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juppy44
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verifier-rm-data

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[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>## Task B-1.3.\n\nA ship travelin...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mat...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Prove that number $1$ can be repr...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>4. Given the three sides of an obtuse triangle are 3, $4, x$, then the...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>A right-angled triangle has side lengths that are integers. What could...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>A right-angled triangle has side lengths that are integers. Wha...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>11. Given that the internal angles $A, B, C$ of $\\triangle ABC...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>11. Given that the internal angles $A, B, C$ of $\\triangle ABC...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros betwe...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Task B-4.2. Let $n$ be the number obtained by writing 2013 zero...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>6. As shown in Figure 2, let $P$ ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>8.59 For each pair of real number...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>8.59 For each pair of real number...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>9.6. Find the minimum value of th...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\co...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\co...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\fr...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\fr...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Example 6 The rules of a \"level-up game\" stipulate: On the $n...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>2. (9th Canadian Mathematical Competition) $N$ is an integer, i...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Problem 6. (8 points) In the plane, there is a non-closed, non-self-in...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Problem 6. (8 points) In the plan...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural number...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural number...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Example 1 (Question from the 13th...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Four, (50 points) In an $n \\times n$ grid, fill each cell with...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Four, (50 points) In an $n \\times n$ grid, fill each cell with...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>1. If the set\n$$\nA=\\{1,2, \\cd...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Augusto has a wire that is $10 \\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>29. Choose any three numbers from $a, b, c, d, e$ and find their sum, ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>29. Choose any three numbers from $a, b, c, d, e$ and find their sum, ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Problem 3. In the school, there are 50 teachers, of whom 29 drink coff...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>12.180. A side of the triangle is equal to 15, the sum of the other tw...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>3. Let $AB$ be a chord of the unit circle $\\odot O$. If the area of $...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>3. Let $AB$ be a chord of the uni...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>On a board, the numbers from 1 to 2009 are written. A couple of...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>4. The number of real solutions t...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>4. The number of real solutions t...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>## Problem Statement\n\nCalculate the limit of the numerical sequence:...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>## Problem Statement\n\nCalculate...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>6. Let $[x]$ denote the greatest ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>40. The sum of three consecutive ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>40. The sum of three consecutive odd numbers is equal to the fourth po...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Two is $10 \\%$ of $x$ and $20 \\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>1. (6 points) Today is January 31...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Consider a regular hexagon with an incircle. What is the ratio ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Consider a regular hexagon with an incircle. What is the ratio ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Let $ABC$ be a triangle with centroid $G$. Determine, with proo...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>1. Four points $A, B, C, D$ in space are pairwise 1 unit apart,...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>The function $f$ maps the set of positive integers into itself,...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>1. A line $l$ intersects a hyperbola $c$, then the maximum numb...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>1. A line $l$ intersects a hyperbola $c$, then the maximum number of i...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>1. Let the universal set be the s...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>1. Let the universal set be the set of real numbers. If $A=\\{x \\mid ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>7.242. $\\left(16 \\cdot 5^{2 x-1...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Example 1 In $\\triangle ABC$, it...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Example 1 In $\\triangle ABC$, it is known that $x \\sin A + y ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>A father wants to divide his property among his children: first...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>A father wants to divide his property among his children: first, he gi...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Example 11 Let $x>0, y>0, \\sqrt{...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>9. Given is a regular tetrahedron of volume 1 . We obtain a sec...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu...