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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>5. For a convex $n$-sided polygon, if circles are constructed w... |
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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>5. For a convex $n$-sided polygon, if circles are constructed w... |
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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>15. Lucia and Carla are playing 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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{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion># Problem 6. (3 points)\n\nThree runners are moving along a circular t... |
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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>## Task 2 - 080522\n\nIn a warehouse, there are three times as ... |
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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>## Task 2 - 080522\n\nIn a warehouse, there are three times as ... |
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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>$17 \\cdot 27$ As shown in the figure, in $\\triangle A B C$, $... |
[
{
"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>$17 \\cdot 27$ 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. Judge from the problem and candidate only.</instruction><current_completion>Example 5. Expand the function\n\n$$\nf(z)=\\frac{z}{z^{2}-2 z-3}\n$$\... |
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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>Example 5. Expand the function\n\n$$\nf(z)=\\frac{z}{z^{2}-2 z-3}\n$$\... |
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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.</instruction><current_completion>7. The set \n$\\left\\{x \\left\\lvert\\,-1 \\leqslant \\log _{... |
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{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>7. The set \n$\\left\\{x \\left\\lvert\\,-1 \\leqslant \\log _{... |
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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. Given $0<x<1$. Simplify\n$$\n=\\quad \\sqrt{\\left(x-\\frac{1}{x}\\... |
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{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The privileged context is an authoritative hint; use it to check the candidate.</instruction><current_completion>2. Given $0<x<1$. Simplify\n$$\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>7. Given the equality\n\n$$\n\\left(a^{m_{1}}-1\\right) \\ldots\\left(... |
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{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>7. Given the equality\n\n$$\n\\left(a^{m_{1}}-1\\right) \\ldots... |
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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 two such common fractions - one with a denominator of 8, the othe... |
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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>Find two such common fractions - ... |
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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>Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums ... |
[
{
"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 $a_1,a_2,a_3,a_4,a_5$ be dist... |
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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. There are ten small balls of the same size, five of which are red 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>3. There are ten small balls of t... |
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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>3. There are ten small balls of the same size, five of which ar... |
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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>|\n\nThere are 85 balloons in the room - red and blue. It is known tha... |
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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>3. Given positive numbers $a, b, c$ satisfying $a+b+c=1$,\nthen... |
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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. Given positive numbers $a, b, ... |
[
{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. Judge from the problem and candidate only.</instruction><current_completion>7. A circle with a radius of 1 has six points, these six points divide... |
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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>7. A circle with a radius of 1 has six points, these six points divide... |
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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 3.\n\nA polynomial $P(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 3.\n\nA polynomial $P(x)$ with integer coefficients h... |
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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>15.15 A paper punch can be placed at any point in the plane, an... |
[
{
"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>15.15 A paper punch can be placed... |
[
{
"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>Bogganov I.I.\n\nGiven an infinite supply of white, blue, and r... |
[
{
"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>Bogganov I.I.\n\nGiven an infinite supply of white, blue, and red cube... |
[
{
"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>$7 \\cdot 74$ There are several l... |
[
{
"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>$7 \\cdot 74$ There are several locks, and now six people each hold a ... |
[
{
"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>Suppose that $c\\in\\left(\\frac{1}{2},1\\right)$. Find the least $M$ ... |
[
{
"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>Suppose that $c\\in\\left(\\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. 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. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$.... |
[
{
"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. Given positive integers $a, b$... |
[
{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu... |
[
{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>1. A three-digit number is 29 times the sum of its digits. Then... |
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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. A three-digit number is 29 times the sum of its digits. Then... |
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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>A unit side regular hexagonal lattice point is reflected to the midpoi... |
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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>A unit side regular hexagonal lattice point is reflected to the midpoi... |
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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>Variant 9.5.2. Sides $A D$ and $D C$ of the inscribed quadrilateral $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... |
[
{
"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 7 How many positive integer factors does 20! have?</current_co... |
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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 7 How many positive integ... |
[
{
"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>The area of the region in the $xy... |
[
{
"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>The area of the region in the $xy... |
[
{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence. The privileged context is an authoritative hint; use it to check the candidate.</instruction><cu... |
[
{
"content": "<instruction>Reward mathematical correctness and self-contained, logically valid reasoning. Accept equivalent derivations and equivalent final answers. The reference solution is authoritative evidence.</instruction><current_completion>1. a) $\\left\\{\\begin{array}{l}y=\\sqrt{x} \\\\ y=\\frac{x^{2... |
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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. On a line, several points were... |
[
{
"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. On a line, several points were... |
[
{
"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 triples $(p, q, r)$ of prime numbers such that all of the fol... |
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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>Find all triples $(p, q, r)$ of p... |
[
{
"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 1978, someone introduced their family by saying: \"I have a son ... |
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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. (8 points) When Tongtong was 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>3. (8 points) When Tongtong was calculating a division with a 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>4.9 On a $10 \\times 10$ grid paper, there are 11 horizontal gr... |
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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 13 Given that $a$, $b$, and $c$ are all positive intege... |
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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 13 Given that $a$, $b$, and $c$ are all positive integers, and... |
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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. (15 points) As shown in the figure, ten points are evenly m... |
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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>7. It is known that the number $a$ satisfies the equation param... |
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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. Math Puzzle $8 / 65$\n\nA crane is being transported to a constr... |
[
{
"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. Math Puzzle $8 / 65$\n\nA c... |
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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>6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f... |
[
{
"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>6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f... |
[
{
"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>. In the decimal writing of $A$, ... |
[
{
"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>. In the decimal writing of $A$, ... |
[
{
"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>30. 20 identical balls are placed... |
[
{
"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>30. 20 identical balls are placed into 4 identical boxes, with ... |
[
{
"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>1.57. The vertices of a rectangle inscribed in a circle divide it into... |
[
{
"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>1.57. The vertices of a rectangle inscribed in a circle divide it into... |
[
{
"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>316. $\\left\\{\\begin{array}{l}(... |
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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>Example 1. Find the general solut... |
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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. Let $a$ and $n$ be natural num... |
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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. Let $a$ and $n$ be natural num... |
[
{
"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>What is the $21^{\\varrho}$ term ... |
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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. Let the function $f(x)=x^{2}+a x+b$, for any $a, b \\in \\mathbf{R}... |
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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>A right truncated pyramid is insc... |
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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 2. In two fruit stores, there were a total of $365 \\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>Problem 2. In two fruit stores, there were a total of $365 \\mathrm{~k... |
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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>F9 (27-5, UK) Let $f$ be a functi... |
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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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