prompt stringlengths 29 3.19k | solution stringlengths 1 22 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
In triangle $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b... | 34 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point outside triangle $... | MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9a9b6eb4-a1cb-49d1-8c1e-62eaf2f74079"
} |
Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. | 113 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the p... | MATH | {
"ground_truth": "113",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b426d104-244d-4831-a2c4-cd756b61700a"
} |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | 37 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ ... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "485b39f5-e71b-42b3-931f-219becbee2a3"
} |
For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square? | 10 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?\n\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "96bd487f-c324-41b4-8015-f8234fe31bb4"
} |
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$. | 509 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.\n\nRemember to p... | MATH | {
"ground_truth": "509",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4c4f3cb8-c7ae-4f8e-b9eb-d2fcfc2dd442"
} |
Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nBurrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d3db6787-eeb1-4abf-8cdd-d26b37114c68"
} |
Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of each fruit and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought? | 15 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDanial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3e477408-f0be-488b-b64b-455c98f01220"
} |
If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.\n\nRe... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0ae83368-3a58-4faa-9ab6-91e989ac15e9"
} |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | 37 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5d21ea89-7190-4dc8-b6c2-540ec349219b"
} |
Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$. | 3 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "34e6dee2-4e04-40c5-b334-ad1d42631c17"
} |
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. | 40 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $\\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\\triangle{ABC}$ intersect $BC$ and $AC$ ... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bf2cddfc-c4c0-4067-b320-3bb506875cf5"
} |
For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\frac{s(n)}{n}$? | 1099 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, whic... | MATH | {
"ground_truth": "1099",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "aea89852-b994-43bd-a1ee-e515555cdac4"
} |
Find a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$ has a rational root. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "859f31a3-1613-46d0-a758-84c3228efd90"
} |
For any real numbers $x$ and $y$ that satisfy the equations:
\[
x + y - xy = 155
\]
and
\[
x^2 + y^2 = 325
\]
Find $|x^3 - y^3|$. | 4375 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any real numbers $x$ and $y$ that satisfy the equations:\n\\[\nx + y - xy = 155\n\\]\nand\n\\[\nx^2 + y^2 = 325\n\... | MATH | {
"ground_truth": "4375",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "54d8c980-0fdd-4dac-b5b5-cd445466234e"
} |
Find all positive integers $k$ for which the equation:
$$ \text{lcm}(m,n) - \text{gcd}(m,n) = k(m-n) $$
has no solution in positive integers $(m,n)$ with $m \neq n$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $k$ for which the equation:\n$$ \\text{lcm}(m,n) - \\text{gcd}(m,n) = k(m-n) $$\nhas no sol... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ff488394-9402-4ab4-ad48-3c57226c86ac"
} |
Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b3d6095c-5105-43bd-ac97-15d3fb5fa1cf"
} |
Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$. | 18 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 +... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f885d323-1b6e-40c4-b485-9dad36eee691"
} |
Find all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular table under the following constraints:
- Each cell contains a distinct divisor.
- The sums of all rows are equal.
- The sums of all columns are equal. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular ta... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fcabd287-8c09-4350-ac36-e157de849b92"
} |
Find all integers $n$ such that $\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer. | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.\n\nRemember to put ... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "058d0f0c-94ca-4653-93bc-9463df11d7e1"
} |
Determine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural number. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural n... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4bc3f815-bd95-4a8c-a379-05d1e0363228"
} |
Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. | 200 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total o... | MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b1af1b6-125d-45b6-a9e1-0d324815a135"
} |
Find all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numbers $a^3 - 1$, $a^2 - 1$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of ... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "461726a9-fa69-4197-9e97-ac0ee198f8ad"
} |
Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real c... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b274cff3-2bed-4b85-8826-0a798a94f3fb"
} |
Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Fi... | 40 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = ... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5a292f7f-f5c1-4a9f-9c81-338a3bf07ae2"
} |
Find all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\n\nRemember to put your answer ... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9aad1783-fc56-44b1-a05d-c561e5a1f4bb"
} |
Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$. | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the sum of all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.\n\nRememb... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a6d38312-86c7-4022-b8d2-adcf19fa0c3a"
} |
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Ada has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.
Ada has written forty haikus so far ... | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to wr... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1ddc5b1e-6874-49af-b44c-012083a8f504"
} |
The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimet... | 10 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additio... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e1d07660-04dd-4306-badc-8a63aefb15e2"
} |
The radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, with $p, q$ being prime numbers and $m, n$ being positive integers. Determine $r$. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, w... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5cdc62b9-bb76-4d87-9200-5901d567078a"
} |
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$. | 107 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the largest positive integer $n$ such that the following statement holds:\n\nIf $a_1, a_2, a_3, a_4, a_5, a_... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4fe69060-c57f-4375-8eb8-3431c2c74a76"
} |
Solve for $x$:
\[
v - w + x - y + z = 79 \\
v + w + x + y + z = -1 \\
v + 2w + 4x + 8y + 16z = -2 \\
v + 3w + 9x + 27y + 81z = -1 \\
v + 5w + 25x + 125y + 625z = 79.
\] | 24 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSolve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "efee17c8-42a5-40ab-85d1-9e8f59213869"
} |
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) ... | 864 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree 12 cm $\\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by j... | MATH | {
"ground_truth": "864",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fe84e2ab-5d14-453f-aa72-5e5b39100c55"
} |
The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of $a + b + c$. | 31 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "784a3bc9-8930-4557-b536-ac623544f517"
} |
In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapant... | 65 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the country Máxico, there are two islands: the island \"Mayor\" and the island \"Menor\". The island \"Mayor\" has ... | MATH | {
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5e0cbdd2-1475-414e-b171-6f5bed784551"
} |
In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. Similarly, some number of the partic... | 102 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there ar... | MATH | {
"ground_truth": "102",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "09734e5f-d739-4545-b317-fdde985ef379"
} |
Let $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$. | 48 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$.... | MATH | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e307deb8-831f-4e23-80d0-ab46c94c8984"
} |
Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\[
x_1(x_1+1) = A \\
x_2(x_2+1) = A \\
x_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.
\] | -7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following t... | MATH | {
"ground_truth": "-7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d94b606a-d5d2-4b3f-98f7-8e717d1d1b9d"
} |
A function $f$ is defined on integers such that:
- $f(n) = n + 3$ if $n$ is odd.
- $f(n) = \frac{n}{2}$ if $n$ is even.
If $k$ is an odd integer, determine the values for which $f(f(f(k))) = k$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA function $f$ is defined on integers such that:\n\n- $f(n) = n + 3$ if $n$ is odd.\n- $f(n) = \\frac{n}{2}$ if $n$ is... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fbd36f8f-5a52-4d3c-be2a-c7c29ba71841"
} |
Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$. | 1594 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of t... | MATH | {
"ground_truth": "1594",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0de60080-7630-487c-a278-ac27265f935e"
} |
10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\f... | 828 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front an... | MATH | {
"ground_truth": "828",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ef7e7024-d992-4f99-9213-e436702ad1c7"
} |
All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string? | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAll six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. Wha... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29ae2e4e-631e-47b3-83fb-77cea98575be"
} |
Let $f(n)$ be a function that fulfills the following properties:
- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.
Find $\displaystyle f(2009^{20... | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ be a function that fulfills the following properties:\n\n- For each natural $n$, $f(n)$ is an integer great... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "210830fd-5398-4859-9874-4364fe85cdde"
} |
Find all positive integers $n$ such that $n = 100 \times d(n)$, where $d(n)$ represents the number of positive divisors of $n$. | 2000 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n = 100 \\times d(n)$, where $d(n)$ represents the number of positive diviso... | MATH | {
"ground_truth": "2000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "319be8e0-e989-4703-9858-3685d625d378"
} |
Solve in the real numbers the equation $3^{x+1} = (x-1)(x-3)$. | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSolve in the real numbers the equation $3^{x+1} = (x-1)(x-3)$.\n\nRemember to put your answer on its own line after \"... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b4478a7-8a73-4d82-8a1b-1a6b7ff27196"
} |
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? | 32 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nVitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all ... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2384be17-9019-4e38-afea-cc7adabca2eb"
} |
What is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for which the rightmost three digits of $2013^n$ are $001$. | 100 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for ... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2fa35708-19ed-42a1-9f04-d3ef3eca9359"
} |
A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$. | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA $8 \\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6f0fdfbf-e1bc-41ec-9ea1-ff5ffdb74566"
} |
Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ ... | 45 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCircle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is ta... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3c2719eb-d362-46f9-9b66-5edc5395ad91"
} |
Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square. | 4 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ for which $2^n + 2021n$ is a perfect square.\n\nRemember to put your answer on its own ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "151026dc-8e8b-4510-b0d7-9d8eeb22a776"
} |
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "approximately equal" to each other? Two numbers are called approximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considere... | 2004 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many different ways are there to write 2004 as a sum of one or more positive integers which are all \"approximatel... | MATH | {
"ground_truth": "2004",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a806f06d-07d3-48f7-aa68-9df4d09dc8db"
} |
A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. Determine, with proof, the maximum number of distinct license plates that the state can use. | 100000 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license ... | MATH | {
"ground_truth": "100000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "41b46ee1-2ff3-428d-ac03-671ae9f82527"
} |
Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square. | 179 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a p... | MATH | {
"ground_truth": "179",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8111beff-b97f-4342-85dd-3265cb49275f"
} |
There is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct coun... | 439 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs,... | MATH | {
"ground_truth": "439",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1203b437-bb39-4cda-a91b-0a79c80aa72d"
} |
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n + 76$ equal squares. Find the value of $n$. | 324 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n + 76$ equal squares.... | MATH | {
"ground_truth": "324",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b43e8a50-3090-47fb-bda6-d10f767f8f70"
} |
In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$. | 23 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, the... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e9ea24d5-87ac-451e-b769-671685ccc6e8"
} |
The difference between the maximal and the minimal diagonals of the regular $n$-gon equals its side ($n > 5$). Find $n$. | 9 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe difference between the maximal and the minimal diagonals of the regular $n$-gon equals its side ($n > 5$). Find $n... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a0a01733-c134-498e-a23e-06df93e8b715"
} |
A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number... | 20 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins i... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "807c6510-7136-441a-8b15-6f966fd03d72"
} |
Find all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ and $1 + n = 5(d_{1} + d_{2} + d_{3} + d_{4})$. | 1519 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ a... | MATH | {
"ground_truth": "1519",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "65d47d6a-598c-40f7-9c27-b767be998ea8"
} |
Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square. | 25 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \\dots, n$ with three differen... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7da9c773-1120-480b-b7f0-7d3bc6ecc6b0"
} |
Find all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number.\n\nRemember to put your ans... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4450ddbe-f0de-409d-938b-fa0fb735a624"
} |
Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal to $a$. | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equ... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a0d3594b-e952-4a6d-a255-95858a37ca21"
} |
Find the smallest positive integer $N$ satisfying the following three properties:
- $N$ leaves a remainder of $5$ when divided by $7$.
- $N$ leaves a remainder of $6$ when divided by $8$.
- $N$ leaves a remainder of $7$ when divided by $9$. | 502 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer $N$ satisfying the following three properties:\n- $N$ leaves a remainder of $5$ whe... | MATH | {
"ground_truth": "502",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "81d92d15-01ef-49f2-bbc4-da2efcabb5ef"
} |
A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$. | 211 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle betw... | MATH | {
"ground_truth": "211",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2cae6a18-2118-487d-a193-cccb7aca494e"
} |
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every subsequent term is obtained by multiplying its preceding term by $a$ and adding $b$.
For example, if $a = 2$ and $b = 4$, the first three terms are $5, 14, 32$.
What is the m... | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a64c2346-6ea1-431e-8803-ec0443dbecc1"
} |
We are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into $n$ sets such that each set has an equal total weight. Determine the maximal value of $n$ for which this partitioning is possible. | 2886 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWe are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into ... | MATH | {
"ground_truth": "2886",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d2bd2ba1-391d-40ba-b917-d0bc9627a570"
} |
There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off.
Every minute, the chips move simultaneously in a circular pattern as follows: chip $1$ moves one booth... | 32 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "138dd196-b522-4db3-9c1c-a142f7a5a7e2"
} |
Consider an isosceles triangle $ABC$ with sides $BC = 30$, $CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. Determine the length of $PQ$. | 25 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider an isosceles triangle $ABC$ with sides $BC = 30$, $CA = AB = 20$. Let $D$ be the foot of the perpendicular fr... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "78e90029-73a6-4f3e-b4fe-261d439cbf35"
} |
$ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$ and $E$ on sides $AB$ and $AC$ respectively such that $AD = DE = EC = n$, where $n$ is also an integer. Find the value of $m$. | 30 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$ and $E$ on ... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "74b85c12-580c-47bb-b956-05ca16f1e5fa"
} |
Find all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than 16. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than 16.\n\nRemem... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1cbfaad2-c093-45c5-bbb2-9ae159445bba"
} |
Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$. | 24 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nInside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cff25ab5-f227-4d26-b311-d04970bc31ce"
} |
Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$. | 86 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.\n\nRemember to put your ... | MATH | {
"ground_truth": "86",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7598d318-025f-4efd-8cdb-72584b9e3956"
} |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$
Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2047 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important... | MATH | {
"ground_truth": "2047",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a7f1519e-30e0-4d05-a03e-bf86fb60fadd"
} |
Find all natural numbers $n > 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$. | 1342 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all natural numbers $n > 1$ for which the following applies: The sum of the number $n$ and its second largest div... | MATH | {
"ground_truth": "1342",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "297cf1a2-1a20-4757-9d67-5a5799573f5a"
} |
Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}... | 8102 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nBob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \\ldots, 9$ (uniformly at ran... | MATH | {
"ground_truth": "8102",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4fa1b801-01df-45d2-ad27-e47cd3489a40"
} |
Hari is obsessed with cubics. He comes up with a cubic polynomial with leading coefficient 1, rational coefficients, and real roots $0 < a < b < c < 1$. He knows the following three facts:
1. $P(0) = -\frac{1}{8}$
2. The roots form a geometric progression in the order $a, b, c$.
3. \[ \sum_{k=1}^{\infty} (a^k + b^k +... | 31 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHari is obsessed with cubics. He comes up with a cubic polynomial with leading coefficient 1, rational coefficients, a... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "39fb2e93-8189-40ac-9f0c-4651d0772ef6"
} |
$A$ and $B$ are on a circle with radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen such that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$? | 20 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$A$ and $B$ are on a circle with radius $20$ centered at $C$, and $\\angle ACB = 60^\\circ$. $D$ is chosen such that $... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b6faee5a-7096-4030-ae96-f53589a2a1ea"
} |
Find all prime numbers $p$ such that both $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all prime numbers $p$ such that both $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\n\nRemember to put your an... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a05a8b19-49bc-4bc7-8b18-41bdde142e93"
} |
Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTriangle $ABC$ with $\\angle A = 90^\\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, i... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f103a263-3891-40c1-84f6-c841af9602f7"
} |
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5 + 2004z = 1$, then $P(z^2) = 0$. Calculate the quotient $\frac{P(1)}{P(-1)}$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5 + 2... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ea8aa31c-a4de-43b2-bf3b-066cf8c188c9"
} |
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1, m_2, n_1, n_2$ such that $a = m_1^2 + n_1^2$, $a^2 = m_2^2 + n_2^2$, and $m_1 - n_1 = m_2 - n_2$. | 261 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1, m_2, n_1... | MATH | {
"ground_truth": "261",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8eb2fa00-b19e-4f16-af10-4c6741e74f06"
} |
Let $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line $AP$, point $B$ determines point $Q$, as seen in the figure. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$. | 20 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dfc4a2d6-1511-4cf0-b558-4c465dd1e159"
} |
There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the ... | 20 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ ... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e620659c-2465-49c2-85df-14e3c82f8c64"
} |
Suppose an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Find the largest value of $p$. | 47 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Fin... | MATH | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9bcfaf82-85cf-4c15-a5c1-7386cad4fd9f"
} |
A triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, where $y < 28$. What is the value of $x + y$? | 56 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, whe... | MATH | {
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "762fcce5-28bb-402e-9871-cf8154ea1fd8"
} |
In the diagram, $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^\circ$ and $\angle BOE = 68^\circ$. Find $\angle AOF$. | 118 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram, $\\angle AOB = \\angle BOC$ and $\\angle COD = \\angle DOE = \\angle EOF$. Given that $\\angle AOD = 8... | MATH | {
"ground_truth": "118",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb2db0fa-2da3-4836-b03a-8f681ae4b10a"
} |
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy and hops till he falls asleep with $0$ ... | 10201 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nKermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to ho... | MATH | {
"ground_truth": "10201",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9086c9cd-ee8a-4f2a-8395-48f9197de8ea"
} |
There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:
1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with... | 12376 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the fol... | MATH | {
"ground_truth": "12376",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "77f85db4-deb3-45dd-a9ad-6fcec1eb94f0"
} |
Let $a, b, c, d$ be the four roots of the polynomial:
\[
x^4 + 3x^3 - x^2 + x - 2.
\]
Given that:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{2}
\]
and
\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} = -\frac{3}{4},
\]
the value of:
\[
\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^... | 39 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a, b, c, d$ be the four roots of the polynomial:\n\\[\nx^4 + 3x^3 - x^2 + x - 2.\n\\]\nGiven that:\n\\[\n\\frac{1... | MATH | {
"ground_truth": "39",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c91b90b8-5e2a-40cf-946c-4c6fc7d9cd78"
} |
If integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have? | 3 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have?\n\nRemember to put... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "90e430ba-222b-4ad0-b555-b0bcf63d0679"
} |
Find the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$. | 2001 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the greatest exponent $k$ for which $2001^k$ divides $2000^{2001^{2002}} + 2002^{2001^{2000}}$.\n\nRemember to pu... | MATH | {
"ground_truth": "2001",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4c382b57-b2cc-4075-9046-66341bfaa464"
} |
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A, B$ and the other at points $C, D$, so that $|AB| = |BC| = |CD| = 14\text{ cm}$. Another line intersects the circles at $E, F$, respectively $G, H$ so tha... | 13 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the pla... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "42505fdf-2e74-454e-9da2-141c5367dcaa"
} |
A table tennis club hosts a series of doubles matches following several rules:
1. Each player belongs to two pairs at most.
2. Every two distinct pairs play one game against each other at most.
3. Players in the same pair do not play against each other when they pair with others respectively.
Every player plays a cert... | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA table tennis club hosts a series of doubles matches following several rules:\n1. Each player belongs to two pairs at... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d43a6240-e03d-4cec-b67d-2ea7a3bb16a4"
} |
$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three succ... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc3a8e5e-a558-4d22-b791-82869df20c1f"
} |
Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at le... | 12 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSeven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ fe... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8335d254-0d03-46f7-8ba8-a9aa34fe1405"
} |
What are the last two digits of $2^{3^{4^{\cdots^{2019}}}}$? | 52 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat are the last two digits of $2^{3^{4^{\\cdots^{2019}}}}$?\n\nRemember to put your answer on its own line after \"A... | MATH | {
"ground_truth": "52",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bc090560-b75a-423a-bff6-7617b496539c"
} |
Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2 - bx + c = 0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$. | 64 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPositive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation... | MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "507026ad-0651-4f02-b444-fd85894e6baa"
} |
Suppose that $x$, $y$, and $z$ are positive real numbers satisfying the following system of equations:
\[
\begin{cases}
x^2 + xy + y^2 = 64 \\
y^2 + yz + z^2 = 49 \\
z^2 + zx + x^2 = 57
\end{cases}
\]
Then, \(\sqrt[3]{xyz}\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime posit... | 69 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $x$, $y$, and $z$ are positive real numbers satisfying the following system of equations:\n\n\\[\n\\begin... | MATH | {
"ground_truth": "69",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f00a19e2-3662-476e-80b4-7ce84bac7ef1"
} |
Find the number of ordered pairs of integers \((p, q)\) satisfying the equation \(p^2 - q^2 + p + q = 2014\). | 16 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered pairs of integers \\((p, q)\\) satisfying the equation \\(p^2 - q^2 + p + q = 2014\\).\n\nR... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8c35a3f6-0a38-4ebf-b378-9d53374a5349"
} |
Let $x$ be a real number in the interval $(0, \frac{\pi}{2})$ such that $\frac{1}{\sin x \cos x} + 2\cot 2x = \frac{1}{2}$. Evaluate $\frac{1}{\sin x \cos x} - 2\cot 2x$. | 8 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x$ be a real number in the interval $(0, \\frac{\\pi}{2})$ such that $\\frac{1}{\\sin x \\cos x} + 2\\cot 2x = \\... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dbc7aa4a-9de6-4080-9449-e6a63c2b6c0b"
} |
Let $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(x) = (x^4 + 2x^3 + 4x^2 + 2x + 1)^5$. Compute the prime $p$ satisfying $f(p) = 418,195,493$.\n\nRemember to put... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "96db93ed-f975-4ca6-b55f-af2f60548997"
} |
End of preview. Expand in Data Studio
dapo-5k
A fixed 4,800-problem subset of open-r1/DAPO-Math-17k-Processed
(config en, 14,116 rows), drawn once so that every run trains on exactly the same problems.
It is the DAPO counterpart of williamium/open-thoughts-5k:
same size, same draw procedure, so the two differ only in the source of the problems.
Why
Our training runs are 150 optimizer steps at 32 prompts per step — 4,800 examples. Loading the full 14,116-row set and letting the dataloader take what it needs means each run sees a different 4,800 problems. Two runs that differ only in one hyperparameter then also differ in their training data, and the run-to-run spread from that alone has been large enough to hide the effect being measured. Freezing the subset makes the knob the only thing that changes.
How it was drawn
Uniform sample without replacement over all 14,116 rows — no filtering, sorting, or stratification (indices are then sorted, so rows keep the source order):
rng = numpy.random.default_rng(20260823)
idx = numpy.sort(rng.choice(14116, size=4800, replace=False))
sample_indices.json records the source, config, seed, the RNG call, and the full index list, so
the draw can be reproduced or audited against the source. Schema is unchanged from the source
(7 columns: prompt, solution, data_source, source_prompt, ability, reward_model,
extra_info).
What the columns hold
| column | content |
|---|---|
prompt |
the problem statement (plain string, mean 279 chars, median 245, max 3,190) |
solution |
the final answer only — 4800/4800 (100.0%) are bare integers, mean 2.4 chars; identical to reward_model.ground_truth in 100% of rows |
data_source |
math_dapo for every row |
source_prompt |
single-turn [{"role": "user", "content": ...}] |
ability |
MATH |
reward_model |
{"style": "rule-lighteval/MATH_v2", "ground_truth": ...} |
extra_info |
{"index": ...} — row index in the source |
Note that unlike open-thoughts-5k, there is no worked solution: solution is just the boxed
label. A supervised run that uses solution as the teacher reference therefore conditions the
teacher on the answer alone, not on a derivation.
Sanity of the draw
Index quantiles of the sample are [0, 3503, 6980, 10643, 14114] against [0, 3528, 7057, 10586, 14115] for the full set — the draw spans the
file rather than favouring any region. All 4,800 indices are distinct.
Usage
from datasets import load_dataset
ds = load_dataset("williamium/dapo-5k")["train"] # 4800 rows
- Downloads last month
- 26