 20 g respectively. In the first operation Q, the amount of salt in the 20 g saltwater taken from container A is 5 × 20/100 = 1 g, thus, in operation Q, the amount of salt in the 20 g saltwater taken from container B is (20+1) × 20/120 = 3.5 g. Therefore, after one operation Q, how much salt is in container A?'
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'In a geometric sequence with a first term of 3 and a common ratio of 2, which term exceeds 1000 for the first time? Also, up to which term does the sum of terms from the first term exceed 10000 for the first time?'
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'Please prove that the sum of squares 1^{2}+2^{2}+3^{2}+...+n^{2}=\\sum_{k=1}^{n} k^{2}=\\frac{1}{6} n(n+1)(2 n+1).'
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'Prove by mathematical induction that the following equation holds for any natural number n: (n+1)(n+2)(n+3) ... (2n) = 2^n * 1 * 3 * 5 ... (2n-1)'
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'Meaning of addition, geometric mean, and polynomial division'
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'To increase the concentration of salt water in container A to 12% or more, how many times must operation Q be performed? Please answer.'
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'Find the value of m when the quadratic equation \ x^{2}+m x+m+2=0 \ has two integer solutions \ \\alpha, \eta \. [Similar to Waseda University]'
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'When 195 k<-5,27<k, there is 1 solution for k=-5,27, 2 solutions for -5<k<27, and 3 solutions'
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'There is a factory producing 2 products, P and Q, from 3 materials, A, B, and C. To make 1kg of product P, it needs 1kg, 3kg, 5kg of materials A, B, C, respectively, while to make 1kg of product Q, it needs 5kg, 4kg, 2kg of materials A, B, C, respectively. The daily limits for purchasing materials A, B, C are 260kg, 230kg, 290kg. If the factory produces x kg of product P and y kg of product Q in one day, answer the following questions. With the condition that x≥0 and y≥0. The additional conditions are x, y.'
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'Let the first term be a, the common difference be d, the last term be l, and the number of terms be n in an arithmetic sequence. The sum of the arithmetic sequence is denoted as Sn. Sn = 1/2 * n(a+l)'
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'(3) Square of real numbers 7. a^2 ≥ 0 Equality holds when a=0 8. a^2 + b^2 ≥ 0 Equality holds when a=b=0'
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'Prove the following proposition by contraposition. Where a, b, and c are integers. If a^{2}+b^{2}+c^{2}-a b-b c-c a is an odd number, then the number of odd integers among a, b, and c is either 1 or 2.'
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'The minimum value is 1/4 when a=1/2, b=1/2, and the minimum value is 12 when x=2, y=1.'
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'Let U be the set consisting of natural numbers from 61 to 49. Define set V as the set of elements from U whose greatest common divisor with 50 is greater than 1, and set W as the set of even elements from U. If A and B are subsets of U that satisfy the following two conditions, find all elements of set A. (i) A ∪ complement of B = V (ii) complement of A ∩ complement of B = W'
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'Proof: For any integer n, if n is odd and not a multiple of 3, or n is not a multiple of 18, then n satisfies the condition. Therefore, the set A ∪ B includes all integers that meet this condition. The set C includes all integers that are not multiples of 18. Since being not a multiple of 3 does not necessarily mean it is not a multiple of 18, the integers included in A ∪ B may partially include integers that are not multiples of 18 themselves, so A ∪ B is a proper subset of C.'
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'For the set of all integers from 1 to 1000, define sets A, B, and C that satisfy the following conditions.'
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'Based on the following table, find the values of n², n³, √n, and √10n when n = 30.'
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'Prove that there do not exist five distinct real numbers that satisfy both propositions (A) and (B).'
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'59 (1) Minimum value is -1 at x=0; maximum value is \\frac{2+2 \\sqrt{6}}{5} at x=\\frac{2+2 \\sqrt{6}}{5} (2) (p, q)=\\left(-1, \\frac{1}{4}\\right),\\left(-1, \\frac{2+2 \\sqrt{6}}{5}\\right)'
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'When real numbers x, y satisfy x^{2}+y^{2}=1, find the maximum and minimum values of 2x^{2}+2y-1, and the values of x and y at that time.'
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'For a real number a, the greatest integer less than or equal to a is denoted by [a].'
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'When a<0, assuming the maximum value of f(x) is 3, then -a=3, so a=-3, satisfying a<0. When 0 ≤ a ≤ 10, assuming the maximum value of f(x) is 3, then'
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'Answer whether the negation of the following propositions is true or false.\n(1) Negation: For all natural numbers n, n^2 - 5n - 6 ≠ 0 is false, the original proposition is true\n(2) Negation: There exists real numbers x, y such that 9x^2 - 12xy + 4y^2 ≤ 0 is true, the original proposition is false\n(3) Negation: For all natural numbers m, n, 2m + 3n ≠ 6 is true, the original proposition is false'
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'Let [a] represent the largest integer that does not exceed the real number a. Find the values of (4) 70 (1) \\\left[\\frac{13}{7}\\right],[-3],[-√7\\]\. (2) Graph \\( y=-[x](-3 ≤ x ≤ 2) \\). (3) Graph \\( y=x+2[x](-2 ≤ x ≤ 2) \\).'
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'Given \ \\triangle \\mathrm{ABC} \ with \ \\mathrm{AB}=x, \\mathrm{BC}=x-3, \\mathrm{CA}=x+3 \.'
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'There are scores of 10 students in a certain subject test. Assuming the value of x is a positive integer. 43, 55, x, 64, 36, 48, 46, 71, 65, 50 (in points). When the value of x is unknown, how many possible values can be the median of this data?'
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'Determine the range of x values for the point (2x-3,-3x+5) to be located in the second quadrant. Also, state in which quadrant the point does not exist regardless of the value of x.'
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'Two brothers have a total of 52 pencils. Now, the older brother gives exactly one-third of his pencils to the younger brother and still has more than the younger brother. Furthermore, if he gives 3 more pencils to the younger brother, then the younger brother will have more. Find out how many pencils the older brother initially had.'
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'For the sets , B=\\{x|x is a natural number less than 4 \\}, C=\\{x|x is a positive divisor of 6 \\}, fill in the most appropriate symbol - ⊆, ⊂, = in the following blanks:'
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'The distance from home to the station is 1.5km. Initially, walking at a speed of 60m per minute, and then running at a speed of 180m per minute. In order to reach the station within 12 minutes of leaving home, how many meters should the initial walking distance be limited to?'
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'Represent the following sets by listing the elements.(a) A={x | -3<x<2, x is an integer}(b) B={x | x is a positive divisor of 32}'
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'When n = 45, find the values of n², n³, and √10n.'
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'There are 5% saline solution and 8% saline solution. To make a saline solution between 6% and 6.5% by mixing 800 g of 5% saline solution with 8% saline solution, how many grams of 8% saline solution should be mixed?'
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'Given 10 data points of two variables x, y as (x1, y1), (x2, y2), ..., (x10, y10), obtained from x1 + x2 + ...... + x10 = 55, y1 + y2 + ...... + y10 = 75, x1^2 + x2^2 + ...... + x10^2 = 385, y1^2 + y2^2 + ...... + y10^2 = 645, x1y1 + x2y2 + ...... + x10y10 = 445. Also, 10 data points of two variables z, w as (z1, w1), (z2, w2), ..., (z10, w10) are obtained as zi = 2xi + 3, wi = yi - 4 (i = 1, 2, ..., 10). (1) Find the means of variables x, y, z, w denoted by x¯, y¯, z¯, w¯. (2) Let the variance of variable x be s_x^2 and the covariance of variables x, y be s_xy. Show that the two equations x1^2 + x2^2 + ...... + x10^2 = 10(s_x^2 + (x¯)^2) and x1y1 + x2y2 + ...... + x10y10 = 10(s_xy + x¯y¯) hold. (3) Determine the covariance s_xy and correlation coefficient r_xy of variables x and y. Also, determine the covariance s_zw and correlation coefficient r_zw of z and w. Round r_xy and r_zw to the third decimal place.'
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'Please provide the necessary and sufficient conditions for the following proposition.'
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'Intersection and union of 3 sets\nIntersection ( A \\cap B \\cap C ) is the set of elements that belong to all of A, B, and C. Union ( A \\cup B \\cup C ) is the set of elements that belong to at least one of A, B, or C.'
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'Properties of square roots: \\(1 a \\geqq 0) \\), then \\((\\sqrt{a})^{2}=a, \\quad(-\\sqrt{a})^{2}=a, \\quad \\sqrt{a} \\geqq 0 \\)\ 2 a \\geqq 0 \, then \ \\sqrt{a^{2}}=a \\ a<0 \, then \ \\sqrt{a^{2}}=-a \thus, \ \\quad \\sqrt{a^{2}}=|a| \ \ a>0, \\quad b>0, k>0 \,'
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'When a>2, what is the condition for 1<\\frac{a+2}{4}? That is to find h(1)≥0, which means 2-a-2+2-a≥0, so a≤1. There is no common range between a>2 and a≤1.'
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'Prove that there do not exist five distinct real numbers that satisfy both propositions (A) and (B).'
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'Considering the set of real numbers as the universal set, and subsets A, B, and C, answer the following questions.'
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'(3) The average length of the 5 longest individuals out of 10 is mouth. The difference with the average value obtained in (2) is because the individual with number \ \\square \ among the 5 longest individuals belongs to species B.'
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"When finding the square root of a positive number, it is common to use a calculator or computer for calculation, especially for large or decimal numbers. However, it is also possible to calculate the square root by hand. The calculation to find the square root is called '開平' (kaihei), and here we will introduce the method by hand with a specific example."
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'Let be the set of all integers, , prove that but .'
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'Let U be the set of natural numbers starting from 1. For subsets A, B, C of U, the following hold true:'
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'(1) Let t=x^{2}+2 x for a real number x. The range of values for t is t≥A□□. Also, expressing the 65 function y=-x^{4}-4 x^{3}-2 x^{2}+4 x+1 in terms of t, we have y=1□□. Therefore, y takes its maximum value at x=□, and E□ is □. (2) Let a be a real number. Suppose the maximum value of the function y=-x^{4}-4 x^{3}+(2 a-4) x^{2}+4 a x-a^{2}+2 in terms of x is □, as determined in (1). In this case, the range of values for a is a≥qq□.'
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'Find the range of possible values for the following expressions under the given conditions (-1 < x < 2, 1 < y < 3):'
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"A cannot see anyone's hat, how did he know the color of his own hat?"
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'Two brothers have a total of 52 pencils. Now, if the older brother gives exactly one third of his pencils to the younger brother, he would still have more. If he gives 3 more pencils, the younger brother would have more. Find out how many pencils the older brother initially had.'
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'Prove the following proposition by contraposition. Let a, b, c be integers. If a^2 + b^2 + c^2 is even, then at least one of a, b, c is even.'
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'(1) Convert the following fractions to decimals and write them in recurring decimal form.'
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'The following calculation is incorrect. Identify all the mistakes in the equation "27=√729=√3^6=√(-3)^6=√{(-3)^3}^2=(-3)^3=-27" and explain the reasons for considering them incorrect.'
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'Let Z be the set of all integers, A={3n+2|n∈Z}, B={6n+5|n∈Z}, prove that A contains B but A is not equal to B.'
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'Find the following values. (A) Find the value of (B). Find the value of (C). Express \ \\\\\\sqrt{ } \ in non-root form.'
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'When x=0, the maximum value is 10; when x=1,3, the minimum value is 1'
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'Prove the following statements. Where Z represents the set of all integers. (1) A={3n-1 | n∈Z}, B={6n+5 | n∈Z} then A ⊇ B (2) A={2n-1 | n∈Z}, B={2n+1 | n∈Z} then A=B'
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'Let x be a real number. Using sets, determine the truth of the following propositions.'
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'For a real number x, let [k] denote the integer k that satisfies k ≤ x < k+1. (1) Find all integers n that satisfy n²-n-5/4<0. (2) Determine the range of real numbers x that satisfy [x]²-[x]-5/4<0. (3) Let x be within the range obtained in (2). Find all values of x that satisfy x²-[x]-5/4=0.'
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'Find the pair of integers x, y that satisfy the equation √(9+4√5)x+(1+3√5)y=8+9√5. [Yokohama National University of Defense Medicine]'
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'Prove that for a real number x and an integer n, [x+n] = [x] + n.'
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'When \\( (b+c):(c+a):(a+b)=4: 5: 6, R=1 \\), find \ A, a, b, c \'
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'When real numbers x, y satisfy , the range of possible values for is from to .'
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'Let the set consisting of natural numbers from 1 to 49 be defined as the universal set U. Define set V as the set of all elements in U whose greatest common divisor with 50 is greater than 1, and define set W as the set of all even numbers in U. If sets A and B are subsets of U and satisfy the following two conditions, find all elements of set A. (i) A∪(complement of B)=V (ii) (complement of A)∩(complement of B)=W [Iwate University]'
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'Prove the following statements. Where Z is the set of all integers.\n(1) A={3n-1 | n ∈ Z}, B={6n+5 | n ∈ Z} implies A ⊇ B\n(2) A={2n-1 | n ∈ Z}, B={2n+1 | n ∈ Z} implies A = B'
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'Find x and y when the positive integers x and y satisfy the equation √(12-√x)=y-√3. [Seiyaku University]'
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'On a number line, the distance between the origin O and the point P(a) is called the absolute value of the real number a, denoted by the symbol |a|.\n1. |a| ≥ 0\n2. |a| = {a (when a ≥ 0), -a (when a < 0)}'
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'Please provide the page links for the following terms: ①Null Set ②Sine ③Regression Line'
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'(2) For real numbers x and y, if x^2 + y^2 < 1, then |x| < 1 and |y| < 1.'
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'(1) Convert the following fractions to decimals and write them in recurring decimal form.'
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'On the relationship between the value of k and h(x)'
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'When 81 a<0, x<3 a, a^{2}<x; when 0<a<3, a^{2}<x<3 a; when a=3, no solution; when 3<a, 3 a<x<a^{2}'
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'Express the following recurring decimals as fractions.'
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'Let U={1,2,3,4,5,6,7,8,9} be the universal set. If A={1,2,4,6,8} and B={1,3,6,9}, find the following sets: (1) Complement of A (2) Intersection of complement of A and complement of B (3) Union of complement of A and complement of B (4) Complement of A intersection B (5) Complement of A union B. When solving set element problems, start by drawing a Venn diagram as shown below, organize the given conditions, and fill in the elements. When filling in the elements, it is best to do so in the following order: (1) (2) Write down the elements that are not part of A intersection B from the elements of A and B, respectively. (3) (4) (5) Create a Venn chart for set problems.'
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'Investigate the truth of the following propositions. Here, m, n are natural numbers, and x, y are real numbers.'
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'For a real number x, sometimes the symbol [x] is used to represent the largest integer that does not exceed x, and this symbol [ ] is called the Gaussian symbol. Consider the following problems: 1. What is [2.7]. 2. What is [3]. 3. What are [−1.5] and [−0.1] respectively.'
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'Examine the truth of the following propositions. Where m, n are natural numbers, x, y are real numbers.'
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'Represent sets A and B on a number line satisfying the following conditions:\nCondition: A ⊂ B and 3 ≤ k ≤ 4'
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'Investigate the truth values of the following propositions and their negations:\n(1) For all real numbers x, x^2 > 0\n(2) There exists a prime number x such that x is even\n(3) For any real numbers x, y, x^2 - 4xy + 4y^2 > 0'
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'Let [a] denote the largest integer that does not exceed a real number a. Determine the values of [13/7], [-3], [-√7].'
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'Find the sign conditions for ax^{2}+bx+c including the case when a=0.'
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'Prove the following proposition by contraposition: For integers a, b, if the product ab is a multiple of 3, then either a or b is a multiple of 3.'
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'For integers m and n, if m ^ 2 + n ^ 2 is odd, then the product m n is even.'
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'Find the range of values for the constant a such that there exist exactly 3 integers x that satisfy the inequalities x^2 - (a+1)x + a < 0 and 3x^2 + 2x - 1 > 0 simultaneously.'
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'Provide the contrapositive (A multiple of 4 is not necessarily a multiple of 2), the inverse (If x does not equal 3, then x^2 does not equal 9), and the converse (If a and b are not both greater than 0, then a+b is not greater than 0) of the propositions, and state their truth values.'
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'Translate the given text into multiple languages.'
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'The following calculation is incorrect. List all the errors in the equalities from (1) to (6) and provide a reason for considering them incorrect.'
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'[2] One of the solutions is in -2 < x < 0, and the other solution is in the range x <-2 or 0 < x, so the condition for this is f(-2)f(0) < 0, thus (-3a+1)(-a+1) < 0. Therefore, (3a-1)(a-1) < 0. Find the range of a.'
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'When x=-2, one of the solutions is f(-2)=0, therefore -3a+1=0, hence a=1/3. In this case, the equation is 3x^2+7x+2=0, so (x+2)(3x+1)=0, thus the solutions are x=-2, -1/3, and they satisfy the condition. Calculate the value of a.'
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'Dividing a certain integer by 20 and rounding to the nearest tenth gives 17. Find the maximum and minimum of such integers.'
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'Find the range of possible values for the following expressions when \ -1<x<2,1<y<3 \ in practice.'
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'Prove that there do not exist five distinct real numbers that satisfy both (A) and (B).'
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'What kind of triangle is △ABC that satisfies the condition (b-c) sin² A=b sin² B-c sin² C?'
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'When 72-6<a<3, 3<a, there are 2 solutions; for a=-6,3 there is 1 solution; for a<-6 there is no solution; for a=3 x=-\\frac{2}{3}, and for a=-6 x=-\\frac{1}{3}'
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'Let k be a constant. Determine the number of distinct real solutions of the equation .'
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'Dividing a certain integer by 20 and rounding to the nearest tenth results in 17. Find the maximum and minimum integers that satisfy this condition.'
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'Let U = {x | x is a real number} be the universal set. For the subsets of U, A = {2,4, a^2+1}, B = {4, a+7, a^2-4a+5}, if A ∩ B^c = {2,5}, find the value of the constant a.'
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'Explain the notation and representation of sets, and use it to represent the following sets: A: Set of natural numbers B: Set of natural numbers that are multiples of 3'
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'Math I\n-223\nFrom (8), (9), we have -\\frac{9}{2} \\leqq 2 x^{2}+x y-y^{2} \\leqq 0\nWhen 2 x+y<0 and 2 x-y<0\nFrom (1), -(2 x+y)-(2 x-y)=4, hence x=-1\n\nIn this case, \\quad 2 x^{2}+x y-y^{2}=2-y-y^{2}=-\\left(y^{2}+y\\right)+2\n\\[-\\left(y+\\frac{1}{2}\\right)^{2}+\\frac{9}{4}\n\\]\n2 x+y<0 and 2 x-y<0 implies \\quad-2+y<0 and -2-y<0\nTherefore,\n-2<y<2 \\qquad\nFrom (11), (12), we get -4<2 x^{2}+x y-y^{2} \\leqq \\frac{9}{4}\n\nThe range of the desired value is obtained by combining (4), (7), (10), (13)\n\\n\\text { ア }-\\frac{9}{2} \\leqq 2 x^{2}+x y-y^{2} \\leqq \\frac{9}{4}\n\\n<- Splitting the points (x, y) into cases [1]〜 [4]; therefore, what we are looking for is the "combined range".'
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"(3) Negation: 'For all natural numbers m, n, 2m + 3n ≠ 6'\nTruth value: When m = 1, n = 1, 2m + 3n = 5 (≠ 6)\nWhen m ≥ 2, 2m + 3n ≥ 2 * 2 + 3 * 1 = 7, hence 2m + 3n ≠ 6\nWhen n ≥ 2, 2m + 3n ≥ 2 * 1 + 3 * 2 = 8, hence 2m + 3n ≠ 6\nTherefore, the proposition is true. By examining the truth value of the negation, the truth value of the original proposition is also false."
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'Answer the set that satisfies the following conditions:\n(1) 1,2,4,8,16,32\n(2) Subset containing condition P\n(3) Prove that it is a subset of condition P.'
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'Prove the inclusion relationship of the given sets.'
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'(3) Solve the inequality |2x+1|≥3 to get 2x+1≤-3 or 2x+1≥3, which leads to 2x≤-4 or 2x≥2, hence x≤-2 or x≥1'
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'Determine the range of constant a so that x² - 2ax + 3a > 0 always holds in the range 0 ≤ x ≤ 2.'
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'When 0 < a < 1, there are 2 solutions. When a = 1 or a = 4/3, there is 1 solution. When a > 4/3, there are 0 solutions.'
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'When the values of two numbers a and b are in the range -2≤a≤1, 0<b<3, find the range of possible values for 1/2 a-3 b.'
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'(1) \ -19 \\sqrt{2}-17 \\sqrt{3} \\\n(2) 8\\n(3) \ 35-12 \\sqrt{6} \\\n(4) \ -34-7 \\sqrt{2} \\\n(5) \ -5 \\sqrt{2} \\\n(6) \ 10+6 \\sqrt{3} \'
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'Find the union of sets A and B. A={1,2,3}, B={1,3,5,6}'
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'For 1 <|a| <2, the solutions are a<-1,1<a and|a|<2 leads to -2<a<2, thus the solutions for 1 <|a| <2 are -2<a<-1, 1<a<2, hence "1 <|a| <2 \\Rightarrow -1 <a<2" is false. (Counterexample: a=-\\frac{3}{2}) Also, "-1 <a<2 \\Rightarrow 1 <|a| <2" is false too. (Counterexample: a=0) Therefore, 1 <|a| <2 is neither a necessary nor a sufficient condition for -1 <a<2.'
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'52 (1) positive (2) negative (3) positive (4) 0 (5) 0 (6) positive'
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'Determine the truth value of the following propositions. Here, a and b are integers. (A) If a² + b² is even, then ab is odd. (B) If a² + b² is even, then a + b is even.'
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'Determine the truth value of the following propositions. Use sets to investigate (2) and (3).'
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'The catch of scallops in 2017 was 235,952 tons. The average catch of scallops from 2006 to 2017 over 12 years is Co tons, rounding to the nearest whole number.'
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'Find the distance between the following two points.'
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'The scallop catch amount in 2017 was 235,952 tons. What is the average catch amount of scallops from 2006 to 2017 over 12 years (rounded to one decimal place)?'
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'Let θ be an acute angle. When one of sin θ, cos θ, tan θ takes the following value, find the values of the other two trigonometric ratios in each case. (1) sin θ=5/13 (2) cos θ=2/3 (3) tan θ=2√2'
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'Find all non-negative integers \ k \ that make the given equation have real solutions.'
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'Find the discriminant D of the following quadratic equations and indicate the number of real solutions: (1) x^2+3x-2=0 (2) 4x^2-20x+25=0 (3) 2x^2-x+1=0'
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'Investigate the truth of the following propositions. Use sets to investigate (2), (3).'
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"When the proposition 'P⇒Q' is true, which of the following options is correct in its contrapositive?"
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'From |x-9|=3, we get |x-9|=±3, which means x=9+3 or x=9-3, therefore x=12,6'
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'(6) In inequality (1), considering the case when a = 0, for b > 0, there exists a real number x that satisfies ①'
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'Person A and person B both work 4 days a week at their part-time jobs. In this case, show that there is at least 1 day each week when person A and person B work together.'
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'Let U={x | 1 ≦ x ≦ 10, x is an integer}. For subsets A and B of U, with A ∩ B = {3,6,8}, the intersection of the complements of A and B is {4,5,7}, and A ∩ the complement of B is {1,10}. Find the sets A, B, and A ∪ B.'
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'When b=0, the solution to the inequality |3 x-6|<a x is that the graph of y=a x is in the range of x values above the graph of y=|3 x-6|. When a>0, the graphs of y=|3 x-6| and y=a x are as shown in the right figure. Therefore, the statement "a>0 ⟹|3 x-6|<a x has a real number x that satisfies it" is true. Additionally, when a=-4, there exist real number x that satisfy |3 x-6|<a x, but a>0 is not satisfied. Hence, the existence of real number x that satisfies |3 x-6|<a x ⟹ a>0 is false. Therefore, being a>0 is a sufficient condition for the existence of a solution to (1), but not a necessary condition.'
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'EXA Corporation is selling chocolates. The number of units sold, y (where y is an integer greater than or equal to 1), is determined by the following formula based on the selling price, p yen (price per 160 units): y = 10 - p.'
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'Let A be the set of integers divisible by 36, and B be the set of integers divisible by 15. When C={x+y | x∈A, y∈B}, prove that C is the set of integers divisible by 3.'
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'Please choose the appropriate options from the given choices.'
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'Find the intersection of sets A and B. A = {1,2,3}, B = {1,3,5,6}'
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'Let a single-digit natural number be the universal set U, and for its two subsets A, B, if \x08ar{A} ∩ B = {3,9}, A ∩ \x08ar{B} = {2,4,8}, \x08ar{A} ∩ \x08ar{B} = {1,5,7} hold, find the sets A and B.'
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'Basic Question 39 Determination of Set Elements Two sets A, B with integers as elements are A={2,5,a^2}, B={4,a-1,a+b,9} and A∩B={5,9}. (1) Find the values of constants a, b. (2) Find A∪B. [Hiroshima Shudo University] p. 68 Basic Information 1 C. HART & I HINKING'
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'PRACTICE 33: (1) Find all positive odd numbers x that satisfy the inequality x + 1/6 > 5/3x - 9/2. (2) Find the range of the constant a for which the inequality 5(x - a) <= -2(x - 3) holds true and the maximum integer is 2.'
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"How to think about the difference between cases where an inequality sign includes an equal sign and where it doesn't in Example 33 (2)?"
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'From \\\sqrt{2} \\cos \\theta+1=0\ we get \\\cos \\theta=-\\frac{1}{\\sqrt{2}}\, point \\\mathrm{P}\ lies on the semicircle of radius 1 where the x-coordinate is \-\\frac{1}{\\sqrt{2}}\. Therefore, \\\theta\ we are looking for is \\\angle \\mathrm{AOP}\.'
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'Let a be a positive constant. For the function f(x)=-x²+6x where 0 ≤ x ≤ a, (1) find the maximum value. (2) find the minimum value.'
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'Given that the average of three positive numbers a, b, c is 14, and the standard deviation is 8, find the values of a^2+b^2+c^2 and ab+bc+ca.'
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'Mathematics I\nThere are 1%, 5%, and 10% aqueous solutions of a certain substance. If these two or three aqueous solutions are mixed to produce a 7.3% aqueous solution weighing 100 g, how many grams of the 1% aqueous solution can be used at most? Also, what restrictions are there on the use of the 10% aqueous solution?\n[Meijo University]'
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'58 (1) \\( (x, y, z)=(2,3,1) \\)\\n(2) \\( (x, y, z)=\\left(1, \\frac{1}{2}, \\frac{1}{4}\\right) \\)'
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'Find the value of \ x^{2}+4 x y+3 y^{2}+z^{2} \ when \ 10^{2} x=199, y=-98, z=102 \.'
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'In triangle ABC, AB=x, BC=2, CA=4-x. Here, 1<x<3.'
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'When a = 0, the solution of the inequality |3x-6|<b in (1) is that the graph of y=b lies in the range of x values above the graph of y=|3x-6|. When b>0, the graphs of y=|3x-6| and y=b are as shown in the right figure. Therefore, b>0 if and only if there exists a real number x that satisfies |3x-6|<b. Hence, b>0 is a necessary and sufficient condition for the existence of a real number x that satisfies (1).'
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'To create a flowerbed with an area between 9π m² and 33π m² around a circular pond with a radius of 4 m, what should be the width of the flowerbed?'
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'128 (1) 18 items (2) a=23, b=20, c=25, d=36 (3) Variance 18, Standard deviation 4.2 items'
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'When real numbers x, y satisfy (x-y)^{2}<2, we consider x and y to be close. Check the truth of the following propositions for real numbers x, y, z.'
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'If you want to arrive at point B, which is 5km away from point A, in less than 42 minutes, how far should you run at a speed of 10km per hour or more?'
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'Express the following fractions as decimals (finite decimals, recurring decimals).'
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'Basic Example 321: Inequalities and Word Problems'
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"Be cautious of the difference between 'common range' and 'combined range'."
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'What is the largest integer that does not satisfy this?'
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'Q10 There are two units for temperature - Celsius (°C) and Fahrenheit (°F), where the temperature in Celsius x°C is converted to Fahrenheit y°F using the formula y=9/5x+32. Consider the data for the maximum temperature in city A for a certain month. When the average value of the maximum temperature data is 20°C, the average value in Fahrenheit for that month is □°F. Furthermore, if the variance of the maximum temperature in Celsius is X and in Fahrenheit is Y, then Y/X=1□.'
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'Find the values or range of values for x that satisfy the following equations and inequalities:'
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'There is a square. If you increase the length of one side of the square by 1 cm and decrease the length of the other side by 2 cm to create a rectangle, the area of the rectangle becomes half of the area of the square. In that case, what is the length of one side of the square in cm?'
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'Supplementary Example 58: Graph of a function containing the Gauss symbol'
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'(1) When or \nThe negation is and \n(2) When and \nThe negation is or '
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'85 divided by 7 equals 12 with a remainder of 1, less than a and less than 7'
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'Solve the inequality. There are 0 solutions when k<-4, 1 solution when k=-4, 2 solutions when -4<k<2 and 9/4<k, 3 solutions when k=2, 9/4, and 4 solutions when 2<k<9/4.'
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'The average of 11 years of scallop data is 296,332 t, and the additional catch for the year 2017 is 235,952 t, therefore, the average for 12 years is (296,332*11+235,952)/12 = 291,300 t, thus, the average for 12 years will be lower than the original average.'
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'Express the following recurring decimals as fractions.'
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'Integer part and decimal part problem. Find the integer part and decimal part of the following number.'
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'Explain the definition of absolute value on the number line.'
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'Mix 1%, 5%, and 10% aqueous solutions to make a 7.3% aqueous solution with a total weight of 100 grams. How many grams of the 1% aqueous solution can be used? What restrictions are there for using the 10% aqueous solution?'
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'Investigate the truth of the following propositions. Where a and b are integers.\n(A) If a^2 + b^2 is even, then ab is odd.\n(B) If a^2 + b^2 is even, then a + b is even.'
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'When buying an item for 500 yen, how many items or more do you need to purchase in order to join the membership and make it more beneficial?'
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'Find the union of sets A, B, C. A={1,2,3}, B={1,3,5,6}, C={1,3,4}'
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'Choose one from the following (1)~(3) propositions and state its truth value.'
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'Mr. A and Mr. B both work part-time, working 4 days a week. Prove that there is at least one day each week when both A and B work together.'
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'Express the following complex numbers in polar form. Ensure that the argument θ is within the range 0 ≤ θ < 2π.'
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'Determine the value of \ x \ so that the two vectors \ \\vec{a} \ and \ \\vec{b} \ are parallel.'
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'Find the distance between the following two points: (1) A(3+2i), B(6+i) (2) C(10/(1+2i)), D(2+i)'
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'Basic example: This is a type of problem that is used to build foundational skills. It mainly consists of problems that are treated as examples and exercises in textbooks.'
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'112 (1) In the order of A, B, C, D for (1) (-√2, √2), (0, -1), (-3, 0), (3, 0) (2) (2√2, π/4), (2, 5/3π), (2√3, 2/3π), (2, π)'
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'When w ≠ −1, from (1) we have zₙ = 1 / (1 + w) {1 − (−w)ⁿ}, therefore z₆₃ = 0 if and only if (−w)⁶³ = 1, which means w⁶³ = −1. From w⁶³ = cos (63aπ / 3 + b) + i sin (63aπ / 3 + b) we get cos (63aπ / 3 + b) + i sin (63aπ / 3 + b) = cos π + i sin π. Comparing the arguments, we have 63aπ / 3 + b = π + 2kπ (where k is an integer), rearranging gives 63a = (b + 3)(2k + 1), thus, finding the pair (a, b) (1 ≤ a ≤ 6, 1 ≤ b ≤ 6) that satisfies (4) and (5) with the existence of an integer k.'
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'Prove that the following sequence oscillates: the sequence 1, -1, 1, -1, ..., (-1)^(n-1), ...。'
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'Determine the values of x, y, u, v so that the following equations hold true.'
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'Substitute \ x=1 \ into both sides of this equation, we get \\[ (1-\\alpha)(1-\\alpha^{2})(1-\\alpha^{3})(1-\\alpha^{4})=1+1+1+1+1 \\]'
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'When |z|=1, the point z lies on a circle with radius 1 centered at the origin. Since z\x08ar{z}=1, we have \x0crac{1}{z}=\x08ar{z}. Therefore, if we let z=x+yi (where x, y are real numbers), then z-\x0crac{1}{z}=z-\x08ar{z}=2yi. From the given conditions, 1≤2y≤\x0crac{10}{3}, thus \x0crac{1}{2}≤y≤\x0crac{5}{3}'
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'We will prove for α and β satisfying the following conditions: |α| = |β| = 2. Question: When α and β are conjugates, and |α+β| = 2, find the value of |α - 1/2β|.'
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'(2) \\( z=\\frac{1}{5}\\left(\\frac{1}{2}+\\frac{\\sqrt{3}}{2} i\\right)=\\frac{1}{5}\\left(\\cos \\frac{\\pi}{3}+i \\sin \\frac{\\pi}{3}\\right) \\)\n\\egin{\overlineray}{c}\n4 r>0 \\text { so } \\\\\n|r+48|=r+48\n\\end{\overlineray}\\n\ 4 \\mathrm{P} \ with radius \ r \\nbeing the center of a sphere tangent to plane \ \\alpha \\ndepends on the condition that \ \\mathrm{PH}=r \\n\nRefer to page 125 of the book, by using the formula for the distance between a point and a plane\nwe obtain\n\ \\mathrm{PH}=\\frac{|3 r+2 r+6 r-48|}{\\sqrt{3^{2}+2^{2}+6^{2}}} \\nor\n\ \\mathrm{PH}=\\frac{|3 r+2 r-6 r-48|}{\\sqrt{3^{2}+2^{2}+6^{2}}} \\nwhich can be immediately derived.'
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'Exercise problem solution 65 (2) \\frac{2}{3} \\pi+\\frac{2}{3}'
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'3(r+3k) + 2(r+2k) + 6(±r+6k) = 48 49k = -11r + 48 or 49k = r + 48 k = (-11r + 48) / 49 or k = (r + 48) / 49 In this case, r > 0, so |PH| = |k||n0| = |-11r + 48| / 7, (r + 48) / 7 P(r, r, r) when P is the center of a sphere with radius r that touches the plane alpha is PH = r so |-11r + 48| / 7 = r (i) When -11r + 48 > 0, -11r + 48 = 7r, so r = 8/3, which satisfies r > 0 and -11r + 48 > 0. (ii) When -11r + 48 < 0, 11r - 48 = 7r, so r = 12, which satisfies r > 0 and -11r + 48 < 0. For P(r, r,-r) when P is the center of a sphere with radius r that touches the plane alpha the condition is PH = r, so (r + 48) / 7 = r, solving it gives r = 8, which satisfies r > 0. From [1], [2], we have P(8/3, 8/3, 8/3), r = 8/3 or P(12, 12, 12), r = 12 or P(8, 8, -8), r = 8'
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'By expressing 1+i and 3+√3i in polar form, find the values of cos π/12 and sin π/12 respectively.'
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'Math \ \\mathbb{I} \ 221 Lake \ 14 \\Rightarrow \ This volume \ p .315 \'
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'Find all the solutions that satisfy \ x \\geqq-2 \.'
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'It can also be derived as in the following way. Let point be on the line , then , so , the condition for the line to be tangent to the curve is and .'
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'Prove that the inequality sqrt{n!(n-1)!} < n^{n} e^{-n+1} holds for n greater than or equal to 2.'
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'93 (2) Maximum speed is 3rω, minimum speed is -rω'
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'Let a>1 be a constant. For the function f(x)=\\frac{a x}{1+a x},'
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'Prove that the following sequence diverges to negative infinity: sequence 6, 3, 0, ..., 9 - 3n, ...。'
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'What is the price of Chart Formula and Exercise Mathematics I + A (Yellow Chart Mathematics I + A)?'
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'126 (1) \\\frac{2}{5}\ (2) 9 (3) \\\frac{3^{p+1}-1}{2^{p+1}}\'
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'(1) Omitted\n(2) \ I_{n}=\\frac{1}{n-1}-I_{n-2}, \\quad I_{3}=\\frac{1}{2}-\\frac{1}{2} \\log 2 \,\n\ I_{4}=\\frac{\\pi}{4}-\\frac{2}{3} \'
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'The condition for the maximum value to be α4 when α is 1 is 1+α=54, 0<α<2'
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'(4) \ n = 6k + 5 \ (where \ k \ is a non-negative integer \\( ) \\)'
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'Prove by mathematical induction that for all natural numbers n, bn≤an≤cn holds.'
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'For the argument of ( ) 1), in order, \\\frac{\\pi}{2}, \\frac{\\pi}{2}, 0, \\frac{\\pi}{3}, 0, \\frac{\\pi}{6}\, let \\\theta_{n}\ be the argument of \z_{n}^{2}\, then \\\theta_{n+1}\ can be one of the following.'
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'Problem for verifying basic matters. Consists of basic questions involving the application of rules and theorems.'
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'Prove that equation (1) holds for all natural numbers n.'
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'160 times \\frac{4}{3} \\pi^{2}-2 \\sqrt{3} \\pi'
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'Additional question\n(2) \\vec{b}=(-1,2,-3), \\vec{c}=(2,1,-1)\n\n\\left|\\vec{b}\\right|=\\sqrt{(-1)^{2}+2^{2}+(-3)^{2}}=\\sqrt{14},\n\\left|\\vec{c}\\right|=\\sqrt{2^{2}+1^{2}+(-1)^{2}}=\\sqrt{6},\n\\vec{b}\\cdot \\vec{c}=-1\\times 2+2\\times 1-3\\times(-1)=3'
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'Translate the given text into multiple languages.'
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'Let a, b, c be positive constants, and x, y satisfy axy - b*x - cy = 0, x>0, y>0. Find the minimum value of x+y.'
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'(1) Prove using mathematical induction that for any non-negative integer n, cos nθ = T_{n}(cosθ) holds true.'
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"(3) Let's call (1), where \ \\left\\lceil 0<a_{n}<2\\right. \"
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'Exercise 17\\n(1) By the binomial theorem, the following inequality holds for \ n \\geqq 2 \.\\n\\[ (1+1)^{n} \\geqq 1+n \\cdot 1+\\frac{n(n-1)}{2} \\cdot 1^{2} \\]'
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'Comparing the absolute values and arguments of both sides'
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'(2) Assuming there exist complex numbers where at least one pair are equal, let . Then, using (1), we can express this with an integer as \\[ \egin{array}{l} q l-r l=k s \\\\ (q-r) l=k s \\end{array} \\] Since and are coprime, is a multiple of . However, from , we have , therefore cannot be a multiple of .'
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'The number of real solutions to the equation is 12'
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'Using number theory, we proved that if a and b are coprime, then there exist integers x and y such that ax + by = 1. We will prove in a similar way that when a and b are coprime, ax + by can take any integer value.'
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'(1) The region represented by |z|>1 is the exterior of the circle centered at the origin with radius 1. Moreover, the region represented by Re(z)<1/2 is to the left of the vertical line l passing through the point 1/2. Therefore, the required region resembles the diagonally shaded area in the right figure. However, it does not include the boundary.\n(2) From w=1/z, it follows that w≠0 and z=1/w. The line l is the perpendicular bisector of the line segment connecting the two points O(0) and A(1), and for a point P(z) to the left of line l, it must satisfy OP<AP, which translates to |z|<|z-1|. Hence, the desired region can be expressed as |z|>1 and |z|<|z-1|. Substituting z=1/w into |z|>1 results in |1/w|>1, hence |w|<1. Substituting z=1/w into |z|<|z-1| yields |1/w|<|1/w-1|, which simplifies to 1/|w|<|1-w|/|w|, or |w-1|>1. Therefore, the sought region is the intersection of the regions represented by (1) and (2), similar to the diagonally shaded area in the right figure. However, it does not include the boundary.\nAlternative approach: Let z=x+yi (x, y are real numbers), then from |z|^2>1 we get x^2+y^2>1. From Re(z)<1/2 we have x<1/2. Visualize the common areas represented by (1) and (2).\nLet w=x+yi (x, y are real numbers). From w=1/z, it follows that w≠0, when (x, y)≠(0, 0). In this case, z=1/w=1/(x+yi)=(x-yi)/(x^2+y^2). The real part of z is the angle x. Simplifying the denominators by converting to real numbers.'
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'137 (1) In order - \\frac{3}{8}, \\frac{11}{16} (2) \\mathc \\frac{\\sqrt{7}}{2} 138 k=3\\cosθ=\\frac{3}{5}, \\cosθ=\\frac{4}{5}.'
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'Additional explanation about real number conditions'
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'In data representing phenomena in the world, which number from 1 to 9 is most likely to appear the most in the highest position?'
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'45 (1) k=6, \\frac{1}{6} (2) \\alpha=-\\frac{1}{2}'
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'For the three points A(-1,-2), B(1,2), C(a, b), find the values of a and b when triangle ABC is an equilateral triangle.'
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'Relationship between arithmetic mean and geometric mean: For two real numbers a, b, (a+b)/2 is called the arithmetic mean of a and b. Also, when a>0, b>0, √(a*b) is called the geometric mean of a and b.\n\nNote that the relationship between arithmetic and geometric means is often expressed in the form of (*) below. Proof: When a>0, b>0\n\na+b-2√(a*b) = (√a)² - 2√a√b + (√b)² = (√a - √b)² ≥ 0\n\nTherefore, a+b ≥ 2√(a*b) ...( * ).\n\nDividing both sides by 2, we get (a+b)/2 ≥ √(a*b)\n\nEquality holds when (√a - √b)² = 0, i.e., √a = √b, or a = b.\n\nThe condition a>0, b>0 is important.'
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'(1) \\ -5 < k < 2 \\quad \\)\n(2) \\ -7 \\leqq k \\leqq-5 \\)'
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'Explain how to perform addition and subtraction of fractions with the denominator of 11.'
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'Regarding the difficulty levels, all examples and exercises are rated on a scale of five levels of difficulty. @@@ (10) ...... Textbook example level @@(3) ...... End-of-chapter exercises level'
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'Simplify the following fractions into irreducible fractions.'
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'Find the maximum and minimum values of the following functions when the point (x, y) moves within the region represented by the system of inequalities: (1) x²+y² (2) x²+(y-8)²'
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'a: Negative, b: Positive, c: Negative, d: Positive'
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'General arithmetic mean, geometric mean, and their relationship'
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'Rewrite the following angles in terms of degrees and radians.'
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'(1) Find the sine, cosine, and tangent values of \ 2 \\alpha \ and \ \\frac{\\alpha}{2} \ when \ 0<\\alpha<\\pi \ and \ \\cos \\alpha=\\frac{5}{13} \.'
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'Find the maximum and minimum values of x + y in the region D represented by the system of inequalities -2 ≤ 2x + y ≤ 2, -2 ≤ 2x - y ≤ 2.'
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'When t = \\frac{1}{2}, the minimum value is \\frac{4}{3}'
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'A positive number N has the first non-zero digit appearing at the k-th decimal place if and only if 1/10^k ≤ N < 1/10^(k-1) if and only if 10^(-k) ≤ N < 10^(-k+1) if and only if -k ≤ log_10 N < -k+1'
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'Provide an identity involving a fraction with 17.'
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'Let integers n, r satisfy n≥2, 1≤r≤n. Prove that r⋅C(n, r) = n⋅C(n-1, r-1).'
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'Let n be a natural number. If both the real and imaginary parts of (-1+sqrt(3)i)^n are integers, then the remainder of n divided by 3 is W.'
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'For a real number a, the integer part of a, which is the largest integer that does not exceed a, i.e., an integer n such that n ≤ a < n+1, is called the integer part of a, and a - n is called the decimal part of a. For x > 1, let f(x) denote the integer part of log base 2 of x, and g(x) denote the decimal part.'
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'\ 30-2 \\sqrt{2} \\leqq a \\leqq 2 \\sqrt{2} and b \\geqq a^{3}-a \'
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'Select committee members from n people (with at least 1 and at most n members), and then choose one person as the chairman from the members.'
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'Prove that for natural numbers n where the remainder is 1 when divided by 3, (x-1)(x^{3n}-1) is divisible by (x^{3}-1)(x^{n}-1).'
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'There are 2 when 92 and (a<-3,1<a), there is 1 when a=-3,1, there are 0 when -3<a<1'
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'A positive number N is a number where the first non-zero digit appears at the 3rd decimal place.'
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'When \a, b, c\ are non-negative real numbers and satisfy \\\frac{a}{1+a} + \\frac{b}{1+b} \\geqq \\frac{c}{1+c}\, does \a+b\\geqq c\ hold? If it holds, prove it; if not, give a counterexample. [Similar to Tohoku Gakuin University]'
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'When θ is -π/4, the minimum value is -1, and there is no maximum value.'
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'[1] When k is in the range 2≤k≤2√5, find the maximum and minimum values of k.'
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'Determine the sign of the sum of squares of real numbers a and b, a^2 + b^2. Take into account the relative magnitude of real numbers and the sign of the difference.'
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'Find the quotient and remainder when integer (n-1)^{3} is divided by integer n^{2}-2 n+2.'
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'For a non-negative real number a, an real number r, such that 0≤r<1 and a-r is an integer, is denoted by {a}. In other words, {a} represents the decimal part of a. (1) Find one positive integer n that satisfies {nlog_{10}2}<0.02. (2) Find one positive integer n such that the highest digit of 2^{n} in decimal representation is 7. Given that 0.3010<log_{10}2<0.3011 and 0.8450<log_{10}7<0.8451.'
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'63 (A) 4\n(B) 9\n(C) 10\n(D) 6\n(E) 3\n(F) 0\n(G) 7\n(H) 24\n(I) 5\n(J) 8'
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'When dividing the polynomial P(x) by x-1, the remainder is -1; when dividing by x+1, the remainder is 3.'
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'Calculate the following. (1) (√3+i)^8 (2) i-i^2+i^3-i^4+⋯+i^49-i^50'
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'On a circle with radius r=2, the coordinates of point P are (1, √3), therefore sin(7/3)π=√3/2, cos(7/3)π=1/2, tan(7/3)π=√3'
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'For positive real numbers x and y such that xy=100, find the minimum value of (log_{10}x)^3 + (log_{10}y)^3 and the corresponding values of x and y.'
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'Find all positive integers n such that n^n + 1 is divisible by 3.'
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'Find the 2 numbers whose sum and product are as follows: (a) Sum is 7, product is 3 (b) Sum is -1, product is 1.'
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'A positive number N has a 3-digit integer part if it lies between 100 and 999'
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'When real numbers x and y satisfy (x-3)^2 + (y-3)^2 = 8, find the possible range of values for x + y and xy.'
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'Find the values \u200b\u200bof the constants 40a, b such that the polynomial P(x) = ax^{4} + bx^{3} + abx^{2} - (a + 3b - 4)x - (3a - 2) is divisible by x^{2} - 1. Also, factorize P(x) over the set of real numbers for the determined values of a, b. To determine if P(x) is divisible by x^{2} - 1, we use the conditions P(1) = 0 and P(-1) = 0.'
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"For arithmetic mean, geometric mean, and harmonic mean, the property (harmonic mean ≤ geometric mean ≤ arithmetic mean) holds. Now, let's look at specific examples of each type of mean.1. Properties of arithmetic mean, geometric mean, and harmonic mean for a>0, b>0, where the arithmetic mean of a and b is m1, the geometric mean is m2, and the harmonic mean is m3."
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'Determine the polynomial that satisfies the equation 21'
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'Find the values of the constant m such that the quadratic equation has only integer solutions, and determine all the integer solutions at that time.'
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'An index used to indicate the brightness of stars is called "magnitude". The lower the magnitude value, the brighter the star. Ancient Greek astronomers classified the apparent brightness of stars into 1 to 6 magnitudes based on naked-eye observations. This classification relied entirely on human perception. However, in the 19th century, it was discovered that the brightness of a 1st magnitude star is about 100 times that of a 6th magnitude star. Based on this fact, the astronomer Pogson defined the classification of magnitudes, which had previously been based on perception, as follows: The difference in magnitudes between a 1st magnitude star and a 6th magnitude star is 5. Therefore, if we let x be the brightness ratio of the 1st magnitude star, then x^{5}=100, or x^{5}=10^{2}. Hence, x=10^{\x0crac{2}{5}}=10^{0.4} is approximately 2.512. In other words, he defined the brightness ratio of a 1st magnitude star as 2.512 times that of a 6th magnitude star.'
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'The quadratic equation x^2 + px + q = 0 has two distinct roots α and β. When the quadratic equation x^2 + qx + p = 0 has two roots α(β-2) and β(α-2), find the values of real constants p and q.'
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'Fundamental Concepts\n2. Odd Function and Even Function\nA function \\( f(x) \\) is an odd function when \\( f(-x)=-f(x) \\) always holds true.\nA function \\( f(x) \\) is an even function when \\( f(-x)=f(x) \\) always holds true.\nFor example, \ y=\\sin \\theta, y=\\tan \\theta \ are odd functions, while \ y=\\cos \\theta \ is an even function.\n\nProblem: Determine if \\( f(x) = x^3 \\) is an odd function or an even function.'
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'Problem 1: Polynomial division and Cube root of 1\nLet n be a natural number. Find the remainder when x^(2n) + x^n + 1 is divided by x^2 + x + 1.'
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'Practice (1) When n is a natural number greater than or equal to 2, find the remainder when x^n is divided by (x-2)^2.'
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'Prove that the following inequalities hold true. Also, determine the conditions under which the equality holds.'
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'When the points A(-2,3), B(1,2), C(3a+4,-2a+2) are collinear, find the value of the constant a.'
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'Consider the number of ways to select n committee members from a total of 2n people, consisting of n men and n women.'
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'Find the values of the natural numbers a, b, c, d that satisfy the equation (3) 107 (1) (1/∛5 - ∛4) = a ∛b + ∛c + ∛d. Additionally, a > 1, c > d.'
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'Prove the following inequality: \\( \\frac{a^{2}+b^{2}}{2} \\geqq\\left(\\frac{a+b}{2}\\right)^{2} \\)'
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'Find the complex number z that satisfies z^3 = 65 + 142i, where both the imaginary and real parts of z are natural numbers.'
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'Translate the given text into multiple languages.'
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'453\n139\n(1) \ \\frac{1}{2} \\n(2) \ -\\frac{\\sqrt{3}}{2} \\n(3) \ \\frac{1}{\\sqrt{3}} \\n(4) \ \\frac{1}{2} \'
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'When given two numbers, their arithmetic mean (sum divided by 2) is greater than or equal to their geometric mean (square root of the product), and the maximum possible result is x = y'
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'Calculate the expression (3) which is an integer. Find its value.'
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'The required condition is when both (1) and (2) have complex roots as 5.'
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'I want to divide 4 a^{2} + 3 ab + 2 b^{2} by a+2b to find the quotient and remainder.'
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'Indicate the relationship between real numbers a and b, and prove the inequality a > b.'
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'When 149 divided by a is greater than 2, there are 0 roots;\nWhen a is -2, there is 1 root;\nWhen -2 < a < 0, there are 2 roots;\nWhen a is 0, there are 3 roots;\nWhen 0 < a < 9/8, there are 4 roots;\nWhen a is 9/8, there are 2 roots'
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'Practice (1) Find the values of the constant for which the solutions of the quadratic equation are all integers and determine the corresponding integer solutions.\n(2) Let be a positive constant. Find all triplets that satisfy the condition that both and are integers when .'
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'Determine the value of the real number x, so that (1+xi)/(3+i) becomes (A) a real number (B) a purely imaginary number.'
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'Find the condition for the curve y=ax^2+bx+1 using real numbers a and b to not have any shared points with the positive part of the x-axis.'
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'Prove that when the ratio a/b = c/d holds true, the equation (a^2 + c^2)/(b^2 + d^2) = (a^2 - c^2)/(b^2 - d^2) also holds true.'
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'Perform the following calculations:\n(1) \\( \\frac{1}{(x-1)x}+\\frac{1}{x(x+1)}+\\frac{1}{(x+1)(x+2)} \\)\n(2) \\( \\frac{2}{(n-2)n}+\\frac{2}{n(n+2)}+\\frac{2}{(n+2)(n+4)} \\)'
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'Make the numerator 13 (degree of the numerator) < (degree of the denominator) and then calculate.'
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'Solving the simultaneous equations (3) and y=2x, we get x=14/3, y=28/3. Therefore, the coordinates of the point Q we are looking for are (14/3, 28/3).'
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'Consider two conditions: p: (x-1)^{2}+(y-1)^{2}≤4, q: |x|+|y|≤r, where r>0. Find the range of values for the constant r that makes q a sufficient condition for p.'
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'Let k be a natural number. Prove that when the remainder of 2^{k} divided by 7 is 4, the remainder when k is divided by 3 is 2.'
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'Prove that the sequence \ \\{a_{n}\\} \ satisfies \\( 0<a_{1}<3, \\quad a_{n+1}=1+\\sqrt{1+a_{n}}(n=1,2,3, \\cdots \\cdots) \\)'
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'When \\(\\left(\\alpha+\\frac{1}{\ar{\\alpha}}\\right)\\left(\ar{\\alpha}+\\frac{1}{\\alpha}\\right)=4\\), find the absolute value of the complex number \\\alpha\.'
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'Convert the following recurring decimals into fractions.\n(1) 0.∴63\n(2) 0.0∴58\n(3) 3.21∴8'
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'When 77 < k < \\frac{2}{5}, \\frac{18}{5} < k, there are 2 solutions; when k = -\\frac{2}{5}, \\frac{18}{5}, there is 1 solution; when -\\frac{2}{5} < k < \\frac{18}{5}, there are 0 solutions'
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'(1) The inequality [3 x] ≤ 3 x < [3 x] + 1 holds. When x > 0, dividing each side by x, we get [3 x]/x ≤ 3 < [3 x]/x + 1. Therefore, 3 < [3 x]/x + 1 simplifies to 3 - 1/x < [3 x]/x. So, 3 - 1/x < [3 x]/x ≤ 3! As x approaches infinity, (3 - 1/x) = 3, hence the limit lim(x→∞) [3 x]/x = 3'
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'In mathematics, -167 = \ \\frac{0-2a}{1+0+1}=-a \ So, \ \\quad -a=3 \ therefore, \ \\quad \\alpha =-3 \'
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'(1) Let a complex number z satisfy |z|=1 (where z ≠ -1). Let the points represented by 0, z, and 1/(z+1) on the complex plane be denoted as O, A, B respectively.'
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'What geometric shape do the set of points that satisfy the equation form?'
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'Practice (3) When real numbers a, b satisfy 0<a<b<1, compare the sizes of (2^a-2a)/(a-1) and (2^b-2b)/(b-1).'
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'When a complex number z satisfies z+1/z=√2, find the value of z^20+1/z^20.'
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'(2) Continuous for \ -1 \\leqq x<0,0<x \\leqq 2 \, discontinuous at \ x=0 \'
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'(5) Achieves a maximum value of 16√3/9 at x = 8/3, and a minimum value of 0 at x = 0'
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'An index of math terminologies learned for the first time is arranged in alphabetical order at the end of the book.'
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'When complex numbers \\\alpha\ and \\eta\ satisfy \|\\alpha|=|\eta|=|\\alpha-\eta|=1\, find the values of \\(|2 \eta-\\alpha|,\\left(\\frac{\eta}{\\alpha}\\right)^{3}\\).'
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'Real numbers satisfy . In the complex plane, let be the locus of points satisfying , and let be the locus of points satisfying .'
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'How to represent the number of digits of a natural number m using Gauss notation?'
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'Practice proving the following inequalities. Where n is a natural number.'
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'In the interval , the table of increasing and decreasing values of is as follows.'
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'Prove that for all natural numbers n, when the sequence {a_{n}} satisfies 0 < a_{1} < 3 and a_{n+1}=(1+sqrt{1+a_{n}})(n=1,2,3, ...) , it holds that a_{n} > 0 and 3 - a_{n} > 0.'
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'Express the following complex numbers in polar form, where the argument θ satisfies 0 ≤ θ < 2π.'
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'What is the complex number representing the point that internally divides the line segment connecting points A(α) and B(β) in the ratio m:n? Also, please explain the derivation method.'
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'Mathematics Table\n173\n(i) When x=2 m π (m is an integer), cos x=1. Since (1) always holds true, the sum is 0 and the series converges.\n(ii) When x=(2 m+1) π (m is an integer), cos x=-1. The condition for (-1)^{k}=1 is that k is an even number.\nTherefore, the condition for the convergence of the series for all real numbers x is that k is even.\n(2) When x=0, 1-\\cos ^{k} x=0.\nSo, f(0)=0.\nWhen x ≠ 0, near x=0, 0<\\cos x<1.\nf(x) =\\frac{1-\\cos ^{k} x}{1-\\cos x} =1+\\cos x+\\cos ^{2} x+\\cdots+\\cos ^{k-1} x\nTherefore, \\lim _{x \\rightarrow 0} f(x)=1+1+1+\\cdots+1=k>0.\nHence, \\lim _{x \\rightarrow 0} f(x) ≠ f(0).\nTherefore, f(x) is not continuous at x=0.\nWhen k is odd, \\cos ^{k} x=(-1)^{k}=-1.\nTherefore, (1) is not valid.\n1-\\cos ^{k} x =(1-\\cos x)(1+\\cos x+\\cos ^{2} x+\\cdots+\\cos ^{k-1} x)\nChapter 4 ∎'
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'When a point z moves on a circle with radius 1 centered at the origin O, what kind of shape does the point w, represented by the following equation, draw? w=frac{z-1}{z+2}'
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'Practice (2107)\nWhen a ball is dropped to the floor, it always bounces back up to \\\frac{3}{5}\ of the falling height. When this ball is dropped from a height of \3 \\mathrm{~m}\, find the total distance the ball moves up and down until it comes to rest.'
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'What kind of shape does the set of points satisfying the given equations represent?\n(1) 3|z|=|z-8|\n(2) 2|z+4 i|=3|z-i|'
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'\ - 2 < \\alpha < 2 \ and on the coordinate plane, the line parallel to the y-axis passing through point \\( \\mathrm{P}(\\alpha, 0) \\) is denoted as \ \\ell \, the intersection points of line \ \\ell \ and ellipse \ \\frac{x^{2}}{4} + y^{2} = 1 \ are \ \\mathrm{Q}, \\mathrm{R} \. It is assumed that the y-coordinate of Q is greater than the y-coordinate of R.'
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'What geometric shape is formed by the set of points z that satisfy the following equations?\n(1) |2z+1|=|2z-i|\n(2) |z+3-4i|=2\n(3) (3z+2)(3\x08ar{z}+2)=9\n(4) z-\x08ar{z}=4i'
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'Practice\nLet n be a positive integer.\nShow that (1 + √(2/n))^n > n using the inequality (*) above.\nUsing the inequality shown in (1), find the value of lim(n -> ∞) n^(1/n).'
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'(3) When the plane PQR intersects with the line OD, let the vector OX=x⋅d (x is a real number), then x can be expressed in terms of q as x=q/□卜□q-□ナ.'
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'When -2 ≤ x ≤ 2, |x-2|=-(x-2)=2-x; when 2 ≤ x ≤ 3, x-2. Therefore, find ∫_{-2}^{3}√(|x-2|)dx.'
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'(Challenge question)\n(1) (a) 0\n(2) (b) 0\n(3) (I) (1)'
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'0 copies when a>1; 1 copy when a=1 or a≤0; 2 copies when 0<a<1'
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'Determine the value of a so that the points A(α=2+i), B(β=3+2i), and C(γ=a+3i) are collinear on the complex plane.'
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'Let i be the imaginary unit, α=√3+i, β=(√3-1)+((√3+1)i). What is the argument of \ \\frac{\eta}{\\alpha}\?'
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'For the complex number z satisfying |z|=5 and |z+5|=2√5, find the following values.'
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'When the complex number z satisfies z+1/z=√3, find the value of z^10+1/z^10.'
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'When , and , it is equal to 0; when , it is equal to 1.'
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'The sequence {an} is defined by a1=2 and an+1=√(4an-3) (n=1,2,3,...)'
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'(1) (complex signs arbitrary)\n(2) (complex signs arbitrary), '
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'For the triangle OAB with vertices O(0), A(α), B(β), what kind of triangle is triangle OAB when the following equations hold true:'
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'The minimum value is -48 when (x, y)=(-√6, √3) and (√6, -√3)'
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'In the complex plane, find the absolute value of the complex number z = 3 + 4i.'
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'146\n(1) \\(\\left(\\frac{n+1}{n} \\cos \\theta - \\frac{1}{n} \\cos (n+1) \\theta\\right.\\), \\(\\frac{n+1}{n} \\sin \\theta - \\frac{1}{n} \\sin (n+1) \\theta\\)\n(2) \\(\\frac{8(n+1)}{n}\\)\n(3) 8'
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'Convert the given recurring decimal to a fraction.'
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'Minimum value is |a| when a ≤ 2; minimum value is 2√(a-1) when a > 2.'
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'Let c be a real number. Consider the quadratic equation in x: having two roots α, β. Suppose on the complex plane, the three points α, β, and c^2 form the vertices of a triangle, with the centroid being at 0. Find the value of c.'
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'The sequence is defined as . (1) Prove that the inequality holds for all natural numbers . (2) Prove that the inequality holds for all natural numbers . (3) Find the limit .'
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"Calculate the time it takes for a particular ship to travel between two ports. The ship travels at a constant speed from port A to port B, and returns at the same speed. However, on the outbound journey, the speed increases by 20% due to a tailwind, and on the return journey, the speed decreases by 20% due to a headwind. When the distance between the two ports and the ship's speed on still water are known, determine the total time for the entire journey."
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'Prove the inequality 1/(2(n + 1)) ≤ In ≤ 1/(n + 1).'
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'Prove the following equation: \ I_{1}=1, n \\geqq 2, then I_{2n-1} = \\frac{2n-2}{2n-1} \\cdot \\frac{2n-4}{2n-3} \\cdot \\ldots \\cdot \\frac{4}{5} \\cdot \\frac{2}{3} \\cdot 1 \'
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'Example 74 | Calculation of the nth power of a complex number (2)'
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'In problem 314 of math, given OA=7, OB=5, AB=8 in triangle OAB with orthocenter H. Also, let OA be vector a and OB be vector b.'
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'Let the sequence {an} satisfy the conditions {a1} = 1/2, {an+1} = 2{an} - {an}^{2}(n ≥ 1).'
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'When 86k < -3, there are 2; k = -√3, there are 3; -√3 < k < 0, there are 4; k = 0, there are 3; 0 < k < √3, there are 4; k = √3, there are 3; √3 < k, there are 2.'
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'Prove that 1/x log(1+x) > 1/y log(1+y) holds when 0<x<y.'
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'Example 56 | Distance between 2 points, area of a triangle'
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'147\n(1) (A) 40\n(2) (B) \\\frac{1}{3}\\n(3) (C) \\\frac{\\sqrt{3}}{18}\'
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'Practice (2) Consider the fraction function f(x)=\\frac{ax-b}{x-2}, with the condition that b\\neq 2a. For all x satisfying 0\\leq x \\leq 1, where 0\\leq f(x) \\leq 1, and f(f(x))=x. Find the values of constants a, b. [Osaka Univ]'
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'Based on the following price list, calculate the total price of Shing Lee Math I+A (Blue Chart Math I+A) and Shing Lee Math II+B (Yellow Chart Math II+B).'
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'Find the value of the following complex numbers.'
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'For the three points A(5+4i), B(3-2i), C(1+2i), find the complex numbers representing the following points.'
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'Example 40 | Internal and external division points, complex numbers representing the centroid\nFor 3 points A(-1+4i), B(-3-2i), C(5+i), find the complex numbers representing the following points:\n(1) Point P that divides line segment AB in the ratio 2:3 internally\n(2) Point Q that divides line segment AC in the ratio 1:3 externally\n(3) Midpoint M of line segment BC\n(4) Centroid G of triangle ABC'
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'Please solve the problem related to binomial coefficients.'
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'Solve {3 |x-1| \\geqq x+3} and find the range of solutions.'
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'阿44 ⇒ This book p.394 (1) Let a be the number to fill in the blank (where a is an integer, 0 ≤ a ≤ 9). When the last three digits are multiples of 8, 746□2 becomes a multiple of 8 because 600 + 10a + 2 = 602 + 10a = 8(a+75) + 2(a+1). Since 2(a+1) is a multiple of 8, a+1 is a multiple of 4. Therefore, a+1 = 4,8, which means a = 3,7. Hence, the number to fill in the blank is 3 or 7. (2) Let N = 10⁵a + 10⁴b + 10³c + 10²d + 10e + f. Then, N = (100001-1)a + (9999+1)b + (1001-1)c + (99+1)d + (11-1)e + f = 11(9091a + 909b + 91c + 9d + e) + (b+d+f) - (a+c+e). Therefore, N is a multiple of 11 when the difference between the sum of digits at even positions (a+c+e) and at odd positions (b+d+f) is a multiple of 11.'
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'Find the number of integer solutions for the given example '
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'Choose 2 colors out of 4 colors and paint in the order of (1) on the figure. Therefore, the desired way of painting is 4P2 = 4 * 3 = 12 (ways)'
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'(1) \1\ \\\\Undefined. When a=1, \\frac{1}{a}+\\frac{1}{b}=1+\\frac{1}{b}>1 is not valid.'
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'For a real number x, let [x] represent the greatest integer that does not exceed x. Also, let {x}=x-[x]. When 0 ≤ x < 1 and n is an integer greater than or equal to 3, find the number of x that satisfy {nx}=x+1/n.'
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'Given a_{1}=1, a_{2n}=2. It is permissible to count the number of sequences that satisfy this condition, but it is better to consider it as a shortest path problem by replacing 1 with → and 2 with 个.'
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'In binary representation, how many natural numbers N are there with 8 digits?'
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'For a class of students, it was investigated whether they have siblings:\n(a) Students without brothers have sisters.\n(b) Students with brothers do not have either brothers or sisters.\n(c) Students without both brothers and sisters have sisters.\nUsing sets, prove the following conclusions:\n(1) Students with brothers also have sisters.\n(2) Students without sisters have brothers.'
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'Consider a grid like the one on the right. To ensure that no row (horizontal) or column (vertical) have the same numbers, find the number of different ways the natural numbers 1 to 4 can be placed in the grid, denoted as K.'
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'[4] Given m-n=27, m^{2}+mn+n^{2}=37\n\nWhen we expand (n+27)^{2}+(n+27)n+n^{2}=37\nwe get 3n^{2}+81n+729=37\n3\\left(n^{2}+27n+243\\right)=37\n\nTherefore, 483\n\nFrom (m-n)^{2}+3mn=37\nwe have 27^{2}+3mn=37\n\nThe left side is divisible by 3, but the right side is not, which is acceptable.\nThe left side is divisible by 3, but the right side is not, hence there is no integer n that satisfies this equation.\nHence,\n\nm=12, n=9'
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'Let a be a negative real number. Find the maximum value of a and the corresponding values of x, y when x and y are adjacent integers that satisfy 4x^2 + 12y^2 - 12xy + 4x - 18y + 7 = a.'
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"Not having balls of the same color adjacent means in the case where white balls are not adjacent, excluding the cases where red balls or blue balls are adjacent. When white balls are not adjacent and red balls are adjacent and blue balls are adjacent, it means two cases when placing R and B in two of the 3 positions between white balls, such as RRBWBW, in permutation. Finding cases where 4 white balls are adjacent to 2 or more. Selecting 2 places from the 3 places of RBO and arranging W' and white. For example, selecting 3 places out of 5 places to place white."
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'Let natural numbers a and b have remainders r and s when divided by 11, respectively. Prove that the remainder of a+b divided by 11 is equal to the remainder of r+s divided by 11, and the remainder of a*b divided by 11 is equal to the remainder of r*s divided by 11.'
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'Prove that if the quadratic equation with integer coefficients has a rational solution , then is an integer.'
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'0 when 78 k>2, 1 when k=2, 2 when -1<k<2, 3 when k=-1, 4 when -2<k<-1, 3 when k=-2, 2 when k<-2'
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'Let event S denote the product of the numbers on three cards being divisible by 3, and event T denote the sum of the numbers on three cards being divisible by 3. Then, the intersection of events S and T means selecting 3 out of 5 elements from set A in question (2). and choosing one element from each of sets A, B, C. These two events are mutually exclusive.'
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'Let a, b, c be positive numbers. The condition for the existence of a triangle |b-c|<a<|b+c| is derived from the inequalities a<b+c, b<c+a, c<a+b. Therefore, for positive numbers a, b, c, if a is the largest, the triangle condition only needs to consider a<b+c. Since a<a+2<a+4, the condition for the existence of a triangle is'
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'When dividing by a negative number, the direction of the inequality changes.'
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'Represent 6 points on a line g with 6 circles and arrange them. By choosing one out of five places between the circles to insert a divider, two parts can be created, which correspond to 2 points on line h. There are 5 ways to insert the divider. As there are 3 ways to choose 2 points from 3 points on line h, the total number of ways is 5×3=15.'
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'(1) As there are 2 parents and 4 children, a total of 6 people in a circular permutation, the total number of ways to arrange them is (6-1)!=5!=120 (ways). (2) Considering the 2 parents as 1 person, in a total of 5 people in a circular permutation, there are 2 ways to arrange the 2 parents, so it is (5-1)!×2=4!×2=24×2=48 (ways). (3) When fixing the 2 parents, the remaining 4 positions for the 4 children to be arranged have 4!=24 (ways). (4) First, the way 3 males form a group is (3-1)!=2!=2 (ways). Then with 3 females in the 3 positions between them, satisfying the condition, the required arrangement is 2×3!=12 (ways).'
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'0 when n is odd, 0 when n is 1 or even, (17/18)^((n-1)/2)/17 when n is odd and greater than or equal to 3'
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'Out of 40 students who were given two problems I and II, 25 students solved problem I, 32 students solved problem II, and 20 students solved both problems. How many students are there who (1) could not solve problem I and (2) could not solve both problem I and problem II?'
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'Find the quadratic function y that passes through the point (-3, -7) and has a maximum value of 2 at x=0.'
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'There are 10 books lined up in a row on the top shelf of the bookshelf. If the books are moved from top to bottom, from left to right, 1 or 2 books at a time, how many ways are there to do this?'
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'There are 2 cards with the number 1, 3 cards with the number 2, and 4 cards with the number 3. Determine the number of 4-digit integers that can be formed using 4 cards.'
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'Using 4 100 yen coins, 4 50 yen coins, and 7 10 yen coins would be sufficient to pay 420 yen. Note that the total number of coins used should not exceed 15.'
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'Please find the condition for the equation to have two distinct real solutions within the range of , where and the discriminant of is denoted as .'
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'There is a product priced at 100 yen. In store A, a discount of 8% is applied to the product regardless of the quantity purchased. On the other hand, store B sells at full price for the first 10 items, but applies a 15% discount from the 11th item onwards. From how many items purchased, buying from store B is cheaper than store A?'
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'Translate the given text into multiple languages.'
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'Find the total number of permutations of two natural numbers that sum up to 30.'
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'Explain the basic properties of sets.\n1. Subset\n2. Equality\n3. Intersection\n4. Union\n5. Complement'
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'Mathematics I (3) |x|=|y| ⇒ x+y=0 is false. (Counterexample) x=1, y=1, thus, the equivalent condition to x+y=0 is (2) x²+2xy+y²=0 CHECK 17 ⇒ Main book p.77 (1) Converse: x²=1 ⇒ x=1 (false) Contrapositive: x²≠1 ⇒ x≠1 (true) Contrary: x≠1 ⇒ x²≠1 (false) (2) Converse: x>0 ⇒ x²>0 (true) Contrapositive: x≤0 ⇒ x²≤0 Contrary: x²≤0 ⇒ x≤0 Contrary: x²≤0 ⇒ x≤0 ∠|x|=|y| ⇔ x=±y The counterexamples of converse and contrary are x=-1 4x²≤0 so x=0 x=0 satisfies x≤0. Though the truth or falsity of converse, contrapositive, and contrary is not required, they are as described above.'
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'(1) \\ \\ left \\ { \\ begin{array} { l} x+y=10 \\\\ xy=7 \\ end{array} \\ right.'
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'There are 4 red beads, 2 white beads, and 1 blue bead. There are □ ways to arrange all 7 beads in a circular manner.'
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'Determine the conditions for which the number of positive integers x that satisfy the inequality is 5.'
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'How many ways are there for 7 people to sit around a round table?'
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'Find all the subsets of set U={a, b, c, d, e} that contain 3 elements.'
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'\ a+2, c-2 \ are both integers, and since \ a+2>0 \, then \ a+2=3, c-2=-1 \ therefore \ a=c=1 \ which does not satisfy the condition that \ a, c \ are prime numbers. Therefore, \ a, b, c \ satisfy the relationship in \1\, and in this case, \ \\triangle ABC \ is an equilateral triangle.'
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'Also, the length of AB is equal to the length of PQ in Figure [2], so AB = 2 * 3 * sqrt(2) = 6 * sqrt(2)'
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'(1) Hypothesis: Assume there exist natural numbers x, y, z that satisfy x^n+2y^n=4z^n.'
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'(3)\\\\[\\n32123_{(4)}=3 \\cdot 4^{4}+2 \\cdot 4^{3}+1 \\cdot 4^{2}+2 \\cdot 4^{1}+3 \\cdot 4^{0} \\n&=768+128+16+8+3=923 \\n41034_{(5)}=4 \\cdot 5^{4}+1 \\cdot 5^{3}+0 \\cdot 5^{2}+3 \\cdot 5^{1}+4 \\cdot 5^{0} \\n&=2500+125+15+4=2644 \\n\\\\]'
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'(1) \ \\frac{1}{k} \\\n(2) \\( \\frac{(2 p-1)(2 q-1)}{p^{2} q^{2}} \\)'
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'Practice with the following two sets containing integers, A∩B={2,7}.'
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'Therefore, if stones are piled on a mountain other than the one with an equal number of stones, and the number of stones on that mountain is m, then there is a winning strategy when m=0, and a similar question for when m≠0 can be found in question 20=>this book p.489'
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'Practice: Using 4 of the 7 numbers 1, 1, 2, 2, 3, 3, 3, create a 4-digit number. There are a total of [] such 4-digit numbers, and [] of them are less than 2200.'
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'From (1), how many triangles are there that are neither right-angled nor isosceles?'
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'Prove that there does not exist a natural number that is 4 digits long when represented in both decimal and quinary systems.'
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'Multiplying 0.375 by 2 and then multiplying the decimal part by 2 repeatedly yields the result on the right. The obtained integer parts are 0, 1, 1 in order, so the result is 0.011 in base 2.'
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'Consider a game where a single die is rolled 1 or 2 times, and the score is based on the result of the final roll. After seeing the result of the first roll, deciding whether to roll a second time, what is the optimal strategy?'
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'For a natural number n, find the number of positive integer triples (x, y, z) satisfying x+y+z=n.'
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'Since math A is 221, find different sets of 3 numbers that satisfy the condition in which they do not simultaneously contain pairs of (1,4),(1,6),(2,3),(2,6),(3,4) as {1,2,5},{1,3,5},{2,4,5},{3,5,6},{4,5,6}. In this case, k is equal to 10, 15, 40, 90, 120.'
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'(1) (ア) 13 ≡ 4 (mod 9), 4^2 ≡ 16 ≡ 7 (mod 9), 4^3 ≡ 64 ≡ 1 (mod 9), 100 = 3 ・ 33 + 1, so 4^100 ≡ (4^3)^33 ・ 4 ≡ 1^33 ・ 4 ≡ 4 (mod 9), therefore 13^100 ≡ 4^100 ≡ 4 (mod 9)'
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'(1) Find the value of (x, y, z). (±1,0,1), (±1,0,-1), (±1,2,0), (±1,-2,0), (±1,2,1), (±1,-2,-1)'
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'Proof regarding the set of multiples in the set of integers Z'
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'(1) Prove that if n is an integer, and n² is a multiple of 3, then n is also a multiple of 3.\n(2) Prove that √3 is irrational.'
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'Provide the real numbers that satisfy the following propositions: Conditions: (1) There exists a real number x such that x ≥ 2 and x³ ≤ 8. (2) There exist real numbers x, y that satisfy x² + y² < 1 and |x| ≥ 1 or |y| ≥ 1. (3) There exists a positive number x such that for certain real numbers a, b, ax + b > 0 and a ≤ 0 and b ≤ 0.'
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'In this chapter, we learn the basics of sets and logic. Sets were first introduced into mathematics by the German mathematician Cantor (G. Cantor, 1845-1918) toward the end of the 19th century, and now serve as the foundation for almost all mathematics.'
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'If neighboring red balls are grouped together as R and neighboring blue balls are grouped together as B, then the permutation we are looking for is the permutation of red balls, blue balls, and three white balls.'
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'Find the total number of solutions for natural numbers x, y, z that satisfy x + y + z = 30.'
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'Find the set that satisfies the following conditions. The universal set is the set of all real numbers.'
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'Find the maximum and minimum values of x-2y^{2} under the condition x+y=1 and 0 ≤ x ≤ 2.'
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'(1) There are a total of sets of natural numbers that satisfy the inequality . Among them, the maximum value of is .\n(2) There are a total of sets of natural numbers that satisfy the equation . Among them, the maximum value of is .'
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'Show that for an integer n, when divided by 3, the remainder of n² is 0 or 1 if the remainder of n is 0, 1, or 2.'
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'(1) Find the set of natural numbers \\((x, y, z)\\) that satisfy the equation \ x y z = x + y + z \.\n(2) How many sets of natural numbers \\((x, y, z)\\) satisfy the equation \ \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 1 \?'
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'Practice for n to be an integer greater than or equal to 2. Find the number of ways to divide the set {1,2, ⋯, n} into two non-empty sets with no common elements.'
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'Let the universal set be U = {n | n is a single-digit natural number}. For the subsets A = {2, 3, 6, 8, 9} and B = {1, 3, 4, 5, 8} of the universal set U, find the following sets.'
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'Discuss the classification based on the remainder when an integer is divided by a positive integer m.'
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'The volume of a cube with each side doubled becomes 8 times the volume of the original cube, so the dedicated altar is not appropriate for the oracle. Assuming that the side length of the original altar is 1, and the side length of the altar with double volume is x, then x satisfies the equation x^3 = 2 as a positive solution. This positive solution is x = 1.25992…, so it was necessary to create an altar enlarged by a factor of 1.25992… from the original altar.'
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'Convert the given decimal number 0.375 to binary and quinary representations.'
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'Find all positive integers x that cannot be represented as 3m + 5n using any non-negative integers m and n, where x is an integer of the form 97 | ax + by.'
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'Practice (1) Find the range of values for the constant a that satisfies the inequality 2a < x < a+3 when the only integer x that satisfies it is 4.'
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'Example 1 | Maximum and Minimum Number of Elements\nFor a set U and its subsets A, B, with n(U)=100, n(A)=80, n(B)=30.\n(1) Determine the maximum and minimum possible values of n(A ∩ B).\n(2) Determine the maximum and minimum possible values of n(∩ {A} ∩ B).\n[Class Kagoshima University] <Example 1, 2'
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'Find all the integer solutions to the following equations.'
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'(2) Let N be a natural number that satisfies the conditions, and assume N is an n-digit decimal number. That is, 10^(n-1) ≤ N < 10^n, which means (2・5)^(n-1) ≤ N < (2・5)^n. Converting N to binary results in n+3 digits, so 2^(n+2) ≤ N < 2^(n+3). (2) Here (2・5)^n - 2^(n+2) = 2^n (5^n - 2^2) > 0. Thus, (2・5)^n > 2^(n+2), and for N to simultaneously satisfy (1) and (2), it must hold that (2・5)^(n-1) < 2^(n+3), or 5^(n-1) < 2^4 is a necessary condition. Since 2^4 = 16, the values of n that satisfy (3) are n=1,2. For n=1, (1) is 1 ≤ N < 10 and (2) is 8 ≤ N < 16, so the values of N that simultaneously satisfy (1) and (2) are N=8,9. For n=2, (1) is 10 ≤ N < 100 and (2) is 16 ≤ N < 32, so the values of N that simultaneously satisfy (1) and (2) are N=16,17,...,31. Hence, the smallest value of N is 8 and the largest is 31.'
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'When 4=-2, the inequality becomes 0≥0, which is a solution.'
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'1) When throwing 3 dice of different sizes, find the number of ways the product of the outcomes is odd. \n2) How many integers from 1 to 9999 contain two 0s?'
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'By representing apples with 8 ○ and separators with 3 |, the total number of methods is equal to the permutation of 8 ○ and 3 |'
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'There are no natural numbers with 1 or 2 digits that include two zeros. Here, a and b represent any of the numbers from 1 to 9, and they can be equal.'
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'Exercise 42: In triangle ABC, with AB = 2, AC = 3, and BC = x. From the condition for the triangle to exist, we have 3 - 2 < x < 3 + 2. Find the range of x.'
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'Prove the following. Where Z represents the set of all integers.'
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'For 2 x-1 <= 3 with 0 <= x <= 2, each case is of the form a <= x < b, including the left end of the graph and excluding the right end.'
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"Arrange 3 squares, 2 c's, and 2 e's in a row, where the 3 squares can be designated as s, i, n from left to right."
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'Important Example 91 | Proof using congruence (1)'
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'Consider the triplets (x, y, z) that satisfy the following conditions.'
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'When 4x ≠ 3, the inequality is (x-3)^2 > 0, and it is not a solution.'
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'Consider a column in which sequences like 122112 or 212121 do not satisfy the condition. Start from the left of the column and look for the first 2 that appears more times than 1. '
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'Find the integer x that satisfies the given system of inequalities'
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'There are 3 possibilities for odd outcomes, namely 1, 3, 5. The product of the outcomes is odd when the outcomes themselves are odd. Therefore, the number of desired cases is 3 x 3 x 3 = 27 (cases).'
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'Rationalize the following expressions. (1) \ \\frac{12}{\\sqrt{5}} \(2) \ \\frac{4}{3 \\sqrt{8}} \(3) \ \\frac{5}{\\sqrt{5}-\\sqrt{3}} \'
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'Exercise 5 III -> Book p .59 \\[ \\sqrt{9 x^{2}-12 x+4} + \\sqrt{x^{2}+4 x+4} - \\sqrt{16 x^{2}-24 x+9} = \\sqrt{(3 x-2)^{2}} + \\sqrt{(x+2)^{2}} - \\sqrt{(4 x-3)^{2}} \\\\\\[ \\frac{2}{3} < x < \\frac{3}{4} \\text{ when } \\\\\egin{aligned} 3 x - 2 & > 0, x + 2 > 0, 4 x - 3 < 0 \\\\\\text{ therefore } \\\\text{ the given expression }) & = (3 x - 2) + (x + 2) - \\{ - (4 x - 3) \\} \\\\\\ & = 8 x - 3 \\\\end{aligned} \\]'
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'The total number of ways to choose 3 points from 9 points is'
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'Let the vertical length of the original board be x cm, then the horizontal length is 2x cm, and the lengths of the adjacent sides of the rectangular base of the container are (x-5*2) cm and (2x-5*2) cm.'
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'Exercise 17 Find integers that satisfy the equation'
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"Can non-human creatures understand the concept of the number '2'? Explain your reasons."
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'Let 9a be a positive constant. Find the minimum value of a such that the number of positive integers x satisfying the inequality |-2x + 3| ≤ a is 5.'
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'Considering the use of any coins, to pay 260 yen: paying 420 - (100 + 50 + 10) = 260 yen (even if there are extra coins). Assuming x, y, and z are the number of 100 yen, 50 yen, and 10 yen coins used to pay 260 yen, respectively, then x, y, and z are non-negative integers and satisfy 100x + 50y + 10z = 260. Therefore, 10x + 5y + z = 26, and x + y + z ≤ 12.'
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'Translate the given text into multiple languages.'
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'Prove that for integers p and q, p-q is odd if and only if p+q is odd.'
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'When using all 3 colors, one color will need to paint 2 faces. There are 3 ways to choose this color, each with 2 ways to paint 2 faces with that chosen color, and 2 ways to paint the remaining 2 faces with the other 2 colors, which are equivalent when rotated. Hence, the total number of ways to paint with all 3 colors is as follows: when there is a color not used, the total number of ways to paint with 2 colors. There are 3 ways to choose these 2 colors: ① using one color to paint 2 faces and another color to paint the remaining 2 faces, with 1 way. ② using one color to paint 3 faces and another color to paint the remaining 1 face, with 2 ways. Therefore, the total number of ways in this case is 3 × (1 + 2) = 9. When painting with one color, there are 3 ways to choose this color, so even when one color is not used, the total number of ways to paint is 3 + 9 + 3 = 15.'
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'In an experiment of throwing a die, event C: getting an even number, event D: getting a 3, then C={2,4,6}, D={3}, so C∩D=∅, therefore, events C and D are mutually exclusive.'
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'Prove that when selecting 26 distinct integers from 1 to 50, there will always be a pair of numbers with a sum of 51, no matter how you choose.'
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'Practice (3) When there are 4 adults and 3 children, how many ways are there to divide a total of 7 people into 3 rooms A, B, C such that each room has at least 1 adult?'
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'Practice\n(1) Let the quotient of ak, al divided by d be s, t respectively, then the following two equations hold true.'
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'Example 12 | Basics of absolute value, distance between 2 points on a number line'
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'47 (A) 72 √3 (1) 72 √2 (B) √6 (I) 24 π (O) 6 √2 (P) -1/3'
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'Find the first 5 multiples of 7 with the smallest absolute values.'
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'The given sequence of numbers matches the sequence of integers 1,2,3,4,5,... in base 4.'
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'(1) Assuming there exists a natural number N that is four digits in both decimal and base-5 representations. Then, we have 10^3 ≤ N < 10^4, 5^3 ≤ N < 5^4. Hence, 1000 ≤ N < 10000, 125 ≤ N < 625. These two inequalities cannot be simultaneously satisfied, leading to a contradiction. Therefore, a natural number that is four digits in both decimal and base-5 representations does not exist.'
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"Buying 1 piece of candy A for 80 yen, 1 candy B for 100 yen, and 1 candy C for 200 yen totaling 50 pieces. The quantity of A is twice the quantity of C and equals the sum of B's quantity, with at least 1 piece of each candy purchased, and the total amount is less than 5400 yen. In this case, what is the maximum number of candy C that can be purchased?"
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'Permutation of taking 3 from 4 different items (2)'
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'How many integers are there when 0 is included in the 4 sets'
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'(1) Express 23/27 in base-3. (2) Express 11/3 in base-2.'
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'Find the value of x that satisfies the following congruences, expressed as x ≡ a( mod m) in each modulus m [a is a natural number smaller than m].'
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'Given sets A and B, if x is in set A, then x must also be in set B. Express the relationship between sets A and B using symbols.'
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'For D=0, that is (sin θ+1)(2sin θ-1)=0. Since 0° ≤ θ ≤ 180° and 0 ≤ sin θ ≤ 1, sin θ+1 is not equal to 0. Therefore, 2sin θ-1=0, which implies sin θ=1/2. Hence, θ=30° or 150° (alternatively, 150° or 30°).'
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"Let a, b be non-zero integers. Define M as {ax+by | x, y are integers}, and let the smallest positive element of M be d=ax'+by'. Prove the following:\n(1) Every element of M is divisible by d.\n(2) If the greatest common divisor of a and b is g, then g=d.\n(3) Let's denote the set of all multiples of g as N, then N=M."
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'Find all positive integer pairs that satisfy the given equation .'
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'Numbers are not about specific objects or phenomena themselves, but rather abstract representations of objects or phenomena.'
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'First, there are 4 adults, and there are 36 ways to divide them into rooms so that each room has at least one adult. There are 27 ways for each of them to divide 3 children A, B, C into 3 rooms. Therefore, there are 972 ways to divide so that each room has at least one adult.'
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"Please provide the 'page number' that meets the following condition: Page where the mathematical term 'factorial' is explained."
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'(2) Solve the following calculation. \\[ \egin{array}{l} \\text { 2) } \egin{aligned} \\sqrt{42+12 \\sqrt{6}} & =\\sqrt{42+2 \\sqrt{36 \\cdot 6}}=\\sqrt{(36+6)+2 \\sqrt{36 \\cdot 6}} \\\\ & =\\sqrt{36}+\\sqrt{6}=6+\\sqrt{6} \\end{aligned} \\\\ 2<\\sqrt{6}<3 \\text { so } 8<6+\\sqrt{6}<9 \\\\ \\text { Therefore, } a=8, b=(6+\\sqrt{6})-a=\\sqrt{6}-2 \\ \\ \\frac{a}{b(b+4)}=\\frac{8}{(\\sqrt{6}-2)(\\sqrt{6}+2)}=\\frac{8}{6-4}=4 \\end{array} \\]'
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'(イ) 3000 ≡ 4 (mod 14) and 4^2 ≡ 16 ≡ 2 (mod 14), 4^3 ≡ 64 ≡ 8 (mod 14), 4^4 ≡ (4^2)^2 ≡ 2^2 ≡ 4 (mod 14), so the remainder of 4^k (where k is a natural number) repeats in cycles of 4, 2, 8, and particularly 4^3k ≡ 8 (mod 14), hence 4^3000 ≡ 4^3 ・ 1000 ≡ 8 (mod 14) therefore 3000^3000 ≡ 4^3000 ≡ 8 (mod 14)'
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"Let the maximum height of a first grader be M1, the minimum height be m1, and the k-th quartile be Q1k (k=1,2,3). Similarly, for a second grader, let the maximum height be M2, the minimum height be m2, and the k-th quartile be Q2k (k=1,2,3). According to the conditions: (1) M1>185, M2<185, hence, correct. (2) Q12<170, Q22>170, hence, correct. (3) Q21>165, hence, incorrect. (4) Q13>175, hence, incorrect. (5) 185<M1<190,150<m1<155, therefore [185-155<M1-m1<190-150] which means 30<M1-m1<40. 180<M2<185,155<m2<160, therefore [180-160<M2-m2<185-155] which means 20<M2-m2<30. Thus, the range of first grader's data is greater than 30 cm but less than 40 cm. Similarly, the range of second grader's data is greater than 20 cm but less than 30 cm. Therefore, it is incorrect. In conclusion, the correct statements are (1) and (2)."
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'Let the set of all students be U, the set of students with personal computers be A, the set of students with mobile phones be B, and the set of students who have private cars be C. Then, n(U) = 100, n(A) = 75, n(B) = 80, n(A ∩ B) = x, n(C) = 60, n(A ∩ B ∩ C) = y. Let the number of students who only have a personal computer but not a mobile phone be a, the number of students who only have a mobile phone but not a personal computer be b, and the number of students who have neither a personal computer nor a mobile phone be c. The following equations can be derived from the conditions.'
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"Answer the 'page number' that meets the following conditions:"
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'If it is permissible to have vacant rooms, then the number of ways to distribute 4 people among 3 rooms is 3^4=81. If there are 2 vacant rooms, one of the remaining non-vacant rooms can be selected in 3 ways. If there is 1 vacant room, there are 3 choices for the vacant room, and then each of the remaining 2 rooms can have 2^4-2 ways to accommodate 4 people, making a total of 3*(2^4-2)=42 ways. Therefore, the required number of scenarios is 81-(3+42)=36.'
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'Find the natural numbers that leave a remainder of 2 when divided by 5.'
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'Dividing a certain integer by 20 and rounding to the first decimal place results in 17. Find the maximum and minimum integers that satisfy this condition.'
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'Rationalize the denominator and simplify the following expressions.'
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'Select one number from 1 to 5 excluding 0, then select 3 numbers from the remaining 5 numbers, find the total number of permutations.'
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"Please show the negation of the following proposition and verify the truth of the proposition and its negation: Proposition: '22 is a closed number'"
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'Find the pair of natural numbers that satisfy the equation .'
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'Find all triples (x, y, z) that satisfy condition (A) and y ≤ 3.'
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'Can (2) 3|x+1|<x+5 be simplified to the following form?'
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'From the condition a ≡ 2 (mod 7) it follows that a^3 ≡ 2^3 ≡ 8 ≡ 1 (mod 7). Since 2023 = 3 ・ 674 + 1, then a^2023 ≡ (a^3)^674 ・ a ≡ 1^674 ・ a ≡ a ≡ 2 (mod 7). Therefore, the remainder when a^2023 is divided by 7 is 2.'
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'Of these, the routes passing through point E are 3 from A to E and 1 from E to D, so a total of 3 routes.'
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'When arranging red beads next to each other and blue beads next to each other, the case where white beads are not adjacent is when placing white beads between R and B and in the three positions at both ends as defined in (2). This is determined by the arrangement of R and B, so there are 2 possibilities.'
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'There are 5! ways to arrange 2 girls in a group and 4 boys.'
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'Find all pairs of two natural numbers a, b that satisfy the following conditions, where a < b. (1) Sum is 320, greatest common divisor is 16'
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"Consider the set of all integers as the universal set, and consider the proposition P regarding a subset X (where X is not empty): 'There exists a minimum element in the set X'. Choose all options from the following choices A to Z that satisfy this condition."
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'Find the value of x that satisfies the following congruence equations, expressing x in the form of x ≡ a(mod m) for each modulus m, where a is a natural number less than m.'
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'When the integers a, b, c satisfy a^2 + b^2 = c^2, at least one of a, b, c is a multiple of 5.'
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'Distribute 10 indistinguishable 100 yen coins to 3 people. In how many ways can you distribute them so that all 3 people receive at least 100 yen?'
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'For conditions p, q, r, s, the following propositions all hold true.'
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'Given sets A and B, where A contains natural numbers a_k (1 ≤ k ≤ 5), and B contains the squares of a_k, such that a1 < a2 < a3 < a4 < a5. It is known that A ∩ B = {a2, a5} and a2 + a5 = 20. Furthermore, the sum of all elements in the union A ∪ B is 444. Therefore, a1 = ア, a2 = 1, a5 = ウ, and a3 + a4 + a3^2 + a4^2 = エ. Also, the remaining elements are a3 = and a4 = カ.'
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'Please solve problems related to remainders and residue classes.'
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'Example 48 | Properties of Remainders in Division\n(1) Let a and b be integers. When a is divided by 7, the remainder is 2, and when b is divided by 7, the remainder is 5. In this case, \n(A) Find the remainder when 2a+b is divided by 7. \n(B) Find the remainder when a^{2023} is divided by 7. \n(2) For any natural number n, find the remainder when 7^n is divided by 5.'
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'Example 101 Integer solutions to equations (2) Find the pair of integers (x, y) that satisfy the following equations. (1) x^{2}-4 y^{2}=4, x ≥ 0, y ≥ 0 (2) x y-3 x-2 y+3=0 [(2) Soka University]'
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'(2) Find the pairs of natural numbers that satisfy .'
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'How many sets of natural numbers (x, y, z) satisfy the equation 1/x + 1/y + 1/z = 1/2?'
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'Translate the given text into multiple languages.'
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'When rolling two dice, how many ways are there to get a sum of 10 or more?'
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'Determine the value of the positive integer a such that there are 10 integers x satisfying the inequality .'
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'Since A ∩ B = {2,7}, we know 7 ∈ A, therefore a^2 - 9a + 25 = 7 or 2a + 3 = 7\n[1] When a^2 - 9a + 25 = 7\na^2 - 9a + 18 = 0\nHence, (a-3)(a-6) = 0, thus a = 3, 6\n\n When a = 3, B = {-2, -13, -5, 9, 16}\nTherefore A ∩ B ≠ {2,7}, which does not meet the condition. When a = 6, A = {-3, 2, 7, 15}, B = {-2, 2, 7, 12, 16}\nTherefore, A ∩ B = {2, 7} satisfies the condition.\n[2] When 2a + 3 = 7, a = 2\nIn this case, B = {-2, -14, -5, 8, 16}, hence A ∩ B ≠ {2,7}, which does not meet the condition. Therefore, a = 6\nIn this case, A = {-3, 2, 7, 15}, B = {-2, 2, 7, 12, 16}\n(1) A ∪ B = {-3, -2, 2, 7, 12, 15, 16}\n(2) The complement of A ∩ B = {-2, 12, 16}'
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'For a set where all dice have the same number, there is only one way the numbers can turn out. Therefore, there are 6 possible ways for the product to be k, which is the case when there is only one set of three different numbers whose product is k. For example, 1 x 4 = 2 x 2, 1 x 6 = 2 x 3, 2 x 6 = 3 x 4.'
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'Find the number of cases where the product of three numbers is even but not a multiple of 4.'
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'Arrange all the 7 characters of C, O, M, P, U, T, E in alphabetical order to create strings.'
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'By (3)(2), the condition for the solution of (1) to be contained in the solution of (2) is 12 ≤ 4+k, hence k ≥ 8'
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'Assume that the triplet (a, b, c) satisfies condition (A). Prove that there exists an element z such that the triplet (b, c, z) also satisfies condition (A).'
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'Consider the following six conditions:\nFor a positive integer n, consider the following six conditions:\n[Seikei University]\n\nCondition 1: n is even.\n\nCondition 0: n leaves a remainder of 1 when divided by 3.\n\nCondition 2: The square of n is a multiple of 4.\n\nCondition 3: The square of n leaves a remainder of 1 when divided by 6.\n\nCondition 4: n(n+1) is a multiple of 6.\nCondition 5: n(n+2) is a multiple of 12.'
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'[1] If a ∈ A, then a = 2m + 3n (where m, n are integers). In this case, a = 3n + 2m = 3n + (5m - 3m) = 3(n-m) + 5m. Since n-m and m are integers, a ∈ B. Therefore, A ⊂ B\n[2] If b ∈ B, then b = 3m + 5n (where m, n are integers). In this case, b = 3m + 5n = 3m + (2n + 3n) = 2n + 3(m+n). Since n, m+n are integers, b ∈ A. Therefore, B ⊂ A. From [1] and [2], we can conclude that A ⊂ B and B ⊂ A, hence A = B'
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'When there are 3 ways to show hands - rock, paper, scissors - with 1 person, how many combinations are there when 4 people show their hands at once?'
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'(1) Reverse: A multiple of 2 is a multiple of 4.\n(False) Counterexample is 6\nContrapositive: If it is not a multiple of 2, then it is not a multiple of 4.\nA number that is not a multiple of 2 is odd.\nInverse: If it is not a multiple of 4, then it is not a multiple of 2.\n(False) Counterexample is 6'
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"(1) The negation of 'x>0 and y≤0' is 'x≤0 or y>0'\n(2) The negation of 'x≥2 or x<-3' is 'x<2 and x≥-3'\nThat is, -3≤x<2"
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'Please check the maximum and minimum values when finding the number of elements in a set.'
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'There are 2 ways for 2 girls to line up at both ends'
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'Let (a, b, c) be a set of positive integers satisfying the equation a^2 + b^2 = c^2.'
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'Mathematics A - 209 is calculated by determining the probability of winning or losing. The probability of 3 people winning is . The probability of 4 people winning is . Therefore, the desired probability, using the results of (1) and (2), is . An alternative solution is that the outcome is determined when 5 people play 2 types of hands, with a probability of . Therefore, the desired probability is .'
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'The number of ways X appears 3 times, Y appears 1 time, and Z appears 2 times out of 6 button presses is \ \\frac{6!}{3!1!2!}=60 \. Therefore, the required probability is \\( 60 \\times\\left(\\frac{1}{6}\\right)^{3}\\left(\\frac{1}{2}\\right)^{1}\\left(\\frac{1}{3}\\right)^{2}=\\frac{60 \\cdot 1^{6}}{6^{3} \\cdot 2 \\cdot 3^{2}}=\\frac{5}{324} \\)'
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"The inequality f(p) f(q)<0 means f(p) and f(q) have opposite signs. There are two cases: (1) f(p) is positive, f(q) is negative (2) f(p) is negative, f(q) is positive. If you are unsure, using the inequality f(p) f(q)<0 is helpful. On the other hand, for example, if you know it's case (1), it is often easier to consider 'f(p)>0 and f(q)<0' because the degree of the inequality becomes lower. It is important to choose the easier approach depending on the problem."
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'A natural number that has 12 digits when expressed in octal (base-8) is how many digits long when expressed in binary (base-2) and hexadecimal (base-16)?'
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'Express the given proposition in the form of p ⇒ q and denote the set of all x satisfying conditions p, q as P, Q.'
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'There are 3 ways to draw a line segment where only one point on line h is connected, and in this case, the 6 line segments do not intersect. Therefore, combining the results from (3) and (5), the desired way is 729 - (3 + 15 + 10) = 701 (ways)'
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'Consider the figure on the right, where a path is drawn between points X and Y.'
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'(イ) A takes two stones from a mountain with eight stones in the beginning. After that, as long as there are stones in two mountains, A will continue to take the same number of stones as B from the other mountain. In this way, A can definitely take the stones last. Therefore, A has a winning strategy.'
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'Using the identity equation, prove the following:'
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'Let 4n be a natural number greater than or equal to 5. Select n characters from the 4 characters a, b, c, d with duplicates allowed, arrange them in a line, and create a sequence of n characters. It is required that adjacent characters must be different. First, when n=5, i.e., considering a sequence of 5 characters, the number of sequences that start with a and end with a, and include one each of b, c, d is Alpha □, and the number of sequences that start with a and end with a, and include only one b is Beta □. Next, when considering a sequence of n characters, the number of sequences that start with a and do not contain d is Gamma □, the number of sequences that do not contain d but contain at least one each of b, c starting with a is Delta □, and the number of sequences that start with a and include at least one each of b, c, d is Epsilon □□.'
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'Prove that there are infinitely many tuples (x, y, z) that satisfy condition (A).'
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'(1) Let be the universal set. \n\n If sets are defined as , find the following sets: \n(A) \n(1) \n(W) \n(I) \n(T) \n(L) '
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'There was a problem that proved an inequality in the title, which became a topic in a previous university entrance exam. At that time, there was also a discussion on the decline in math skills of elementary school students related to calculating the area and perimeter of a circle.'
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'File exercises (1) The minimum natural number \ n \ that is 2014 or above, in which \ n-2 \ is a multiple of 3 and \ n-3 \ is a multiple of 5, is \ \\square \.\n(2) Let \ n \ be a natural number. Prove that \ 2^{n}+1 \ and \ 2^{n}-1 \ are coprime.'
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'Find all integer solutions of the following equations.'
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'Since 1<3/p, which implies p<3, the integer satisfying 2 ≤ p < 3 is p=2. When p=2, (1) becomes 1/q + 1/r ≥ 1/2. From (2), we get 1/2 ≤ 1/q + 1/r < 1/q + 1/q = 2/q, hence 1/2 < 2/q, i.e., q<4. The integer range for q that satisfies 2<q<4 is q=3. Substituting p=2, q=3 into (1) and simplifying, we obtain 1/r ≥ 1/6, i.e., r ≤ 6. The integer range for r that satisfies 3<r ≤ 6 is r=4,5,6. Therefore, the solutions are (p, q, r)=(2,3,4),(2,3,5),(2,3,6)'
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'Prove that if the sum and product of two natural numbers a and b are coprime, then a and b are also coprime.'
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'Since the size of the data is 6, the median is the average of the 3rd and 4th values. The values other than x when arranged in ascending order are 2, 5, 8, 10, 13. The median of these 5 values is 8. Therefore, the median of the 6 values including x is (5+8)/2 = 6.5, (8+10)/2 = 9, (8+x)/2 (5 ≤ x ≤ 10) will be the median. The median becomes 7 when (8+x)/2 = 7. Hence, x = 6, which satisfies 5 ≤ x ≤ 10.'
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'Express y as a function of t for the range 0 ≤ t ≤ 1: y=2t^2−8t+5=2(t−2)^2−3. At t=0, y has a maximum value of 5, and at t=1, a minimum value of -1. Since 0◦ ≤ θ ≤ 90◦, sinθ=0 corresponds to t=0, hence θ=0◦ corresponds to sinθ=0, and sinθ=1 corresponds to t=1, hence θ=90◦ corresponds to sinθ=1. Therefore, y has a maximum value of 5 at θ=0◦ and a minimum value of -1 at θ=90◦. It follows that y =(1−sin^2θ)−2sinθ−1 = −sin^2θ−2sinθ using cos^2θ =1−sin^2θ. Let sinθ=t, then y=−t^2−2t=−(t+1)^2+1 for the range described in (1), with y having a maximum value of 0 at t=0, and a minimum value of -3 at t=1 for 0◦ ≤ θ ≤ 180◦. Corresponding to t=0 for sinθ=0, we have θ=0◦ or 180◦, and corresponding to t=1 for sinθ=1, we have θ=90◦. Hence, y has a maximum value of 0 at θ=0◦ or 180◦, and a minimum value of -3 at θ=90◦.'
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'There are 2^7 = 128 ways to divide 7 people into groups A and B. Out of these, there are 2 ways to divide them into either group A or group B. Furthermore, considering that there is no distinction between groups A and B, the required number of ways to divide is (128-2) / 2 = 63.'
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'The condition for at least one of (1) or (2) to have no real solutions is or .'
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'Let N = 11 * 14^n + 1.\n[1] When n is even, 14 ≡ -1 (mod 3), so ...\nTherefore, 11 * 14^n + 1 is a multiple of 3 when n is even, and a multiple of 5 when n is odd.'
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'In mathematics A-181, if A and B can be empty sets, then each of 1, 2, 3,..., n belongs to either A or B, resulting in 2^n possible ways of partitioning. Excluding the scenario where all elements belong to either A or B, there are 2^n-2 ways. Removing the distinction between A and B, we are left with (2^n-2)/2=2^(n-1)-1 ways.'
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'Choose the most suitable term for the blank space from (A) to (I).'
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TRAINING 60 Let be a real number \} be the universal set. For the subsets of A=\left\{2,4, a^{2}+1
ight\} , B=\left\{4, a+7, a^{2}-4a+5
ight\} , if A \cap \overline{B}=\{2,5\} , find the value of the constant . [Toyama Prefecture University]
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What do you call the combination of integers and numbers represented as finite or infinite decimals?
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Basic Example 55
Let n be an integer, and proposition A be defined as 'n being a multiple of 4 ⟹ n being a multiple of 8'. (1) State the converse and contrapositive of proposition A and check their truth values. (2) State the inverse of proposition A.
The converse, contrapositive, and inverse of proposition p ⟹ q
(1) The converse of proposition p ⟹ q is q ⟹ p. Additionally, by forming the negations ar{p}, ar{q}, the contrapositive of proposition p ⟹ q is
(2) The inverse of proposition p ⟹ q is
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How should we determine whether something is true or false? Here, let's learn the way of thinking necessary for making that judgment.
Propositions and Conditions
In general, a sentence or expression whose truth or falsehood is clearly determined is called a proposition. When a proposition is true, we say the proposition is true, and when it is false, we say the proposition is false.
For example, 'Japan is large' is not a proposition. (Reason: Some people may think it's large, others may think it's small, so its truth or falsehood cannot be clearly determined.) 'If two lines are parallel, then the corresponding angles are equal' is a proposition and is true. Expressions containing a variable x, such as x=1 or x^2=1, which can determine truth by assigning a value to x, are called conditions related to x. When considering conditions, it is necessary to clarify which set the variable belongs to. This set is called the universal set of the condition.
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(1) Let the set of all natural numbers less than 10 be the universal set , and let be subsets of with and . Find the following sets: (a)
(b)
(c) \overline{A}
(d) \overline{A} \cap B
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If set represents 'the set of all positive divisors of 20', enumerate the elements of set .
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(1) Let the set of all positive integers not greater than 10 be the universal set , and let be subsets of defined as . Find the following sets.
(a)
(b)
(c) \overline{A}
(d) A \cap \overline{B}
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Prove the following proposition.
(1) If is a multiple of 3, then is a multiple of 3.
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Let be the set of all positive divisors of 24. Fill in the appropriate symbol or in the following .
(ア) 6 A
(イ) 9 A
(ウ) -2 A
(2) Express the relationship between the following two sets and using the symbol or .
(ア) is a natural number less than or equal to 5
(个) \( A=\{5 n \mid n=1,2\}, \quad B=\{x \mid(x-5)(x-10)=0\} \)
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Consider the set of all elements that satisfy the condition 'negation of p or q'.
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Determine the truth value of the following propositions P. Also, state the negation of proposition P and determine its truth value. (1) P: “For all integers x, x^2 > 0.” (2) P: “There exists a prime number x such that x is even.”
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Let be the set of all one-digit positive even numbers. If , then .
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When the range of the function is , state its domain.
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Find the smallest natural number that satisfies the inequality \( \frac{n+1}{7}+n \leqq \frac{3(n-1)}{2} \).
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Find the range of the constant when the quadratic equation has two distinct real roots.
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For the set , find the complement \overline{A} of the subset .
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The number of integers that satisfy the inequality is .
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For example, when dividing 12 candies among 3 people, A, B, and C, at least one person will receive 4 or more. If this proposition is proven using the method of contradiction, it would be as follows.
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Supplement 1. A concrete example of De Morgan's laws
Let the universal set U be the set of natural numbers from 1 to 9, and let the subsets of U be .
In this case, , thus
Moreover, \overline{A}=\{2,4,5,6,7,8\}, \overline{B}=\{1,2,4,5,7,8\}
thus \overline{A} \cup \overline{B}=\{1,2,4,5,6,7,8\}
and , thus
Furthermore, \overline{A}=\{2,4,5,6,7,8\}, \overline{B}=\{1,2,4,5,7,8\} , thus
\overline{A} \cap \overline{B}=\{2,4,5,7,8\}
Indeed, \overline{A \cap B}=\overline{A} \cup \overline{B}, \overline{A \cup B}=\overline{A} \cap \overline{B} holds true.
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Express 'The sum of two numbers x and y is positive and not more than 6' as an inequality.
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A school has decided to create a pamphlet for the school festival. The printing costs are 4000 yen for up to 100 copies, and it costs 27 yen per copy for each additional copy beyond 100. To keep the printing cost per copy at 30 yen or less, how many copies should be printed at a minimum? Note that this calculation does not include consumption tax.
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(1) Represent the recurring decimals as fractions respectively. (2) (a) rac{5}{37} , (b) rac{1}{26} when expressed as decimals, find the digit at the 200th decimal place.
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Let A be the set of all rational numbers, then . Choose one suitable symbol from to fill in .
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Consider the set of all elements that satisfy the condition 'not p or not q'. How can this set be represented?
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Given 2 < x < 5 and -1 < y < 3, find the range of values for the following expressions.
(1) x-5
(2) 3y
(3) x+y
(4) x-2y
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Let A={n | n is a positive divisor of 12}, B={n | n is a positive divisor of 18}, C={n | n is a natural number less than or equal to 7}. Find the following sets:
(1) A ∪ B ∪ C
(2) A ∩ B ∩ C
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Which of the following has a range for its domain?
(1)
(2)
(3)
(4)
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Let m and n be integers. Prove the following propositions using contrapositive.
(1) If n^2 + 4n + 3 is a multiple of 4, then n is odd.
(2) If mn is even, then at least one of m or n is even.
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Let the set of all real numbers be the universal set, and let its subsets and be
is a real number is a real number . Find the sets and .
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What do you call set A when all elements of set A are also elements of set B?
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Please explain the range of numbers and the four basic arithmetic operations.
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Find the sum and difference of the complex numbers and .
Plot them on the complex plane.
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Let the set of natural numbers less than 5 be the universal set , and let the subsets of be and . Find the sets and \overline{A} respectively.
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Find all integer values of that satisfy the system of inequalities \( \left\{egin{array}{l}2 x-1<3(x+1) \ x-4 \leqq-2x+3\end{array}\right\} \).
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Question: Find the intersection and the union of the sets and .
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Regarding example (2) above, find the sets \overline{A} and \overline{A} \cap B .
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For the following sets A, B, and C, find A ∩ B ∩ C and A ∪ B ∪ C. A={1,3,4,5,7}, B={1,3,5,9}, C={2,3,5,7}
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Find the values of the natural numbers that satisfy the equation \( (i-\sqrt{3})^{m}=(1+i)^{n} \) with being the smallest.
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In the complex plane, given two points A (α) and B (β) which are different from the origin O, where 3α² - 6αβ + 4β² = 0 holds. Let C be the circle passing through O, A, and B.
(1) Express α/β in polar form, with the argument θ in the range -π < θ ≤ π.
(2) Express the center and radius of the circle C using α.
(3) Express |3α - 2β| in terms of β.
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Let k be a constant. Find the number of intersection points between the ellipse 4x^2 + y^2 = 4 and the line y = -x + k.
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Do the following quadratic curve and line have common points? If so, state whether they are intersection points or tangency points, and find the coordinates of those points. (2) and
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In the plane, there is a triangle , with a circumcircle radius of 1 and circumcenter O. When this triangle satisfies 4 \overrightarrow{\mathrm{OA}}+4 \overrightarrow{\mathrm{OB}}+\overrightarrow{\mathrm{OC}}=\overrightarrow{0}, the value of the dot product is A, and the area of is B times the area of .
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Math C
The value of \frac{eta}{\alpha} when considered on the complex plane, can be determined as follows. On the complex plane, the points represented by \alpha and eta are A and B respectively. According to the conditions
\mathrm{OA}=\mathrm{AB}=1, \quad \mathrm{OB}=\sqrt{2}\n\nTherefore, \triangle \mathrm{OAB} is a right isosceles triangle with \angle \mathrm{A} as the right angle, as shown in the right diagram.\n \frac{eta}{\alpha}=\frac{eta-0}{\alpha-0} has a positive imaginary part, so point B is the point obtained by rotating point A by \frac{\pi}{4} around the origin O and then scaling the distance from point O by \sqrt{2}.\nTherefore, eta=\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \alpha\nThat is, \frac{eta}{\alpha}=\sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} i\right)=1+i\n(2) (1) Hence eta=(1+i) \alpha
egin{aligned}\n\|\alpha+eta\| & =\|\alpha+(1+i) \alpha\| \\
& =|2+i||\alpha| \\
& =\sqrt{2^{2}+1^{2}} \cdot 1=\sqrt{5}\n\end{aligned}\n\negin{array}{c}\n-\frac{eta}{\alpha}=1+i\ \text{ so }\neta=(1+i) \alpha\end{array}\
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For the complex number , the number of natural numbers such that holds true, and satisfies , is .
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122 minimum value 6, coordinates of point R (3/√2, √2)
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The minimum value of 59 t=\frac{1}{5} is \frac{\sqrt{345}}{5}
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68 β=3+5i, γ=-3+5i, δ=-3-5i The pairs that are conjugates are α and β, γ and δ
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Point (-√6-√2 i) z represents the point z after what movement. Assume the range of the rotation angle θ is -π<θ≤π.
Find the complex number w when the point z=2√2+√2 i is rotated around the origin by -π/4.
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Answers for EXERCISES
11 Omitted
12 (a) 3 (b) 5 (c) (I)
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Simplify \( \left(\frac{1+\sqrt{3}i}{2}\right)^{n} + \left(\frac{1-\sqrt{3}i}{2}\right)^{n} \) when is a non-negative integer.
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Let lpha, eta be complex numbers. Given that |lpha| = |eta| = |lpha - eta| = 2 , find the value of |lpha + eta| .
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75 (1) Rotate around the origin by -5/6π and then scale the distance from the origin by 2√2 (2) 3-i
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Let point α be rotated by π/3 around the origin to become point β. If β = 2 + 2i, find the complex number representing point α.
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Find the dot product of vectors BA and BC, and the angle θ of ∠ABC for the triangle with vertices A(4,3,-3), B(3,1,0), and C(5,-2,1).
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Plot the points representing the following complex numbers on the complex plane.
(a)
(b)
(c) -2
(d) 1
(e)
(f)
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Let z be a non-zero complex number. If z + rac{1}{z} is a real number, then must be a real number or .
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In a square on the complex plane, if one pair of adjacent vertices are 0 and , find the complex numbers representing the other two vertices.
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52 \overrightarrow{\mathrm{OS}}=\frac{1}{2} \vec{a}+\frac{1}{4} \vec{b}+\frac{1}{4} \vec{c}
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TRAINING 11 (2) Given \( ec{a}=(2,3), ec{b}=(-2,2), ec{c}=(5,5) \), find the values of the real numbers and that satisfy ec{c}=x ec{a}+y ec{b} .