Datasets:
juppy44
/
verifier-rm-data

Dataset card Data Studio Files
xet
 Community
1
messages
listlengths
2
2
[ { "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>Find values of the parameter $u$ ...
[ { "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>Task 3. Find the smallest possible value of\n\n$$\nx y+y z+z x+\\frac{...
[ { "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.</instruction><current_completion>16. If a positive integer cannot be written as the difference o...
[ { "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>16. If a positive integer cannot ...
[ { "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. Given a convex quadrilateral $...
[ { "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 a convex quadrilateral $...
[ { "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. It is known that for some posi...
[ { "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. It is known that for some positive coprime numbers $m$ and $n$, 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>12. The maximum value of the function $S=\\sqrt{x^{4}-5 x^{2}-8 x+25}-...
[ { "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>Problem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle 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.</instruction><current_completion>Problem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle 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>7.2. Find the GCD of all numbers ...
[ { "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>7.2. Find the GCD of all numbers obtained by all possible permu...
[ { "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>Zaslavsky A.A.\n\nTwo ants each crawled along their own closed path on...
[ { "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>Zaslavsky A.A.\n\nTwo ants each c...
[ { "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>5. Let $a_{0}, a_{1}, a_{2}, \\ldots$ be integers such that $a_{0}=19,...
[ { "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>B. As shown in Figure 2, in the square $A B C D$ with side leng...
[ { "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>14. Let $k$ be a positive integer, such that for any positive 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. Judge from the problem and candidate only.</instruction><current_completion>For which real numbers $x$ is it true that\n\n$$\n\\left\\{\\frac{1}{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>Find consecutive integers bounding the expression \\[\\frac{1}{...
[ { "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>Find consecutive integers bounding the expression \\[\\frac{1}{x_1 + 1...
[ { "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>Find consecutive integers bounding the expression \\[\\frac{1}{x_1 + 1...
[ { "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>Find consecutive integers boundin...
[ { "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-2.1. (20 points) For complex numbers $z, w$ such that $|...
[ { "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. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)\nIf ...
[ { "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 $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)\nIf ...
[ { "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>## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\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 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>Example 1-21 Laying a $1 \\times ...
[ { "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>5. Given a sphere with a radius of 6. Then the maximum volume of a reg...
[ { "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>5. Given a sphere with a radius of 6. Then the maximum volume of a reg...
[ { "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>## Task 1 - 191231\n\nLet $n$ and...
[ { "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 - 191231\n\nLet $n$ and $m$ be natural numbers 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>Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ 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>Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ po...
[ { "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 rectangle can be divided into $n$ equal squares. The same rectangle ...
[ { "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 rectangle can be divided into $...
[ { "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>11. Given $\\frac{x y}{x+y}=2, \\...
[ { "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 $\\frac{x y}{x+y}=2, \\frac{x z}{x+z}=3, \\frac{y z}{...
[ { "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 $\\frac{x y}{x+y}=2, \\frac{x z}{x+z}=3, \\frac{y z}{...
[ { "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>8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2...
[ { "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>8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ...
[ { "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>8. Let $a>0, b>0$. Then the inequality that does not always hol...
[ { "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 5: On the ground, there are 10 birds pecking, and among any 5 ...
[ { "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 5: On the ground, there are 10 birds pecking, and among any 5 ...
[ { "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 5 - 211235\n\n37 cards, each of which is colored red on...
[ { "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 5 - 211235\n\n37 cards, e...
[ { "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>9. In the diagram, $P Q=P R=Q S$ and $\\angle Q P R=20^{\\circ}$. What...
[ { "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>Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers suc...
[ { "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>20.26 In the attached figure, $T P$ and $T^{\\prime} Q$ are two parall...
[ { "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>20.26 In the attached figure, $T P$ and $T^{\\prime} Q$ are two parall...
[ { "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>[ [ equations in integers ]\n\nSolve the equation $12 a+11 b=2002$ in ...
[ { "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>Exercise 8. The price (in euros) of a diamond corresponds to 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>Exercise 8. The price (in euros) ...
[ { "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>32. How many five-digit numbers are there that read the same from left...
[ { "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>32. How many five-digit numbers are there that read the same from left...
[ { "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>Leibniz Gottfried Wilhelm\n\nConsider the numerical triangle:\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.</instruction><current_completion>For some positive integer $n$, a coin will be flipped $n$ times ...
[ { "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>For some positive integer $n$, a coin will be flipped $n$ times to obta...
[ { "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>24. Find all positive functions defined on $(0,+\\infty)$ that ...
[ { "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>Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC ...
[ { "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>Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \\equal...
[ { "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. Divide a circle into a group o...
[ { "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>Tomongho A.K.\n\nFind all increas...
[ { "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. Given that the three vertices $A, B, C$ of the right triangl...
[ { "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\nWrite 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.</instruction><current_completion>## Problem Statement\n\nWrite the equation of the plane passing...
[ { "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 cuboid has an integer volume. Three of the faces have different area...
[ { "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>11. The solution set of the equat...
[ { "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...