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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.</instruction><current_completion>$9 \\cdot 6$ For $a>b>c>0, \\quad m>n>0(m, n$ are integers), wh... |
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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.</instruction><current_completion>$9 \\cdot 6$ For $a>b>c>0, \\quad m>n>0(m, n$ are integers), wh... |
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{
"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>376. Given a triangle $A B C$, the angles of which are $\\alpha, \\bet... |
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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.</instruction><current_completion>1. One car covers a distance of 120 km 18 minutes faster than t... |
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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>1. One car covers a distance of 1... |
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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.</instruction><current_completion>1. What are the last two digits of $1 \\times 2 \\times 3 \\tim... |
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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... |
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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.</instruction><current_completion>Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and ma... |
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{
"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>Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make firs... |
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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... |
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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... |
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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... |
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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.</instruction><current_completion>In the figure shown below, $ABCDE$ is a regular pentagon and $A... |
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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>Let there be a figure with $9$ di... |
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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... |
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{
"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>669. For which numbers $x$ does the equation $x+|x|=0$ hold?</current_... |
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{
"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>669. For which numbers $x$ does the equation $x+|x|=0$ hold?</current_... |
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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... |
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{
"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>14.18 Indicate all points $(x, y)$ in the $(x, y)$ plane that satisfy ... |
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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.</instruction><current_completion>Example 12 Given $a, b, c, d \\in \\mathbf{R}^{+}$, try to find... |
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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... |
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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>Example 4 Find the integer soluti... |
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{
"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 Find the integer solution of the equation $\\left[\\frac{x}{... |
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{
"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>321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.\n\nTranslate the text above into E... |
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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... |
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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... |
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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.</instruction><current_completion>13. (20 points) As shown in Figure 2, planes $M$ and $N$ inters... |
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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>## Problem Statement\n\nFind the ... |
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{
"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\nFind the coordinates of point $A$, which is eq... |
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{
"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 2. Find the number of the subsets $B$ of the set $\\{1,2, \\ld... |
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{
"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 2. Find the number of the subsets $B$ of the set $\\{1,2, \\ld... |
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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>2、D Teacher has five vases, these... |
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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>2、D Teacher has five vases, these... |
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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>11. Choose three different digits... |
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{
"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>11. Choose three different digits from $0,1, \\cdots, 9$ to form a fou... |
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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... |
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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... |
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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... |
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{
"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. Function $f$ is defined on the set of integers, satisfying\n$$\nf(n... |
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{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_... |
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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 privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>2. A helicopter and an airplane f... |
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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... |
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{
"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 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+... |
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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.</instruction><current_completion>Problem 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{... |
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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>3. The monotonic increasing inter... |
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{
"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. The monotonic increasing interval of the function $f(x)$ $=\\frac{(... |
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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.</instruction><current_completion>1. Without using calculators or other computational tools, dete... |
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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.</instruction><current_completion>1. Without using calculators or other computational tools, dete... |
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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>6. (10 points) As shown in the fi... |
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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>6. (10 points) As shown in the fi... |
[
{
"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 the set $\\{1,2,3,5,8,13,21... |
[
{
"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 the set $\\{1,2,3,5,8,13,21,34,55\\}$, how many integers between... |
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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.</instruction><current_completion>9. Given that $M N$ is a moving chord of the circumcircle of eq... |
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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... |
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{
"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. Determine all pairs $m>n$ of positive integers such that\n\n$$\n1=\... |
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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... |
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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.</instruction><current_completion>For example, in the sequence $x_{1}, x_{2}, \\cdots, x_{n}, \\c... |
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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.</instruction><current_completion>For example, in the sequence $x_{1}, x_{2}, \\cdots, x_{n}, \\c... |
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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>13.223. Point $C$ is located 12 k... |
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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.</instruction><current_completion>13.223. Point $C$ is located 12 km downstream from point $B$. A... |
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{
"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 11 Find the minimum value of the bivariate function $f(x, y)=(... |
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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.</instruction><current_completion>Example 11 Find the minimum value of the bivariate function $f(... |
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{
"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. Zhao, Qian, Sun, and Li each prepared a gift to give to one of the ... |
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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.</instruction><current_completion>4. Zhao, Qian, Sun, and Li each prepared a gift to give to one ... |
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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.</instruction><current_completion># Problem 5. (3 points)\n\nIn trapezoid $A B C D$, the lateral ... |
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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.</instruction><current_completion># Problem 5. (3 points)\n\nIn trapezoid $A B C D$, the lateral ... |
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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>3. As shown in Figure 1, $EF$ is ... |
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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>3. As shown in Figure 1, $EF$ is ... |
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{
"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 the invertibles modulo 8, then their inverses. Do the same modulo... |
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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>II. (25 points) If $2 x+y \\geqsl... |
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{
"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>II. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value... |
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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-4.6.\n\nDetermine the a... |
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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... |
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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.</instruction><current_completion>The letter $A$ has a vertical line of symmetry and the letter $... |
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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.</instruction><current_completion>The letter $A$ has a vertical line of symmetry and the letter $... |
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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.</instruction><current_completion>II. Find the last four digits of $2^{1999}$.\n\nTranslate the t... |
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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>II. Find the last four digits of ... |
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{
"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>II. Find the last four digits of $2^{1999}$.\n\nTranslate the text abo... |
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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... |
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{
"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>18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of\n$$\... |
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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.</instruction><current_completion>## Task B-4.1.\n\nIn the set of complex numbers, solve the equa... |
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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... |
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{
"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>Henry starts with a list of the first 1000 positive integers, and perf... |
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{
"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>Henry starts with a list of the first 1000 positive integers, and perf... |
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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 reference solution is authoritative evidence.</instruction><current_completion>5. Fyodor starts writing pairs of natural numbers $(a, b)$ on t... |
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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>5. Fyodor starts writing pairs of... |
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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 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... |
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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.</instruction><current_completion>13. As shown in the figure, natural numbers starting from 1 are... |
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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... |
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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>Problem 4. A segment is divided b... |
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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... |
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