prompt stringlengths 19 3.19k | solution stringlengths 1 23 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | content listlengths 1 1 |
|---|---|---|---|---|---|---|---|
Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | 141 | 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 $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area ... | MATH | {
"ground_truth": "141",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a4c713de-34ae-48ba-bf66-9c6bef94a614"
} | [
{
"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 $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area ... |
If the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\text{th}}$ digit to be written? | 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 the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\\text{th}}$ di... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c85089aa-de89-483b-aa01-e5b5f1539012"
} | [
{
"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 the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\\text{th}}$ di... |
What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors? | 36 | 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 perfect square larger than $1$ with a perfect square number of positive integer factors?\n\nRemem... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d16a933b-4fe3-4e38-9a33-79b4f57b4728"
} | [
{
"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 perfect square larger than $1$ with a perfect square number of positive integer factors?\n\nRemem... |
Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the m... | 17 | 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 triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\\triangle ABC$. The... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c8a458c7-e614-4c91-81f9-8c3aa6e97fa1"
} | [
{
"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 triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\\triangle ABC$. The... |
How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 15\}$ have the following two properties?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
Find the number of such subsets. | 405 | 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 non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 15\\}$ have the following two properties?\n\n1. No two consecu... | MATH | {
"ground_truth": "405",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "42cb9bf3-c676-4d05-9274-7840099bd83b"
} | [
{
"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 non-empty subsets $S$ of $\\{1, 2, 3, \\ldots, 15\\}$ have the following two properties?\n\n1. No two consecu... |
To every pair of numbers $x, y$ the Rule assigns a number $x ∗ y$. Find $1993 ∗ 1935$ if it is known that
$x ∗ x = 0,$ and $x ∗ (y ∗ z) = (x ∗ y) + z$ for any $x, y, z.$ | 58 | 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\nTo every pair of numbers $x, y$ the Rule assigns a number $x ∗ y$. Find $1993 ∗ 1935$ if it is known that\n$x ∗ x = 0,... | MATH | {
"ground_truth": "58",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8e8017bf-1ff5-42e6-9a63-56e5644a35bb"
} | [
{
"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\nTo every pair of numbers $x, y$ the Rule assigns a number $x ∗ y$. Find $1993 ∗ 1935$ if it is known that\n$x ∗ x = 0,... |
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 large 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": "44cf0921-0efa-46bc-ba07-6ca74be490c8"
} | [
{
"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... |
Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$. | 7294927 | 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 $A$ denote the set of all integers $n$ such that $1 \\leq n \\leq 10000$, and the sum of the decimal digits of $n$... | MATH | {
"ground_truth": "7294927",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5e9c6f9a-dffe-49ca-af69-2399aefaf29d"
} | [
{
"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 $A$ denote the set of all integers $n$ such that $1 \\leq n \\leq 10000$, and the sum of the decimal digits of $n$... |
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as:
\[
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\]
with $P, Q, R \in \mathcal{A}$. Fin... | 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\nLet $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\ma... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5b5665ef-cc5c-43f0-aec2-e846dd0758f9"
} | [
{
"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 $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\ma... |
Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \leq p < q < r \leq 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$. | 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\nLet $x_1, x_2, \\ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "93d07483-b4d3-406c-8953-f30b698a7d5c"
} | [
{
"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 $x_1, x_2, \\ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some... |
An airline serves a dinner to all the passengers on an airplane. They get their choice of steak or fish. Three steak meals and three fish meals are set aside for the six-member crew. If the meals are distributed to the crew members randomly, what is the probability that both pilots get the fish?The answer is in the for... | 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\nAn airline serves a dinner to all the passengers on an airplane. They get their choice of steak or fish. Three steak m... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2d2a1112-7652-416b-bf99-56a2dcd0d34f"
} | [
{
"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\nAn airline serves a dinner to all the passengers on an airplane. They get their choice of steak or fish. Three steak m... |
How many three-digit positive integers $N$ satisfy the following properties?
The number $N$ is divisible by $7$.
The number formed by reversing the digits of $N$ is divisible by $5$.
Provide the number of such integers. | 14 | 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 three-digit positive integers $N$ satisfy the following properties? \nThe number $N$ is divisible by $7$. \... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "21e8f19e-1ebf-4b1e-b76b-6bc72496f6c2"
} | [
{
"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 three-digit positive integers $N$ satisfy the following properties? \nThe number $N$ is divisible by $7$. \... |
Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | 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\nCall an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6050fe63-df6c-4ce6-b348-1eb45b2c4fd1"
} | [
{
"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\nCall an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n... |
Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF},$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct e... | 51 | 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 $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such t... | MATH | {
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1fad0198-f997-4357-9e4f-626373b786c4"
} | [
{
"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 $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such t... |
What is the greatest integer value of $b$ such that $-4$ is not in the range of $y=x^2+bx+12$? | 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\nWhat is the greatest integer value of $b$ such that $-4$ is not in the range of $y=x^2+bx+12$?\n\nRemember to put your... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7a5b8633-9e75-4aa7-adf7-88c1f71a0867"
} | [
{
"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 greatest integer value of $b$ such that $-4$ is not in the range of $y=x^2+bx+12$?\n\nRemember to put your... |
In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^2=2/5$. The ratio of the area of triangle $DEF$ ... | 61 | 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 $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\\overline{AB}$, $E$ is on $\\overline{BC}$, and $F$... | MATH | {
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "575cc594-e543-4aa3-9b7b-c7fa75668600"
} | [
{
"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 $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\\overline{AB}$, $E$ is on $\\overline{BC}$, and $F$... |
In $\triangle ABC$, $AC=BC$, and $m\angle BAC=40^\circ$. What is the number of degrees in angle $x$? [asy]
size(80);
draw((0,0)--(7,0));
draw((0,0)--(3,2)--(6,0));
label("$A$",(0,0),W);
label("$B$",(6,0),S);
label("$C$",(3,2),N);
label("$x$",(6,0),NE);
[/asy] | 140 | 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 $\\triangle ABC$, $AC=BC$, and $m\\angle BAC=40^\\circ$. What is the number of degrees in angle $x$? [asy]\nsize(80... | MATH | {
"ground_truth": "140",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1aa9bd52-f8f0-427b-ab73-d0093f5d974f"
} | [
{
"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 $\\triangle ABC$, $AC=BC$, and $m\\angle BAC=40^\\circ$. What is the number of degrees in angle $x$? [asy]\nsize(80... |
Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square. | 829 | 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 positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect squar... | MATH | {
"ground_truth": "829",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "664f45ce-16bd-416d-a1a4-17f34beff74d"
} | [
{
"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 positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect squar... |
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, find the value of $n$. | 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\nTriangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertice... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "40a6f07a-42d0-4a87-a9f2-d985be1504ef"
} | [
{
"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 $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertice... |
Let $a_1$, $a_2$, \ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given seq... | 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 $a_1$, $a_2$, \\ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\\ell$, an... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e3e6139e-83f3-4bd2-bfc3-216c08747dda"
} | [
{
"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 $a_1$, $a_2$, \\ldots, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\\ell$, an... |
For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers. | 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\nFor any set $A = \\{a_1, a_2, a_3, a_4\\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f921e68c-4d32-4a43-bdbb-79ce8762492d"
} | [
{
"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 set $A = \\{a_1, a_2, a_3, a_4\\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, ... |
Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls. | 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\nSix boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six ... | MATH | {
"ground_truth": "86",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "33dc29e1-6845-4e0e-bff2-2ce7e9d1379e"
} | [
{
"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\nSix boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six ... |
Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$. | 369 | 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 $S = \\{1,2,\\cdots,2013\\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \\dots, S_9)$ such that ... | MATH | {
"ground_truth": "369",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "09e186bf-4020-449d-8ea8-c615d796359d"
} | [
{
"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 $S = \\{1,2,\\cdots,2013\\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \\dots, S_9)$ such that ... |
Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated). | 1769580 | 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\nGiven the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the ... | MATH | {
"ground_truth": "1769580",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "26685f6d-408f-485e-977f-bab21359242b"
} | [
{
"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\nGiven the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the ... |
For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\sum_{n=1}^{2022} f(n) \mod 1000$. | 242 | 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 positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the... | MATH | {
"ground_truth": "242",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5aa95896-49de-47e7-a43b-a1661c74bf51"
} | [
{
"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 positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the... |
Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which ... | 2048 | 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\nSix distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against ev... | MATH | {
"ground_truth": "2048",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9cbbbad0-62c4-4a5c-8e5a-a3444f16f235"
} | [
{
"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\nSix distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against ev... |
Call a positive integer $N$ a $7$-$10$ double if the digits of the base- $7$ representation of $N$ form a base- $10$ number that is twice $N$ . For example, $51$ is a $7$-$10$ double because its base- $7$ representation is $102$ . What is the largest $7$-$10$ double? | 315 | 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\nCall a positive integer $N$ a $7$-$10$ double if the digits of the base- $7$ representation of $N$ form a base- $10$ n... | MATH | {
"ground_truth": "315",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1567bc9c-0f8d-4222-b89c-fd5aadf2b61d"
} | [
{
"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\nCall a positive integer $N$ a $7$-$10$ double if the digits of the base- $7$ representation of $N$ form a base- $10$ n... |
For any positive integer $n$, we define the integer $P(n)$ by:
$$P(n) = n(n+1)(2n+1)(3n+1)\ldots(16n+1).$$
Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3), \ldots, P(2016)$. | 510510 | 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 positive integer $n$, we define the integer $P(n)$ by:\n\n$$P(n) = n(n+1)(2n+1)(3n+1)\\ldots(16n+1).$$\n\nFind... | MATH | {
"ground_truth": "510510",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cdd48a9e-b953-4c53-9c3b-6b49b50a91cd"
} | [
{
"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 positive integer $n$, we define the integer $P(n)$ by:\n\n$$P(n) = n(n+1)(2n+1)(3n+1)\\ldots(16n+1).$$\n\nFind... |
A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines? | 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 rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c26b09d0-b3e4-4172-8b82-d117e05040e2"
} | [
{
"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, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)... |
Find the number of even integers \( n \) such that \( 0 \le n \le 100 \) and \( 5 \mid n^2 \cdot 2^{2n^2} + 1 \). | 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\nFind the number of even integers \\( n \\) such that \\( 0 \\le n \\le 100 \\) and \\( 5 \\mid n^2 \\cdot 2^{2n^2} + 1... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "825df3cd-2191-4253-ad68-3ce2d74964ab"
} | [
{
"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 even integers \\( n \\) such that \\( 0 \\le n \\le 100 \\) and \\( 5 \\mid n^2 \\cdot 2^{2n^2} + 1... |
In parallelogram $ABCD$ , point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$ . Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$ . Find $\frac {AC}{AP}$ . | 17 | 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 parallelogram $ABCD$ , point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4435a921-d6cf-49df-a1f3-3cbd98c455ed"
} | [
{
"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 parallelogram $ABCD$ , point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$... |
Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes th... | 71 | 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 math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \\{1, 3, 4... | MATH | {
"ground_truth": "71",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8a3eaf60-84c9-4ac1-bd97-3c9f8b28b792"
} | [
{
"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 math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \\{1, 3, 4... |
Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects \(\overline{BO}\) at \(D\). The original answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\). | 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\nPoints \\(A\\) and \\(C\\) lie on a circle centered at \\(O\\), each of \\(\\overline{BA}\\) and \\(\\overline{BC}\\) ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "43c7be7e-1897-43aa-a374-cf1d30782120"
} | [
{
"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\nPoints \\(A\\) and \\(C\\) lie on a circle centered at \\(O\\), each of \\(\\overline{BA}\\) and \\(\\overline{BC}\\) ... |
What is the remainder when $7^{8^9}$ is divided by $1000?$ | 801 | 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 remainder when $7^{8^9}$ is divided by $1000?$\n\nRemember to put your answer on its own line after \"Answ... | MATH | {
"ground_truth": "801",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5b8fd727-b176-4e42-9b3d-f43f4510941c"
} | [
{
"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 remainder when $7^{8^9}$ is divided by $1000?$\n\nRemember to put your answer on its own line after \"Answ... |
For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number? | 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\nFor how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number?\n\nRemember to put your answer on its ow... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "feb6475b-3a4a-47b2-99cd-9feb6b0f8ac8"
} | [
{
"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 how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number?\n\nRemember to put your answer on its ow... |
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the porti... | 323 | 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 right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$,... | MATH | {
"ground_truth": "323",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "287dd05a-7544-4790-8c75-5eba46f4057f"
} | [
{
"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 right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$,... |
Find a positive integer $n$ with five non-zero different digits such that $n$ is equal to the sum of all the three-digit numbers that can be formed using the digits of $n$. | 35964 | 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 positive integer $n$ with five non-zero different digits such that $n$ is equal to the sum of all the three-dig... | MATH | {
"ground_truth": "35964",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ddfe877e-a68f-479d-9ac1-200c43b879f8"
} | [
{
"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 positive integer $n$ with five non-zero different digits such that $n$ is equal to the sum of all the three-dig... |
Let $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline
$$I_k=\int_{0}^{\infty}\frac{x^k}{p(x)}dx$$.
For which $k$ is $I_k$ smallest. | 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 $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline\n$$I_k=\\int_{0}^{\\infty}\\frac{x^k}{p(x)}dx$$... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "710aead4-48aa-4c8d-850d-d78ddd0bd8d3"
} | [
{
"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 $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline\n$$I_k=\\int_{0}^{\\infty}\\frac{x^k}{p(x)}dx$$... |
What real values of $x$ are not in the domain of
$f(x)=\frac{1}{|x^2-6x+8|+|x^2+x-6|}$? | 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\nWhat real values of $x$ are not in the domain of\n\n$f(x)=\\frac{1}{|x^2-6x+8|+|x^2+x-6|}$?\n\nRemember to put your an... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7eda10cf-df84-4450-95f6-9cb175efe3c9"
} | [
{
"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 real values of $x$ are not in the domain of\n\n$f(x)=\\frac{1}{|x^2-6x+8|+|x^2+x-6|}$?\n\nRemember to put your an... |
Square $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are coplanar. Line segment $AB$ has a length of 6 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MC$? Express your answer in simplest radical form. [asy]
size(4cm);
dotfac... | 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\nSquare $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are copla... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5bd441c6-07f6-403f-ad9f-f6b310bf9b3f"
} | [
{
"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\nSquare $ABCD$ is constructed along diameter $AB$ of a semicircle, as shown. The semicircle and square $ABCD$ are copla... |
The foci of a certain ellipse are at $(3,10 + \sqrt{105})$ and $(3,10 - \sqrt{105}).$ The endpoints of one of the axes are $(-5,10)$ and $(11,10).$ Find the semi-major axis. | 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\nThe foci of a certain ellipse are at $(3,10 + \\sqrt{105})$ and $(3,10 - \\sqrt{105}).$ The endpoints of one of the a... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4de4f8d6-c8b4-40b7-88a6-1a28b9cb1b4b"
} | [
{
"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 foci of a certain ellipse are at $(3,10 + \\sqrt{105})$ and $(3,10 - \\sqrt{105}).$ The endpoints of one of the a... |
Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one ... | 1010 | 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\nDexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that,... | MATH | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "55d21e01-08c9-4d43-b828-c0c670e4710c"
} | [
{
"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\nDexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that,... |
Suppose the function $f(x)$ is defined on the domain $\{x_1,x_2,x_3\}$, so that the graph of $y=f(x)$ consists of just three points. Suppose those three points form a triangle of area $32$.
The graph of $y = 2f(2x)$ also consists of just three points. What is the area of the triangle formed by those three points? | 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\nSuppose the function $f(x)$ is defined on the domain $\\{x_1,x_2,x_3\\}$, so that the graph of $y=f(x)$ consists of ju... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c024b94d-0f90-47ec-8389-86514c889101"
} | [
{
"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 the function $f(x)$ is defined on the domain $\\{x_1,x_2,x_3\\}$, so that the graph of $y=f(x)$ consists of ju... |
Compute the smallest positive integer $a$ for which
$$\sqrt{a + \sqrt{a + \ldots}} - \frac{1}{a + \frac{1}{a+\ldots}} > 7$$ | 43 | 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 smallest positive integer $a$ for which \n$$\\sqrt{a + \\sqrt{a + \\ldots}} - \\frac{1}{a + \\frac{1}{a+\\... | MATH | {
"ground_truth": "43",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8326d608-9960-4461-88ef-319dd12fe97b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the smallest positive integer $a$ for which \n$$\\sqrt{a + \\sqrt{a + \\ldots}} - \\frac{1}{a + \\frac{1}{a+\\... |
Find the number of noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neither equilateral, isosceles, nor right triangles. | 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 the number of noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neith... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f9105936-97e0-4fd9-a423-fd97809e887a"
} | [
{
"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 noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neith... |
Given $A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, find how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$. | 51 | 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\nGiven $A = \\{1, 2, 3, 5, 8, 13, 21, 34, 55\\}$, find how many of the numbers between $3$ and $89$ cannot be written a... | MATH | {
"ground_truth": "51",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "160f17e1-da11-4324-870e-ff861d3b3a53"
} | [
{
"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\nGiven $A = \\{1, 2, 3, 5, 8, 13, 21, 34, 55\\}$, find how many of the numbers between $3$ and $89$ cannot be written a... |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, "My favorite number is a two-digit perfect square!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?"... | 36 | 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\nCat and Claire are having a conversation about Cat’s favorite number. Cat says, \"My favorite number is a two-digit pe... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "44cdb387-cf4a-404a-98d1-f916e19b577b"
} | [
{
"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\nCat and Claire are having a conversation about Cat’s favorite number. Cat says, \"My favorite number is a two-digit pe... |
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB} || \overline{EF}$. The area of $ABCD$ is $2016$, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the differe... | 840 | 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\nSquares $ABCD$ and $EFGH$ have a common center and $\\overline{AB} || \\overline{EF}$. The area of $ABCD$ is $2016$, a... | MATH | {
"ground_truth": "840",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "98c5af2a-4102-4d82-9e79-29154f09129d"
} | [
{
"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\nSquares $ABCD$ and $EFGH$ have a common center and $\\overline{AB} || \\overline{EF}$. The area of $ABCD$ is $2016$, a... |
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Find $\rho^2.$The answer is in the form rac{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\nLet $a$ and $b$ be positive real numbers with $a\\ge b$. Let $\\rho$ be the maximum possible value of $\\frac {a}{b}$ ... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "944895fb-dc2c-4719-87b0-d99440e719c8"
} | [
{
"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 real numbers with $a\\ge b$. Let $\\rho$ be the maximum possible value of $\\frac {a}{b}$ ... |
Find the sum of all positive integers $n$ such that $\frac{2n+1}{n(n-1)}$ has a terminating decimal representation. | 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\nFind the sum of all positive integers $n$ such that $\\frac{2n+1}{n(n-1)}$ has a terminating decimal representation.\n... | MATH | {
"ground_truth": "52",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5fe76fe4-8902-40f1-a99e-0d4dd5215146"
} | [
{
"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 $n$ such that $\\frac{2n+1}{n(n-1)}$ has a terminating decimal representation.\n... |
Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \cdot s_k + b \cdot s_{k-1} + c \cdot s_{k-2}$ for $k = 2$, $3$, $\ldots$ Find the value of $a + b + c$. | 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\nLet $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "27c6421e-ac94-4720-a371-9a7953c599e1"
} | [
{
"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 $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_... |
Around a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their difference is $1$ or $2$. We will say that a number is *huge* if it is greater than its two neighbors, and that it is *tiny* if it is less than its two neighbors. The sum of all the huge numbers is equal t... | 1005 | 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\nAround a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their dif... | MATH | {
"ground_truth": "1005",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fdf9b816-3753-4028-b1f2-9d1d2ae8cb67"
} | [
{
"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\nAround a circle are written $2009$ integers, not necessarily distinct, so that if two numbers are neighbors, their dif... |
If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?The answer is in the form r... | 19 | 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 $0 \\le p \\le 1$ and $0 \\le q \\le 1$, define $F(p, q)$ by\n\\[\n F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-... | MATH | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ae35f480-f4c5-4f18-9057-4cfa52e82972"
} | [
{
"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 $0 \\le p \\le 1$ and $0 \\le q \\le 1$, define $F(p, q)$ by\n\\[\n F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-... |
Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 21 | 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\nBilly is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How ... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "88ebb206-1285-4416-8288-cfe78508afa2"
} | [
{
"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\nBilly is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How ... |
The diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with side length $4$. The distance between the two vertices marked $A$ and $B$ can be written as $\sqrt{m}+\sqrt{n}$ for two positive integers $m$ and $n$. Find $m + n$. | 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\nThe diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with ... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "140db4c0-682c-4001-afae-7c8ecf7fb714"
} | [
{
"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 diagram shows two equilateral triangles with side length $4$ mounted on two adjacent sides of a square, also with ... |
A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$.
[img]https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/201... | 90 | 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 with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so th... | MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ccd40bf6-60e7-43d3-a6df-a3a7f1148d1b"
} | [
{
"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 with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so th... |
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end... | 351 | 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 $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$... | MATH | {
"ground_truth": "351",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e3ea3737-5cbb-4170-9a1f-dcea2e9d384a"
} | [
{
"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 $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$... |
What is the product of all solutions to the equation
$$\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023?$$ | 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\nWhat is the product of all solutions to the equation \n$$\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023?$$\n... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "03397ac4-f821-471c-b44f-bc26cce8123f"
} | [
{
"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 product of all solutions to the equation \n$$\\log_{7x}2023\\cdot \\log_{289x}2023=\\log_{2023x}2023?$$\n... |
The number $10!$ (where $10$ is written in base $10$), when converted to the base $12$ system, ends in exactly $k$ zeroes. What is the value of $k$? | 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\nThe number $10!$ (where $10$ is written in base $10$), when converted to the base $12$ system, ends in exactly $k$ zer... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e8f0aa2f-e709-4830-afe7-45657baea6f5"
} | [
{
"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 number $10!$ (where $10$ is written in base $10$), when converted to the base $12$ system, ends in exactly $k$ zer... |
Suppose we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have that $\sum_{k=1}^{n} a_ka_{\lfloor \sqrt{k} \rfloor} = n^2$. Determine the first value of $k$ so $a_k > 100$. | 1018 | 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 we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have tha... | MATH | {
"ground_truth": "1018",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a986294c-d5c1-445d-acb9-0b960a7c86a3"
} | [
{
"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 we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have tha... |
A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy]
unitsize(1mm);
defaultpen(linewidth(0.7pt));
draw((0,0)--(20,0)--(20,20)--(0,20)--cycle);
draw(Circle((5,5),5));
draw(Circle((15,5),5));
draw(Circle((5,15),5));
draw(Circle((15,15)... | 400 | 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 square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, ... | MATH | {
"ground_truth": "400",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d25e0e4e-0277-4332-aeb0-2b2051ca73d9"
} | [
{
"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 square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, ... |
Given three equilateral triangles $\triangle HOW$, $\triangle BOW$, and $\triangle DAH$ in a plane:
- $WO = 7$
- $AH = 2$
It is also given that the points $D$, $A$, and $B$ are collinear in that order.
Find the length of $BA$. | 11 | 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\nGiven three equilateral triangles $\\triangle HOW$, $\\triangle BOW$, and $\\triangle DAH$ in a plane:\n- $WO = 7$\n- ... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29201a6c-9ce5-496b-9a01-5765f287c662"
} | [
{
"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\nGiven three equilateral triangles $\\triangle HOW$, $\\triangle BOW$, and $\\triangle DAH$ in a plane:\n- $WO = 7$\n- ... |
A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner. | 504 | 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 paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to ev... | MATH | {
"ground_truth": "504",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2bd16f1d-2eeb-4125-9478-929a9b1ef4e4"
} | [
{
"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 paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to ev... |
Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$. | 1344 | 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 $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers.... | MATH | {
"ground_truth": "1344",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dceb69ae-2ce3-43fc-885e-be0bd858b93a"
} | [
{
"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 $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers.... |
One million bucks (i.e., one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are honest bucks,... | 1200000 | 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\nOne million bucks (i.e., one million male deer) are in different cells of a $1000 \\times 1000$ grid. The left and rig... | MATH | {
"ground_truth": "1200000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "11a8c85f-b656-44f0-9a42-ffe41c14b28a"
} | [
{
"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\nOne million bucks (i.e., one million male deer) are in different cells of a $1000 \\times 1000$ grid. The left and rig... |
Let $a,b$ be positive integers satisfying
$$\sqrt{\frac{ab}{2b^2-a}}=\frac{a+2b}{4b}$$.
Find $|10(a-5)(b-15)|+8$. | 2018 | 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 $a,b$ be positive integers satisfying\n$$\\sqrt{\\frac{ab}{2b^2-a}}=\\frac{a+2b}{4b}$$.\nFind $|10(a-5)(b-15)|+8$.... | MATH | {
"ground_truth": "2018",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "61bd11b4-4409-43ac-b989-817f1b1b1c0c"
} | [
{
"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 $a,b$ be positive integers satisfying\n$$\\sqrt{\\frac{ab}{2b^2-a}}=\\frac{a+2b}{4b}$$.\nFind $|10(a-5)(b-15)|+8$.... |
In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA, AB, AC, BB, BC, AD, CC, BD, CD, DD$$ in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A+A, A+B, A+C, B+B$, etc., corr... | 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 assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA, AB, AC, BB, BC, AD,... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e75e78c5-4bdd-42d1-b6d3-17e891c6acb3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA, AB, AC, BB, BC, AD,... |
Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have? | 905 | 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 a subset of $\\{1, 2, 3, \\cdots, 1989 \\}$ in which no two members differ by exactly $4$ or by exactly $7$... | MATH | {
"ground_truth": "905",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c685635d-26d2-4abf-8aa4-48841dacc32b"
} | [
{
"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 a subset of $\\{1, 2, 3, \\cdots, 1989 \\}$ in which no two members differ by exactly $4$ or by exactly $7$... |
How many positive integers $a$ less than $100$ satisfy that $4a^2 + 3a + 5$ is divisible by $6$? | 33 | 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 positive integers $a$ less than $100$ satisfy that $4a^2 + 3a + 5$ is divisible by $6$?\n\nRemember to put yo... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d7fcb593-8648-4b4c-b269-6e55f10116ba"
} | [
{
"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 positive integers $a$ less than $100$ satisfy that $4a^2 + 3a + 5$ is divisible by $6$?\n\nRemember to put yo... |
Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\). | 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\nReal numbers \\(a\\) and \\(b\\) are chosen with \\(1 < a < b\\) such that no triangle with positive area has side len... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f03c8b8e-ba9a-4569-8c22-a11e561c9772"
} | [
{
"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\nReal numbers \\(a\\) and \\(b\\) are chosen with \\(1 < a < b\\) such that no triangle with positive area has side len... |
For positive integers $m$ and $n$, find the smallest possible value of $|2011^m - 45^n|$. | 14 | 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 $m$ and $n$, find the smallest possible value of $|2011^m - 45^n|$.\n\nRemember to put your answ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c24da97b-783c-49c1-acca-72d0b8424ce7"
} | [
{
"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 $m$ and $n$, find the smallest possible value of $|2011^m - 45^n|$.\n\nRemember to put your answ... |
What are the rightmost three digits of $5^{1993}$? | 125 | 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 rightmost three digits of $5^{1993}$?\n\nRemember to put your answer on its own line after \"Answer:\".",... | MATH | {
"ground_truth": "125",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "02411046-40d3-4d66-a5bb-fe43fc91b426"
} | [
{
"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 rightmost three digits of $5^{1993}$?\n\nRemember to put your answer on its own line after \"Answer:\".",... |
Suppose that I have an unlimited supply of identical math books, history books, and physics books. All are the same size, and I have room on a shelf for 8 books. In how many ways can I arrange eight books on the shelf if no two books of the same type can be adjacent? | 384 | 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 I have an unlimited supply of identical math books, history books, and physics books. All are the same s... | MATH | {
"ground_truth": "384",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "469e1f06-b042-415a-bf8a-03b791461346"
} | [
{
"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 I have an unlimited supply of identical math books, history books, and physics books. All are the same s... |
For how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$ ? | 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 how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$ ? \n\nReme... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3627145d-9873-4f78-927b-fd645a56aa46"
} | [
{
"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 how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$ ? \n\nReme... |
Of the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number $m$ (possibly greater than $9$) and the direction of reference were determined. People began to be counted, starting from the first. Each one ... | 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\nOf the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "26d93e52-bdac-40ba-b9c6-de2fdb3636da"
} | [
{
"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\nOf the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were... |
Compute the number of real numbers $t$ such that \[ t = 50 \sin(t - \lfloor t \rfloor). \] Here, $\lfloor \cdot \rfloor$ denotes the greatest integer function. | 42 | 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 number of real numbers $t$ such that \\[ t = 50 \\sin(t - \\lfloor t \\rfloor). \\] Here, $\\lfloor \\cdot... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0930600f-35dc-4543-987d-5b497c9d7eb2"
} | [
{
"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 number of real numbers $t$ such that \\[ t = 50 \\sin(t - \\lfloor t \\rfloor). \\] Here, $\\lfloor \\cdot... |
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times 30 \times 30$ cube. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra. | 4500 | 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 $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \\times 30 \\times 30$ cube. The two figures... | MATH | {
"ground_truth": "4500",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28cf64ee-cd94-4970-a065-972845a9e568"
} | [
{
"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 $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \\times 30 \\times 30$ cube. The two figures... |
The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? | 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\nThe least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the le... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6b4a2d8e-dbaa-423f-ae30-d8c8f4aeb109"
} | [
{
"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 least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the le... |
Let $p, q, r$ be distinct prime numbers and let
\[ A = \{p^a q^b r^c \mid 0 \le a, b, c \le 5\} \]
Find the least $n \in \mathbb{N}$ such that for any $B \subset A$ where $|B| = n$, there exist elements $x$ and $y$ such that $x$ divides $y$. | 28 | 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 $p, q, r$ be distinct prime numbers and let \n\\[ A = \\{p^a q^b r^c \\mid 0 \\le a, b, c \\le 5\\} \\]\nFind the ... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2db09c52-5d56-4930-bb39-a78ab9b1a303"
} | [
{
"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 $p, q, r$ be distinct prime numbers and let \n\\[ A = \\{p^a q^b r^c \\mid 0 \\le a, b, c \\le 5\\} \\]\nFind the ... |
A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?
[asy]
size(4cm,4cm);
pair A,B,C,D,E,F,G,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r;
A=(0.5,0... | 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 wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cu... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c1ecc3de-268f-4072-81e5-471e580154b2"
} | [
{
"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 wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cu... |
For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$. | 2021 | 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 some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer po... | MATH | {
"ground_truth": "2021",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1606d6e5-a84f-41ac-a2e3-599c58dae42c"
} | [
{
"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 some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer po... |
A robot is standing on the bottom left vertex $(0,0)$ of a $5 \times 5$ grid and wants to go to $(5,5)$, only moving to the right $(a,b) \mapsto (a+1,b)$ or upward $(a,b) \mapsto (a,b+1)$. However, this robot is not programmed perfectly and sometimes takes the upper-left diagonal path $(a,b) \mapsto (a-1,b+1)$. As the ... | 1650 | 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 robot is standing on the bottom left vertex $(0,0)$ of a $5 \\times 5$ grid and wants to go to $(5,5)$, only moving ... | MATH | {
"ground_truth": "1650",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f3f1b494-6801-42da-8601-9c368cce4412"
} | [
{
"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 robot is standing on the bottom left vertex $(0,0)$ of a $5 \\times 5$ grid and wants to go to $(5,5)$, only moving ... |
Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$
| 50 | 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 positive integers less than $1000$ that can be expressed as the difference of two integral powers o... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9aac97f3-c2ff-490b-84ee-137ab06e3449"
} | [
{
"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 positive integers less than $1000$ that can be expressed as the difference of two integral powers o... |
In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \... | 53 | 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\\), \\(AB = BC\\), and \\(\\overline{BD}\\) is an altitude. Point \\(E\\) is on the extension of ... | MATH | {
"ground_truth": "53",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b41cbcb9-a488-488b-bec7-84b0ff1fc55e"
} | [
{
"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\\), \\(AB = BC\\), and \\(\\overline{BD}\\) is an altitude. Point \\(E\\) is on the extension of ... |
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 450 | 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 $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\\overline{AB}$ at $D.$ The lengths of ... | MATH | {
"ground_truth": "450",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6b11b009-9276-44e5-8f58-a92a76503494"
} | [
{
"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 $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\\overline{AB}$ at $D.$ The lengths of ... |
In how many ways can $47$ be written as the sum of two primes? | 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\nIn how many ways can $47$ be written as the sum of two primes?\n\nRemember to put your answer on its own line after \"... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "42ce5d24-a317-43d7-9184-15aa2e9f3a94"
} | [
{
"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 how many ways can $47$ be written as the sum of two primes?\n\nRemember to put your answer on its own line after \"... |
Determine all positive integers $n$ for which the equation
\[
x^n + (2+x)^n + (2-x)^n = 0
\]
has an integer as a solution. | 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 positive integers $n$ for which the equation\n\\[\nx^n + (2+x)^n + (2-x)^n = 0\n\\]\nhas an integer as a... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a40a9f04-fe48-4e8e-acd0-718143ecf5ab"
} | [
{
"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 positive integers $n$ for which the equation\n\\[\nx^n + (2+x)^n + (2-x)^n = 0\n\\]\nhas an integer as a... |
For a positive integer $n$, let $f(n)$ denote the number of ways to pair $n$ objects, considering only which objects are paired together. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023. | 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\nFor a positive integer $n$, let $f(n)$ denote the number of ways to pair $n$ objects, considering only which objects a... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0787e381-83e8-4fa1-8470-d0d933af8aa5"
} | [
{
"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 a positive integer $n$, let $f(n)$ denote the number of ways to pair $n$ objects, considering only which objects a... |
Let $f_n(x) = n + x^2$. Evaluate the product $gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\} \times gcd\{f_{2021}(2022), f_{2021}(2023)\}$, where $gcd\{x, y\}$ is the greatest common divisor of $x$ and $y$ | 125 | 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 $f_n(x) = n + x^2$. Evaluate the product $gcd\\{f_{2001}(2002), f_{2001}(2003)\\} \\times gcd\\{f_{2011}(2012), f_... | MATH | {
"ground_truth": "125",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d5b3d86d-139d-4320-874a-a1d87b60c02b"
} | [
{
"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 $f_n(x) = n + x^2$. Evaluate the product $gcd\\{f_{2001}(2002), f_{2001}(2003)\\} \\times gcd\\{f_{2011}(2012), f_... |
There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws ... | 405 | 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 $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visi... | MATH | {
"ground_truth": "405",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "96deafce-85b5-463d-ab96-b61cf104dd60"
} | [
{
"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 $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visi... |
The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer. | 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\nThe number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.\n\nRe... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c2c7f62c-9f1f-4fe5-8582-35f8fa16a5c1"
} | [
{
"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 number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.\n\nRe... |
In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.
If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain? | 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\nIn base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ st... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e6d44b2a-385c-4948-a4ac-1791c5536bc4"
} | [
{
"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 base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ st... |
Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word... | 111110 | 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\nRuney is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words... | MATH | {
"ground_truth": "111110",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6c1ff59c-606f-4a14-90ed-1bf16c73dcee"
} | [
{
"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\nRuney is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words... |
Let $x, y, z, w$ be integers such that $2^x + 2^y + 2^z + 2^w = 24.375$. Find the value of $xyzw$. | 72 | 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 $x, y, z, w$ be integers such that $2^x + 2^y + 2^z + 2^w = 24.375$. Find the value of $xyzw$.\n\nRemember to put ... | MATH | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e32f25fe-70c7-4d6d-b7ba-0918c44078ad"
} | [
{
"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 $x, y, z, w$ be integers such that $2^x + 2^y + 2^z + 2^w = 24.375$. Find the value of $xyzw$.\n\nRemember to put ... |
Let $N,O$ be functions such that $N(x) = 2\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(3))))))$? | 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\nLet $N,O$ be functions such that $N(x) = 2\\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(3))))))$?\n\nRemember to ... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "10f282f7-d410-4c6c-ab15-e87afc8a06a7"
} | [
{
"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 $N,O$ be functions such that $N(x) = 2\\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(3))))))$?\n\nRemember to ... |
A man buys a house for $10000$ dollars and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep, pays $325$ dollars a year in taxes, and realizes $5\frac{1}{2}\%$ on his investment. If the monthly rent is $x$ dollars, what is the value of $100x$? | 8333 | 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 man buys a house for $10000$ dollars and rents it. He puts $12\\frac{1}{2}\\%$ of each month's rent aside for repair... | MATH | {
"ground_truth": "8333",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a59dd3cb-7be7-4be5-8604-62880c398e05"
} | [
{
"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 man buys a house for $10000$ dollars and rents it. He puts $12\\frac{1}{2}\\%$ of each month's rent aside for repair... |
There are $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd-numbered geese are standing behind them in the order $1,3,5,\dots,19,$ where $19$ is at the end of the line. The geese want to rearr... | 55 | 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 $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the orde... | MATH | {
"ground_truth": "55",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a02fcf35-b35a-4afa-84a0-28dbdeea3e99"
} | [
{
"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 $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the orde... |
Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone? | 270 | 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\nFive bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours t... | MATH | {
"ground_truth": "270",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "472d9ebd-1e06-4f3c-9ba5-db12079a2cd8"
} | [
{
"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\nFive bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours t... |
Let \(ABC\) be a triangle such that \(|AB| = 7\), \(|BC| = 8\), \(|AC| = 6\). Let \(D\) be the midpoint of side \([BC]\). If the circle through \(A\), \(B\), and \(D\) cuts \(AC\) at \(A\) and \(E\), find the length of \(|AE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + 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 \\(ABC\\) be a triangle such that \\(|AB| = 7\\), \\(|BC| = 8\\), \\(|AC| = 6\\). Let \\(D\\) be the midpoint of s... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6c4169ad-8e47-4be4-9fa1-daf015d0198e"
} | [
{
"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 such that \\(|AB| = 7\\), \\(|BC| = 8\\), \\(|AC| = 6\\). Let \\(D\\) be the midpoint of s... |
Find, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_1$, $R_2$, $R_3$ and three distinct columns $C_1$, $C_2$, $C_3$, there exists an uncolored cell $c$ and integers $1 \le i, j \le 3$ so that $c$ lies in... | 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, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \\times 6$ grid ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f85ab70-da84-4b71-b483-2bb6f6838ecb"
} | [
{
"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, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \\times 6$ grid ... |
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