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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 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 privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>13.238. The shooter at the shooti...
[ { "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>Assuming that the roots of $x^3 + p \\cdot x^2 + q \\cdot x + r = 0$ a...
[ { "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>Example 3 Fill the numbers $1 \\sim 8$ into the 8 squares surro...
[ { "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>Example 3.15. Find the partial derivatives of the function\n\n$$\nf(x,...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>Example 3.15. Find the partial derivatives of the function\n\n$$\nf(x,...
[ { "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>110. Find the general solution of the equation $y^{\\prime \\prime}=4 ...
[ { "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>110. Find the general solution of...
[ { "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>Example 4 Given $\\cdots 1 \\leqslant a \\leqslant 1$, the inequality ...
[ { "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 4 Given $\\cdots 1 \\leqs...
[ { "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>3A. Given the set of quadratic fu...
[ { "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>3A. Given the set of quadratic functions $f(x)=a x^{2}+b x+c$, ...
[ { "content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>16. In an $\\mathrm{m}$-row $\\mathrm{n}$-column grid, it is st...
[ { "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>[b]p1.[/b] You can do either of two operations to a number written on ...
[ { "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. Judge from the problem and candidate only.</instruction><current_completion>A rectangular cuboid has the sum of the three edges starting from one ...
[ { "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. In the Cartesian coordinate system $x O y$, the ellipse $C$: $\\fra...
[ { "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. In the Cartesian coordinate sy...
[ { "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. Given the function $f(x)=\\fra...
[ { "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>3. Given the function $f(x)=\\frac{2+x}{1+x}$. Let\n$$\n\\begin...
[ { "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>12. The teacher is buying souvenirs for the students. There are...
[ { "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. 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 privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>Three vertices of parallelogram $...
[ { "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>In an equilateral triangle $A B C$ with a side length of $8 \\m...
[ { "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>Three generous friends, each with some money, redistribute the money a...
[ { "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>Three generous friends, each 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. 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>Solve the following system of equ...
[ { "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>5. In the sequence $\\left\\{a_{n}\\right\\}$, $a_{1}=2, a_{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>5. In the sequence $\\left\\{a_{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>From a barrel, we pour out 4 lite...
[ { "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>From a barrel, we pour out 4 lite...
[ { "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 area of a rectangle is 180...
[ { "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 area of a rectangle is 180...
[ { "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>160. Solve the equation in integers\n\n$$\nx^{2}+x=y^{4}+y^{3}+...
[ { "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>160. Solve the equation in integers\n\n$$\nx^{2}+x=y^{4}+y^{3}+...
[ { "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>6. For natural numbers $a, b, c$, define the operation * such that it ...
[ { "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. For natural numbers $a, b, 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>7. In a free-throw test, a person...
[ { "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 \\cdot 103$ real numbers $\\al...
[ { "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>$4 \\cdot 103$ real numbers $\\alpha, \\beta$ satisfy the syste...
[ { "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>20. Ruth and Sarah decide to have...
[ { "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>20. Ruth and Sarah decide to have...
[ { "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>## Task 1 - 220831\n\nOn a day in 1981, Cathrin asks her grandf...
[ { "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>## Task A-1.5.\n\nHow many four-digit numbers divisible by 3 do not 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>Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} ...
[ { "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. Using this picture we can observe that\n$$\n1+3+5+7=4 \\time...
[ { "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. Using this picture we can observe that\n$$\n1+3+5+7=4 \\time...
[ { "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 [i]normal magic square[/i] of order $n$ is an arrangement 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>A [i]normal magic square[/i] of o...
[ { "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-4.3. (20 points) What are the minimum and maximum of the functi...
[ { "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-4.3. (20 points) What are the minimum and maximum of 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. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>(9 Given the function $f(x)=\\cos...
[ { "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>$\\frac{1}{4}$ of 100 is equal to\n(A) $20 \\%$ of 200\n(B) $10...
[ { "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>$\\frac{1}{4}$ of 100 is equal to...
[ { "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>## 162. Math Puzzle $11 / 78$\n\n...
[ { "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>## 162. Math Puzzle $11 / 78$\n\nAssume that a fly lays 120 eggs at 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.</instruction><current_completion>3. Let $a_{1}, a_{2}, \\cdots, a_{n}$ represent any permutation...
[ { "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>3. Let $a_{1}, a_{2}, \\cdots, a_{n}$ represent any permutation...
[ { "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>Given an integer $\\mathrm{n} \\g...
[ { "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. In the regular triangular frustum $A^{\\prime} B^{\\prime} C^{\\pri...
[ { "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. In the regular triangular frustum $A^{\\prime} B^{\\prime} C^{\\pri...
[ { "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>3.20. Calculate\n\n$$\n\\int_{1}^{i} \\frac{\\ln ^{2} z}{z} d z...
[ { "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. Given $a, b, c \\in [0,1]$. Th...
[ { "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. Given $a, b, c \\in [0,1]$. Th...
[ { "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>17. What is the value of the sum:...
[ { "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. As shown in the figure, in the right triangle $A B C$, $\\an...
[ { "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>9. As shown in the figure, in the right triangle $A B C$, $\\angle A 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.</instruction><current_completion>## Task 4 - 240624\n\nRita is experimenting with a balance scal...
[ { "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>2. (a) In Figure 1, trapezoid $A B E D$ is formed by joining $\\triang...
[ { "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>2. (a) In Figure 1, trapezoid $A B E D$ is formed by joining $\\triang...
[ { "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>Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are root...
[ { "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>Natural numbers $k, l,p$ and $q$ ...
[ { "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. [4] A hundred friends, includi...
[ { "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>10.243. Inside an angle of $60^{\...
[ { "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>10.243. Inside an angle of $60^{\...
[ { "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. [7 points] Let $M$ be a figure...
[ { "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 privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>3. [7 points] Let $M$ be a figure...
[ { "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>An airplane traces out a regular $n$-sided polygon, and does so at spe...