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"content": "# 5.1. Condition:\n\nPolina came to the cafeteria and saw that 2 puff pastries cost 3 times more than 2 pies. Polina didn't have enough money for 2 puff pastries, but she did have enough for 1 pie and 1 puff pastry. After the purchase, she wondered how many times more money she spent buying 1 puff... | 5C |
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"content": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{1 / \\sqrt{2}} \\frac{d x}{\\left(1-x^{2}\\right) \\sqrt{1-x^{2}}}\n$$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by... | 5C |
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"content": "## Task 1.\n\nLet $a, b, c, d$ be real numbers such that\n\n$$\n\\begin{gathered}\na+b+c+d=10 \\\\\nab+bc+cd+da=25 \\\\\nabc+bcd+cda+dab=50\n\\end{gathered}\n$$\n\nProve that then $a^{2}+b^{2}+c^{2}+d^{2}=30$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question makin... | 5C |
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"content": "## Task 11/88\n\nDetermine all natural numbers $n$ for which $n^{4}+4$ is a prime number!",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n### Und... | 5C |
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"content": "## Task 3 - 300513\n\nFritz, Hans, and Petra have collected a bag full of shells on the Baltic Sea beach. They do not know how many shells they have in the bag.\n\nFritz says: \"If you take 12 shells out of the bag seven times in a row, more than 6 shells will remain.\"\n\nHans says: \"But if you ... | 5C |
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"content": "## Task 4 - 100934\n\nWe are looking for all ordered triples of real numbers $(x, y, z)$ that are solutions to the system of equations:\n(1) $x+y=2$\n(2) $x y-z^{2}=1$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve ... | 5C |
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"content": "## Task 6 - 070836\n\nA two-digit natural number is sought with the following properties:\n\n1) The difference between its digits is three.\n2) If its digits are swapped, the new number is nine less than twice the original number.\n\nBegin your response with \"Rewritten Question: \" and by rewriti... | 5C |
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"content": "## Task B-3.1.\n\nDetermine the value of the expression $A=1+\\operatorname{tg} 35^{\\circ}+\\operatorname{tg} 10^{\\circ}+\\operatorname{tg} 35^{\\circ} \\cdot \\operatorname{tg} 10^{\\circ}$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain onl... | 5C |
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"content": "$1,1,2,3,5,8,13, ?, 34, \\cdots \\ldots$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</a... | 5C |
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"content": "$16(x-1)-(x-1)^{2}+(x-1)^{3}-(x-1)^{4}+(x-1)^{5}$ The coefficient of the $x^{2}$ term in the expanded form is $\\qquad$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your r... | 5C |
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"content": "$18t = 18000kg$． (Judge True or False)",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and going really deep into detail about how I should solve this.\n\nStep 1: Understand the given equation.\nThe equation pr... | 5C |
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"content": "$2 \\cdot 131$ The number of roots of the equation $\\sqrt{x^{2}+12 x+36}+x+6=0$ is\n(A) 0.\n(B) 2.\n(C) 4.\n(D) Infinitely many.\n(3rd \"Five Sheep Cup\" Junior High School Mathematics Competition, 1991)\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it ... | 5C |
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"content": "$7.30 \\lg (\\lg x)+\\lg \\left(\\lg x^{3}-2\\right)=0$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.\n\n$7.30 \\lg (\\lg x)+\\lg \\left(\\lg x^{3}-2\\right)=0$.\n\nBegin your response with \"Rewritt... | 5C |
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"content": "0.35 \\cdot 160+0.1 x=0.2 \\cdot 160+0.2 x, 0.15 \\cdot 160=0.1 x, x=240 \\text {. }\n$$\n\nIn total, it results in $160+240=400$ g of solution.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by ... | 5C |
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"content": "1. For every pair of real numbers $x, y$, the function $f$ satisfies the equation\n$$\nf(x+y)-f(x)-f(y)=1+xy,\n$$\n\nand $f(1)=1$. Then, the number of integers $n$ for which $f(n)=n (n \\neq 1)$ is ( ) .\n(A) 0\n(B) 1\n(C) 2\n(L) 3\n\nBegin your response with \"Rewritten Question: \" and by rewrit... | 5C |
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"content": "1. Given real numbers $a$, $b$, $x$, $y$ satisfy $x y=2008^{2008}$,\n$$\n\\frac{1}{1+2008^{a} x}+\\frac{1}{1+2008^{b-2009} y}=1 \\text {. }\n$$\n\nThen the value of $2008^{a+b}$ is\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is nee... | 5C |
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"content": "1. Given that the total surface area of a rectangular prism is 11, and the sum of its 12 edges is 24, find the length of the diagonal of the rectangular prism.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then... | 5C |
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"content": "1. Pleasant Goat, Pretty Goat, and Warm Goat represent the Sheep Village in a mental relay race, with each match having 100 questions. In the first match, Pleasant Goat plays, and his accuracy rate is $91 \\%$; in the second match, Pretty Goat plays, and she answers 85 questions correctly; in the ... | 5C |
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