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 |
|---|---|---|---|---|---|---|---|
Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. What is $m+n$? | 455 | 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\nEach number in the list $1, 2, 3, \\ldots, 10$ is either colored red or blue. Numbers are colored independently, and b... | MATH | {
"ground_truth": "455",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d281c508-c811-4e81-b05a-cf8da525fc53"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach number in the list $1, 2, 3, \\ldots, 10$ is either colored red or blue. Numbers are colored independently, and b... |
What is the radius of a circle inscribed in a rhombus with diagonals of length \(10\) and \(24\)? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please find the value of \(k + m\). | 73 | 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 radius of a circle inscribed in a rhombus with diagonals of length \\(10\\) and \\(24\\)? Express your ans... | MATH | {
"ground_truth": "73",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d0877888-b357-44c7-8b84-5e0573765cc1"
} | [
{
"content": "Solve the following math problem step by step. 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 radius of a circle inscribed in a rhombus with diagonals of length \\(10\\) and \\(24\\)? Express your ans... |
A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$ . | 41 | 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 rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are r... | MATH | {
"ground_truth": "41",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cf04cb6a-1743-43fd-a0b6-9a4f3965379f"
} | [
{
"content": "Solve the following math problem step by step. 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 rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are r... |
Q. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$. | 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\nQ. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along w... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b96a23be-617d-4047-b651-0d5ba7223d0d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nQ. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along w... |
Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor $. | 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\nCalculate $ \\lfloor \\log_3 5 +\\log_5 7 +\\log_7 3 \\rfloor $.\n\nRemember to put your answer on its own line after ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "34bae0a6-9c55-4792-b584-f2d5fc57ed0b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCalculate $ \\lfloor \\log_3 5 +\\log_5 7 +\\log_7 3 \\rfloor $.\n\nRemember to put your answer on its own line after ... |
Assume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period, what is the maximum possible number of leap years? | 38 | 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\nAssume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period,... | MATH | {
"ground_truth": "38",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7522b0d1-6ccf-4c28-be36-3f383ef154b8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAssume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period,... |
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ such that $k^2_1 + k^2_2 + ... + k^2_n = 2002?$ | 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\nWhat is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ such that ... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "766a5953-ef8b-44d2-a1de-d00a2e676399"
} | [
{
"content": "Solve the following math problem step by step. 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 maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ such that ... |
Find the number of positive integers $x$ such that:
\[
\left\lfloor \frac{x}{99} \right\rfloor = \left\lfloor \frac{x}{101} \right\rfloor.
\] | 2499 | 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 $x$ such that:\n\\[\n\\left\\lfloor \\frac{x}{99} \\right\\rfloor = \\left\\lfloo... | MATH | {
"ground_truth": "2499",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "51a3a0ec-5e36-4776-b5d1-a2ca0323b6cd"
} | [
{
"content": "Solve the following math problem step by step. 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 $x$ such that:\n\\[\n\\left\\lfloor \\frac{x}{99} \\right\\rfloor = \\left\\lfloo... |
The diagram shows a $20 \times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions. | 200 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe diagram shows a $20 \\times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The p... | MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "937266a7-7d0c-47fc-99c0-58a36c43db58"
} | [
{
"content": "Solve the following math problem step by step. 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 a $20 \\times 20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The p... |
A function $f$ is defined on the positive integers by \[ \left\{ \begin{array}{rcl} f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array} \right. \] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$. | 92 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA function $f$ is defined on the positive integers by \\[ \\left\\{ \\begin{array}{rcl} f(1) &=& 1, \\\\ f(3) &=& 3, \... | MATH | {
"ground_truth": "92",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7cdf33cd-02c4-4e71-aa7d-0bc14f2fddb8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA function $f$ is defined on the positive integers by \\[ \\left\\{ \\begin{array}{rcl} f(1) &=& 1, \\\\ f(3) &=& 3, \... |
What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$? | 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\nWhat is the area of the region defined by the inequality $|3x-18|+|2y+7|\\le3$?\n\nRemember to put your answer on its ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a99dcd1d-017c-4bdd-b6a1-e422febe35dd"
} | [
{
"content": "Solve the following math problem step by step. 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 area of the region defined by the inequality $|3x-18|+|2y+7|\\le3$?\n\nRemember to put your answer on its ... |
In an isosceles triangle, the altitudes intersect on the incircle. Compute the cosine of the vertex angle.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of 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\nIn an isosceles triangle, the altitudes intersect on the incircle. Compute the cosine of the vertex angle.The answer ... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0434d6f9-30c3-4834-972c-6d385a388d09"
} | [
{
"content": "Solve the following math problem step by step. 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 an isosceles triangle, the altitudes intersect on the incircle. Compute the cosine of the vertex angle.The answer ... |
A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$.
Find the expression that represents the initial kinetic energy of the truck (i.e., the kinetic energy before the driver starts braking). The answer should be in the form of $kF \cdot mx$, where $k$ and $m$ are integers. Please give the value of $k + m$. | 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\nA truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The br... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6b461f67-dcad-4b0f-b33b-9e68555fe404"
} | [
{
"content": "Solve the following math problem step by step. 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 truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The br... |
What is the smallest possible value of \( x^2 + y^2 - x - y - xy \)? | -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 smallest possible value of \\( x^2 + y^2 - x - y - xy \\)?\n\nRemember to put your answer on its own line ... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "52dd2b05-0c5c-48bf-95ea-07ea92866a80"
} | [
{
"content": "Solve the following math problem step by step. 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 possible value of \\( x^2 + y^2 - x - y - xy \\)?\n\nRemember to put your answer on its own line ... |
A deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of five cards has cards numbered $1, 2, 3, 4, 5$, in that order. The two decks are riffle-shuffled together to form a deck with $13$ cards with the cards from each deck in the same order as they were originally. Thus, numbers on the cards might end up in the order $1122334455678$ or $1234512345678$ but not $1223144553678$. Find the number of possible sequences of the $13$ numbers. | 572 | 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 deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of \ffive cards has cards... | MATH | {
"ground_truth": "572",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1e751410-8233-42c3-bb22-cf462ea5a59b"
} | [
{
"content": "Solve the following math problem step by step. 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 deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of \ffive cards has cards... |
For $n \geq 1$, let $a_n$ be the number beginning with $n$ 9's followed by 744; for example, $a_4 = 9999744$. Define \[ f(n) = \max \{ m \in \mathbb{N} \mid 2^m \text{ divides } a_n \}, \] for $n \geq 1$. Find $f(1) + f(2) + f(3) + \cdots + f(10)$. | 75 | 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 $n \\geq 1$, let $a_n$ be the number beginning with $n$ 9's followed by 744; for example, $a_4 = 9999744$. Define ... | MATH | {
"ground_truth": "75",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dc982546-51de-4d46-b972-b072b41f0d5c"
} | [
{
"content": "Solve the following math problem step by step. 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 $n \\geq 1$, let $a_n$ be the number beginning with $n$ 9's followed by 744; for example, $a_4 = 9999744$. Define ... |
Let $f(x)$ be a third-degree polynomial with real coefficients satisfying
$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.$
Find $|f(0)|$ . | 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 $f(x)$ be a third-degree polynomial with real coefficients satisfying \n$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|... | MATH | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "80fcb024-be04-42a1-9e79-50a9112f7490"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(x)$ be a third-degree polynomial with real coefficients satisfying \n$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|... |
On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?
For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$. | 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\n\n\t On the Cartesian plane $\\mathbb{R}^2$, a circle is said to be \\textit{nice}\nif its center is at the origin $(0... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b660dfda-0fd5-449b-9c74-5d836c667e73"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n\n\t On the Cartesian plane $\\mathbb{R}^2$, a circle is said to be \\textit{nice}\nif its center is at the origin $(0... |
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$. | 91 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x$ be a real number between 0 and $\\tfrac{\\pi}{2}$ for which the function $3\\sin^2 x + 8\\sin x \\cos x + 9\\... | MATH | {
"ground_truth": "91",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c46faf23-0f45-465c-8fe6-5a8ab7c2890e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x$ be a real number between 0 and $\\tfrac{\\pi}{2}$ for which the function $3\\sin^2 x + 8\\sin x \\cos x + 9\\... |
(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits. | 1944 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that d... | MATH | {
"ground_truth": "1944",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1d9259cd-053f-45a1-bd03-f45a0600684f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that d... |
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$. | 30 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFrist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall t... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2ac12efc-4e1e-4b23-b23a-0aee01ecd549"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFrist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall t... |
Find the largest prime factor of $9951$. | 107 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the largest prime factor of $9951$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4eee24b2-69dd-4556-b22d-f623b0f2f056"
} | [
{
"content": "Solve the following math problem step by step. 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 largest prime factor of $9951$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role... |
Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183$ and $a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac{9}{5} (a + b)$. | 73 | 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 positive real numbers such that $a\\sqrt{a} + b\\sqrt{b} = 183$ and $a\\sqrt{b} + b\\sqrt{a} = 182$... | MATH | {
"ground_truth": "73",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "26a8e0fe-1d35-4bbd-93c4-4aff7313dffa"
} | [
{
"content": "Solve the following math problem step by step. 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 positive real numbers such that $a\\sqrt{a} + b\\sqrt{b} = 183$ and $a\\sqrt{b} + b\\sqrt{a} = 182$... |
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square, it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2 \times 4$ rectangle instead of $2 \times 3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.
A trip of the length $n$ of the knight is a sequence of $n$ squares $C_1, C_2, \ldots, C_n$ which are all distinct such that the knight starts at the $C_1$ square and for each $i$ from $1$ to $n-1$, it can use the movement described before to go from the $C_i$ square to the $C_{i+1}$.
Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5 \times 5$ board. | 12 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEsmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight ... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d2bd5597-131c-4a6b-a15e-c10ae9b618de"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEsmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight ... |
How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?
| 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 ordered pairs $(a, b)$ of positive integers with $1 \\le a, b \\le 10$ are there such that in the geometric s... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1451d018-c002-423b-a3f0-3c7ee9d5e766"
} | [
{
"content": "Solve the following math problem step by step. 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 ordered pairs $(a, b)$ of positive integers with $1 \\le a, b \\le 10$ are there such that in the geometric s... |
The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion:
\[1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, \dots\]
Find the $2014$th number in this sequence. | 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 first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by thre... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "13a636eb-a96e-4566-9a18-6f5fe19bb5c4"
} | [
{
"content": "Solve the following math problem step by step. 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 first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by thre... |
Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day? | 100 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAl and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's sche... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e55f6fe9-c8c9-46f0-8194-56e578beeb84"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAl and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's sche... |
A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex. | 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 parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordi... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8375c95c-b0c6-4376-a422-447278c4782d"
} | [
{
"content": "Solve the following math problem step by step. 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 parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordi... |
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$. | 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 remainder when $26!^{26} + 27!^{27}$ is divided by $29$.\n\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0afdae4e-2e85-4b13-8e37-537ac5e4e135"
} | [
{
"content": "Solve the following math problem step by step. 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 remainder when $26!^{26} + 27!^{27}$ is divided by $29$.\n\nRemember to put your answer on its own line after... |
Let $n$ be an arbitrary positive integer. Calculate \( S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n} \). | 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\nLet $n$ be an arbitrary positive integer. Calculate \\( S_n = \\sum_{r=0}^n 2^{r-2n} \\binom{2n-r}{n} \\).\n\nRemember... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6d164198-9c4c-43fa-8e0e-4a57dbe50198"
} | [
{
"content": "Solve the following math problem step by step. 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 $n$ be an arbitrary positive integer. Calculate \\( S_n = \\sum_{r=0}^n 2^{r-2n} \\binom{2n-r}{n} \\).\n\nRemember... |
Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$? | 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\nLet $f$ be the function defined by $f(x) = -2 \\sin(\\pi x)$. How many values of $x$ such that $-2 \\le x \\le 2$ sat... | MATH | {
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f118e3e6-4a13-41ac-a86a-c0490efbf3ff"
} | [
{
"content": "Solve the following math problem step by step. 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 $f$ be the function defined by $f(x) = -2 \\sin(\\pi x)$. How many values of $x$ such that $-2 \\le x \\le 2$ sat... |
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 | 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\nAlice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with t... | MATH | {
"ground_truth": "101",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "188cdcac-e290-4e39-9075-832ef26c75c7"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAlice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with t... |
Define $f(n) = \frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \leq 1000$ and $f(n)$ is the product of two prime numbers. | 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\nDefine $f(n) = \\frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \\leq 1000$ and $f(n)$... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6505b583-8c7d-49e8-8f7d-96f233be10b3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine $f(n) = \\frac{n^2 + n}{2}$. Compute the number of positive integers $n$ such that $f(n) \\leq 1000$ and $f(n)$... |
For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?
[asy]
draw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("8",(2.5,2.5),NW);
label("10",(8.5,3),NE);
label("16",(9.5, -3.5),SE);
label("12",(3.5,-4),SW);
[/asy] | 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\nFor the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the d... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "66d1d761-43c4-46ef-85d0-bc72e77312e1"
} | [
{
"content": "Solve the following math problem step by step. 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 the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the d... |
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | 997 | 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 $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in ... | MATH | {
"ground_truth": "997",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6591c811-425f-4480-a0b2-5ef7558e4db2"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in ... |
Let $a$, $b$, and $c$ be real numbers satisfying the equations:
\[
a^3 + abc = 26 \\
b^3 + abc = 78 \\
c^3 - abc = 104
\]
Find $a^3 + b^3 + c^3$. | 184 | 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 $a$, $b$, and $c$ be real numbers satisfying the equations:\n\\[\na^3 + abc = 26 \\\\\nb^3 + abc = 78 \\\\\nc^3 - ... | MATH | {
"ground_truth": "184",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8b88e563-491b-4f9d-a034-c3c7b1f05ae8"
} | [
{
"content": "Solve the following math problem step by step. 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 $a$, $b$, and $c$ be real numbers satisfying the equations:\n\\[\na^3 + abc = 26 \\\\\nb^3 + abc = 78 \\\\\nc^3 - ... |
Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying $$\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.$$ What is the maximal value of $a_1$? | 550 | 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 $a_1, a_2, \\ldots, a_{100}$ be positive integers, satisfying $$\\frac{a_1^2+a_2^2+\\ldots+a_{100}^2} {a_1+a_2+\\l... | MATH | {
"ground_truth": "550",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "66b160f6-5c6b-46db-b9ef-8142646a305a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a_1, a_2, \\ldots, a_{100}$ be positive integers, satisfying $$\\frac{a_1^2+a_2^2+\\ldots+a_{100}^2} {a_1+a_2+\\l... |
A sequence of real numbers $\{a_n\}_{n = 1}^\infty$ with $n=1,2,...$ has the following property:
\[
6a_n + 5a_{n-2} = 20 + 11a_{n-1} \quad (\text{for } n \geq 3).
\]
The first two elements are $a_1=0$ and $a_2=1$. Find the integer closest to $a_{2011}$. | 40086 | 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 sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty$ with $n=1,2,...$ has the following property:\n\\[\n6a_n + 5a_{n... | MATH | {
"ground_truth": "40086",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc51f432-79ac-4b72-aef0-d1e05c9ad155"
} | [
{
"content": "Solve the following math problem step by step. 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 sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty$ with $n=1,2,...$ has the following property:\n\\[\n6a_n + 5a_{n... |
Let $r$ be the remainder when $1342$ is divided by $13$.
Determine the smallest positive integer that has these two properties:
$\bullet~$ It is a multiple of $1342$.
$\bullet~$ Its remainder upon being divided by $13$ is smaller than $r$. | 6710 | 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 $r$ be the remainder when $1342$ is divided by $13$.\n\nDetermine the smallest positive integer that has these two... | MATH | {
"ground_truth": "6710",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2c052a28-5d9d-4314-93a0-fef57c372900"
} | [
{
"content": "Solve the following math problem step by step. 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 $r$ be the remainder when $1342$ is divided by $13$.\n\nDetermine the smallest positive integer that has these two... |
2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 18 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside ... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b176dbac-19fa-4676-8434-879e0bd83ce1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside ... |
A collection of $8$ cubes consists of one cube with edge - length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by $1000$? | 458 | 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 collection of $8$ cubes consists of one cube with edge - length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower i... | MATH | {
"ground_truth": "458",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "986c5e28-6d8a-49df-81bd-ee167e0c3ad1"
} | [
{
"content": "Solve the following math problem step by step. 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 collection of $8$ cubes consists of one cube with edge - length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower i... |
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$. | 38 | 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 $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minim... | MATH | {
"ground_truth": "38",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2d55a883-7acd-4f68-b452-280581e6836a"
} | [
{
"content": "Solve the following math problem step by step. 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 $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minim... |
A cubic block with dimensions $n \times n \times n$ is composed of $1 \times 1 \times 1$ unit cubes. Determine the smallest value of $n$ such that, after removing the outer two layers of unit cubes, more than half of the original unit cubes remain. | 20 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA cubic block with dimensions $n \\times n \\times n$ is composed of $1 \\times 1 \\times 1$ unit cubes. Determine the... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9b3df0f1-51fd-4ca1-aa30-23f572da1378"
} | [
{
"content": "Solve the following math problem step by step. 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 cubic block with dimensions $n \\times n \\times n$ is composed of $1 \\times 1 \\times 1$ unit cubes. Determine the... |
Find the number of two-digit positive integers whose digits total $7$. | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of two-digit positive integers whose digits total $7$.\n\nRemember to put your answer on its own line ... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8fc763b9-40c4-466c-8ee9-6ba61073865c"
} | [
{
"content": "Solve the following math problem step by step. 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 two-digit positive integers whose digits total $7$.\n\nRemember to put your answer on its own line ... |
Circles with centers \(A\) and \(B\) have radius \(3\) and \(8\), respectively. A common internal tangent intersects the circles at \(C\) and \(D\), respectively. Lines \(AB\) and \(CD\) intersect at \(E\), and \(AE=5\). Find the length of \(CD\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 47 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCircles with centers \\(A\\) and \\(B\\) have radius \\(3\\) and \\(8\\), respectively. A common internal tangent inte... | MATH | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "aafd2477-11fa-450f-9b29-2a72a826c2e6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCircles with centers \\(A\\) and \\(B\\) have radius \\(3\\) and \\(8\\), respectively. A common internal tangent inte... |
Find all natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers. | 7 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers.\n\... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ee7300fb-b25e-4c2c-a62e-b0cff84d90a7"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers.\n\... |
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\cdots f}_k(n)\cdots)).$ Provide the value of $f(1) + g(1)$. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6f213b19-882f-4abf-b32d-457d65571da8"
} | [
{
"content": "Solve the following math problem step by step. 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 pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(... |
Let $a$ and $b$ be real numbers satisfying \[a^4 + 8b = 4(a^3 - 1) - 16 \sqrt{3}\] and \[b^4 + 8a = 4(b^3 - 1) + 16 \sqrt{3}.\] Find $a^4 + b^4$. | 56 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be real numbers satisfying \\[a^4 + 8b = 4(a^3 - 1) - 16 \\sqrt{3}\\] and \\[b^4 + 8a = 4(b^3 - 1) + 1... | MATH | {
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "30ae83fb-09ab-4e48-b338-f26a2853bb2a"
} | [
{
"content": "Solve the following math problem step by step. 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 real numbers satisfying \\[a^4 + 8b = 4(a^3 - 1) - 16 \\sqrt{3}\\] and \\[b^4 + 8a = 4(b^3 - 1) + 1... |
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 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\nDave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent... | MATH | {
"ground_truth": "52",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "47968ecd-4c02-49ba-bb80-ef35a6fb3e44"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent... |
Find the sum of all positive integers $n$ such that $$ \frac{n+11}{\sqrt{n-1}} $$ is an integer. | 216 | 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 $$ \\frac{n+11}{\\sqrt{n-1}} $$ is an integer.\n\nRemember to put ... | MATH | {
"ground_truth": "216",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "76783701-ddc3-4ac5-a283-7c5281350551"
} | [
{
"content": "Solve the following math problem step by step. 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 $$ \\frac{n+11}{\\sqrt{n-1}} $$ is an integer.\n\nRemember to put ... |
A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\triangle ADG$. The original answer is in the form k + m\sqrt{2}, where k and m are integers. Please find the value of k + m. | 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\nA regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\\triangle ADG$. The original answer is in the... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2d6e056a-f69a-4ff9-80a5-c800fb0ea158"
} | [
{
"content": "Solve the following math problem step by step. 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 regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\\triangle ADG$. The original answer is in the... |
What is the positive difference between the greatest and the least member of the set $\left\{\frac{3}{7},\frac{4}{3},\frac{11}{8},\frac{6}{16}\right\}$? Express your answer in simplest form. | 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 positive difference between the greatest and the least member of the set $\\left\\{\\frac{3}{7},\\frac{4}{... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc7496b8-b606-4e4c-9fa6-7f9f87976032"
} | [
{
"content": "Solve the following math problem step by step. 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 positive difference between the greatest and the least member of the set $\\left\\{\\frac{3}{7},\\frac{4}{... |
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? The original answer is in the format $a - \frac{b}{\pi}$, please provide the value of $a + b $. | 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\nSuppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on th... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ca27cce9-8f8b-442d-b12d-c6fbe37402ae"
} | [
{
"content": "Solve the following math problem step by step. 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 the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on th... |
A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius? Provide your answer as a number. | 100 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in th... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8df23f6c-ffa6-4642-8fcc-275c37bbc966"
} | [
{
"content": "Solve the following math problem step by step. 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 with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in th... |
Pratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, while Payel multiplies his number by $2$ and then adds $2$ to it. In other words, on the first day, their numbers will be $(2n - 2)$ and $(2m + 2)$ respectively. Find the minimum integer $x$ with proof such that if $n - m \geq x$, then Pratyya's number will be larger than Payel's number every day. | 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\nPratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his numbe... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "24babe9c-c6d8-4f88-8216-21c01a8c6751"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPratyya and Payel each have a number, $n$ and $m$ respectively, where $n > m$. Every day, Pratyya multiplies his numbe... |
Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 904 | 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\nHenry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability ... | MATH | {
"ground_truth": "904",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "02c84339-6bdd-47c5-a84f-e8088e13bf89"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHenry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability ... |
A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For every distinct pair of lights $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 < x_2$ and $y_1 > y_2$, all lights strictly between them (i.e., $x_1 < x < x_2$ and $y_2 < y < y_1$) are toggled. After this procedure is done, how many lights are on? | 1022121 | 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 $2023 \\times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For ev... | MATH | {
"ground_truth": "1022121",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc780155-dc32-40c2-9b1d-d200fe62043a"
} | [
{
"content": "Solve the following math problem step by step. 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 $2023 \\times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x, y)$. For ev... |
Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$. | 1009 | 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\nEach of the squares in a $2 \\times 2018$ grid is to be colored black or white such that in any $2 \\times 2$ block, a... | MATH | {
"ground_truth": "1009",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3c715d66-66a0-416b-8cdc-b35a219949a5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEach of the squares in a $2 \\times 2018$ grid is to be colored black or white such that in any $2 \\times 2$ block, a... |
In the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs "The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.
Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$.
Let $N$ be the smallest positive integer so that
\[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]
is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$. | 1040 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs \"The Process.\" I... | MATH | {
"ground_truth": "1040",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5b979ae2-dc7f-4570-9c5b-d48f3d2fb29a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs \"The Process.\" I... |
Find $A+B$ (in base 10), given the following addition problem \[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}& & & 4 & A & B_{6}\\ &+& & & 4 & 1_{6}\\ \cline{2-6}& & & 5 & 3 & A_{6}\\ \end{array} \] | 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\nFind $A+B$ (in base 10), given the following addition problem \\[ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}& & & 4 & A... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0bfb68f3-b3b0-4b31-9979-e4a8eac06335"
} | [
{
"content": "Solve the following math problem step by step. 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 $A+B$ (in base 10), given the following addition problem \\[ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}& & & 4 & A... |
Jay has a $24 \times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e., it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, such that after each toggle, he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules expended by all the lights after all $48$ toggles. | 9408 | 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\nJay has a $24 \\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a swit... | MATH | {
"ground_truth": "9408",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49620ede-6d1b-4763-ab93-c5b448b86854"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJay has a $24 \\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a swit... |
On the refrigerator, MATHEMATICS is spelled out with $11$ magnets, one letter per magnet. Two vowels and four consonants fall off and are put away in a bag. If the T's, M's, and A's are indistinguishable, how many distinct possible collections of letters could be put in the bag? | 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\nOn the refrigerator, MATHEMATICS is spelled out with $11$ magnets, one letter per magnet. Two vowels and four consonan... | MATH | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e8e52ee1-2c79-40fc-ba8b-b01ac246d1af"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nOn the refrigerator, MATHEMATICS is spelled out with $11$ magnets, one letter per magnet. Two vowels and four consonan... |
What is the $1000^{\rm th}$ positive integer with an odd number of digits? | 10090 | 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 $1000^{\\rm th}$ positive integer with an odd number of digits?\n\nRemember to put your answer on its own ... | MATH | {
"ground_truth": "10090",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "33db6363-ad35-4a5b-b869-bdc88ae973d3"
} | [
{
"content": "Solve the following math problem step by step. 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 $1000^{\\rm th}$ positive integer with an odd number of digits?\n\nRemember to put your answer on its own ... |
Sristan Thin is walking around the Cartesian plane. From any point \((x, y)\), Sristan can move to \((x+1, y)\) or \((x+1, y+3)\). How many paths can Sristan take from \((0, 0)\) to \((9, 9)\)? | 84 | 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\nSristan Thin is walking around the Cartesian plane. From any point \\((x, y)\\), Sristan can move to \\((x+1, y)\\) or... | MATH | {
"ground_truth": "84",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b5cd1031-2482-41dc-b714-341da89c2a14"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSristan Thin is walking around the Cartesian plane. From any point \\((x, y)\\), Sristan can move to \\((x+1, y)\\) or... |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 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\nRegular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?The answer is... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "59988403-02a9-4211-9c1c-a531bf857fa6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRegular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?The answer is... |
Ana has \(22\) coins. She can take from her friends either \(6\) coins or \(18\) coins, or she can give \(12\) coins to her friends. She can perform these operations as many times as she wants. Find the least number of coins Ana can have. | 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\nAna has \\(22\\) coins. She can take from her friends either \\(6\\) coins or \\(18\\) coins, or she can give \\(12\\)... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3aa18105-9190-4611-a6cc-6463336b44f0"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAna has \\(22\\) coins. She can take from her friends either \\(6\\) coins or \\(18\\) coins, or she can give \\(12\\)... |
Assume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of Lena, Wisconsin, is at $45^{\circ}$ North Latitude, exactly halfway between the equator and the North Pole. What is the number of miles in the circumference of the circle on Earth parallel to the equator and through Lena, Wisconsin? Express your answer to the nearest hundred miles. (You may use a calculator for this problem.)
[asy]
size(4.5cm,4.5cm);
draw(unitcircle);
draw((-1,0)..(0,-0.2)..(1,0));
draw((-0.95,0.05)..(0,0.2)..(0.97,0.05),1pt+dotted);
draw((-0.7,0.7)..(0,0.6)..(0.7,0.7));
draw((-0.65,0.75)..(0,0.8)..(0.66,0.75),1pt+dotted);
dot((0,0));
draw((0,0)--(1,0));
draw((0,0)--(0.7,0.7));
dot((0.7,0.7));
dot((0,0.72));
label("Lena",(0.7,0.7),ENE);
label("$45^\circ$",shift(0.3,0.1)*(0,0));
[/asy] | 17700 | 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\nAssume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of ... | MATH | {
"ground_truth": "17700",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "153d8e7f-cd07-4a82-8486-3d76707868af"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAssume that the length of Earth's equator is exactly 25,100 miles and that the Earth is a perfect sphere. The town of ... |
Find the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \dots, a_n$, the product of all differences $a_i - a_j$ for $i < j$ is divisible by $1991$. | 182 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \\dots, a_n$, the product of... | MATH | {
"ground_truth": "182",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb152935-7af3-4dab-9fee-7ad1721af3d8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer $n$ such that for any $n$ distinct integers $a_1, a_2, \\dots, a_n$, the product of... |
Compute the value of the expression
\[ 2009^4 - 4 \times 2007^4 + 6 \times 2005^4 - 4 \times 2003^4 + 2001^4 \, .\] | 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\nCompute the value of the expression\n\\[ 2009^4 - 4 \\times 2007^4 + 6 \\times 2005^4 - 4 \\times 2003^4 + 2001^4 \\, ... | MATH | {
"ground_truth": "384",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5547763a-a35e-493b-8672-05f464474923"
} | [
{
"content": "Solve the following math problem step by step. 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 value of the expression\n\\[ 2009^4 - 4 \\times 2007^4 + 6 \\times 2005^4 - 4 \\times 2003^4 + 2001^4 \\, ... |
Let $A_{12}$ denote the answer to problem $12$. There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}>B>C>D>0$ and \[\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C},\] where $\overline{A_{12}BCD}$ denotes the four digit base $10$ integer. Compute $B+C+D$. | 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\nLet $A_{12}$ denote the answer to problem $12$. There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f99c1671-eb39-4fd8-a259-1ad7367f6dcc"
} | [
{
"content": "Solve the following math problem step by step. 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 $A_{12}$ denote the answer to problem $12$. There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}... |
The function defined by
\[f(x) = \left\{
\begin{array}{cl}
x + k & \text{if $x < 4$}, \\
2x - 3 & \text{if $x \ge 4$}
\end{array}
\right.\]has an inverse, and the inverse is defined for all real numbers. Enter all possible values of $k,$ separated by commas. | 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\nThe function defined by\n\\[f(x) = \\left\\{\n\\begin{array}{cl}\nx + k & \\text{if $x < 4$}, \\\\\n2x - 3 & \\text{if... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ccca0755-76fa-4b39-aad2-1137db0a7818"
} | [
{
"content": "Solve the following math problem step by step. 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 function defined by\n\\[f(x) = \\left\\{\n\\begin{array}{cl}\nx + k & \\text{if $x < 4$}, \\\\\n2x - 3 & \\text{if... |
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. | 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\nTwo distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ suc... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2570e737-8b4b-4b73-a9d7-846c61d189ef"
} | [
{
"content": "Solve the following math problem step by step. 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 distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ suc... |
Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8$, $4$, $2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What is the minimum value for $n$? | 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\nHalfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ p... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "eeb129bf-062f-494e-b517-1a4f63c7d596"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHalfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ p... |
a) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b) How many distinct ways are there of painting the faces of a dodecahedron with 12 different colors? (Colorations are considered distinct if they do not coincide when the dodecahedron is rotated.) Please provide the sum of the distinct ways for both a) and b). | 7983390 | 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) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered ... | MATH | {
"ground_truth": "7983390",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0f6a80b1-4ba5-4589-8b67-f22f7973e147"
} | [
{
"content": "Solve the following math problem step by step. 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) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered ... |
Let $S$ be the set of points in the Cartesian plane that satisfy
$\Big|\big||x|-2\big|-1\Big|+\Big|\big||y|-2\big|-1\Big|=1.$
If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime number. Find $a+b$.
| 66 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n\nLet $S$ be the set of points in the Cartesian plane that satisfy \n$\\Big|\\big||x|-2\\big|-1\\Big|+\\Big|\\big||y|-... | MATH | {
"ground_truth": "66",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0f6b41c8-7d16-4e25-949e-b9c886154709"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n\nLet $S$ be the set of points in the Cartesian plane that satisfy \n$\\Big|\\big||x|-2\\big|-1\\Big|+\\Big|\\big||y|-... |
You roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of the three dice can be written as $\frac{a}{b}$ in simplest form. Find $a+b$. | 706 | 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\nYou roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of ... | MATH | {
"ground_truth": "706",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc7bcbba-8f48-4973-845e-cf514dd94c40"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nYou roll three fair six-sided dice. Given that the highest number you rolled is $5$, the expected value of the sum of ... |
Parallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? | 784 | 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\nParallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. ... | MATH | {
"ground_truth": "784",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "60c61eaa-cbe2-4dc0-a2a3-b594fbb509df"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nParallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. ... |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $247, 475$, and $756$ and end with the term $824$. Let $S$ be the sum of all the terms in the sequence. Find the largest prime factor that always divides $S$. | 37 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA finite sequence of three-digit integers has the property that the tens and units digits of each term are, respective... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "81da3db9-2790-4c45-86d4-13428e95698e"
} | [
{
"content": "Solve the following math problem step by step. 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 finite sequence of three-digit integers has the property that the tens and units digits of each term are, respective... |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of 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\nAmong all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, y \\le 10 \\... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f54cc5c0-90a4-4eb5-8275-870bafac0e08"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAmong all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, y \\le 10 \\... |
If $x$, $y$, $z$ are real numbers satisfying:
\[
\begin{align*}
(x + 1)(y + 1)(z + 1) &= 3, \\
(x + 2)(y + 2)(z + 2) &= -2, \\
(x + 3)(y + 3)(z + 3) &= -1,
\end{align*}
\]
find the value of
\[ (x + 20)(y + 20)(z + 20). \] | 6748 | 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 $x$, $y$, $z$ are real numbers satisfying:\n\\[\n\\begin{align*}\n(x + 1)(y + 1)(z + 1) &= 3, \\\\\n(x + 2)(y + 2)(... | MATH | {
"ground_truth": "6748",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "744526b6-c8d5-4846-82ae-e58487319719"
} | [
{
"content": "Solve the following math problem step by step. 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 $x$, $y$, $z$ are real numbers satisfying:\n\\[\n\\begin{align*}\n(x + 1)(y + 1)(z + 1) &= 3, \\\\\n(x + 2)(y + 2)(... |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | 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\nSuppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$,... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0853b3f1-4c51-4709-ab10-139bf4acb99a"
} | [
{
"content": "Solve the following math problem step by step. 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 the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$,... |
How many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$? | 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\nHow many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?\n\nRemember to put your ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3ff3161b-3a93-4502-92fe-c97f93395ce7"
} | [
{
"content": "Solve the following math problem step by step. 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 integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?\n\nRemember to put your ... |
Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$, where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required? | 660 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcabc... | MATH | {
"ground_truth": "660",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7ab565ce-fa09-4382-8a0f-c3276bcd71f1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.abcabc... |
Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\triangle ABC$ out of paper and then folding $\triangle AFD$ up along $FD$, folding $\triangle BED$ up along $DE$, and folding $\triangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$. | 80 | 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 $\\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of... | MATH | {
"ground_truth": "80",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "24106a15-7dc9-4127-96c8-03220d31707e"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of... |
What is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$? | 1010527 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$?\n\nRemember to put your answer... | MATH | {
"ground_truth": "1010527",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "355dd92d-df0d-4fab-b76f-87b96702340c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$?\n\nRemember to put your answer... |
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. Find the difference between the largest and smallest numbers possible in the top cell. | 26 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added... | MATH | {
"ground_truth": "26",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4beca746-6e4c-4d9c-931e-c08b176bd4c8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added... |
A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes. | 258 | 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 rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengt... | MATH | {
"ground_truth": "258",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2579fd32-fdb4-46c5-8a9d-2aa462caf1d5"
} | [
{
"content": "Solve the following math problem step by step. 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 rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengt... |
For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursively as $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$. | 256 | 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 given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursiv... | MATH | {
"ground_truth": "256",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bd00a106-0efb-4097-be39-7beeedb0f575"
} | [
{
"content": "Solve the following math problem step by step. 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 given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursiv... |
Define the function $f: \mathbb{R} \setminus \{-1,1\} \to \mathbb{R}$ as follows:
\[ f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} . \]
Suppose that $f(y) - f\left(\frac{1}{y}\right) = 2016$. Then, express $y$ in its simplest form as $\frac{p}{q}$. Find $p+q$. | 4033 | 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\nDefine the function $f: \\mathbb{R} \\setminus \\{-1,1\\} \\to \\mathbb{R}$ as follows:\n\\[ f(x) = \\sum_{a,b=0}^{\\i... | MATH | {
"ground_truth": "4033",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2af78de9-9097-4b1e-a708-1fe6f1f70878"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine the function $f: \\mathbb{R} \\setminus \\{-1,1\\} \\to \\mathbb{R}$ as follows:\n\\[ f(x) = \\sum_{a,b=0}^{\\i... |
What is the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|?\) Express your answer in the form \(a\pi + b\), where \(a\) and \(b\) are constants. Please provide the value of \(a + b\). | 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\nWhat is the area of the region enclosed by the graph of the equation \\(x^2+y^2=|x|+|y|?\\) Express your answer in the... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f22a0547-7c9e-4c61-910d-e8e55b2a2708"
} | [
{
"content": "Solve the following math problem step by step. 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 area of the region enclosed by the graph of the equation \\(x^2+y^2=|x|+|y|?\\) Express your answer in the... |
For how many pairs of consecutive integers in $\{1000,1001,1002,\ldots,2000\}$ is no carrying required when the two integers are added? | 156 | 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 pairs of consecutive integers in $\\{1000,1001,1002,\\ldots,2000\\}$ is no carrying required when the two... | MATH | {
"ground_truth": "156",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6f0df948-8086-4160-a9da-4773c04ea06b"
} | [
{
"content": "Solve the following math problem step by step. 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 pairs of consecutive integers in $\\{1000,1001,1002,\\ldots,2000\\}$ is no carrying required when the two... |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered? | 80 | 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\nMrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spread... | MATH | {
"ground_truth": "80",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "66932207-ccbc-4fed-b006-a80f7f3c1fbd"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spread... |
Lupe went to the store and paid for her purchase with a $\$ 10$ bill. She found that the digits making the amount of her purchase could be rearranged to make the amount she received back in change. If her purchase amount and her change amount were different and each amount was at least $\$1 $, how many possible amounts of change could she have received? | 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\nLupe went to the store and paid for her purchase with a $\\$ 10$ bill. She found that the digits making the amount of ... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "eb432674-959a-4ee9-a2b7-5c8496a3bf86"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLupe went to the store and paid for her purchase with a $\\$ 10$ bill. She found that the digits making the amount of ... |
Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$
| 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\nConsider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f399477f-55ff-4801-9232-98464b12577e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that... |
In the figure below, if the area of $\triangle ABC$ is 27, what is the value of $p$? [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (2, 12); pair t = (2, 0); pair c = (0, 9);
draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(q--a--b);
//draw(a--t);
draw(a--c--b);
label("$Q(0, 12)$", q, W);
label("$A(2, 12)$", a, NE);
label("$B(12, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
//label("$T(2, 0)$", t, S + 0.6 * E);
label("$C(0, p)$", c, W);
[/asy] | 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\nIn the figure below, if the area of $\\triangle ABC$ is 27, what is the value of $p$? [asy]\nsize(5cm);defaultpen(font... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c0df1b6d-bb8a-4c0e-bbf4-07892131597d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the figure below, if the area of $\\triangle ABC$ is 27, what is the value of $p$? [asy]\nsize(5cm);defaultpen(font... |
Determine the exact value of
\[\sqrt{\left( 2 - \sin^2 \frac{\pi}{7} \right) \left( 2 - \sin^2 \frac{2 \pi}{7} \right) \left( 2 - \sin^2 \frac{3 \pi}{7} \right)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of 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\nDetermine the exact value of\n\\[\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1d0efdab-3709-4c01-954e-f841c32bbbc2"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the exact value of\n\\[\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi... |
How many fractions in the form $\frac{n}{99}$, with $0<n<99$, are in lowest terms? | 60 | 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 fractions in the form $\\frac{n}{99}$, with $0<n<99$, are in lowest terms?\n\nRemember to put your answer on ... | MATH | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f62c398d-4f03-47e2-9f14-421d1a64d817"
} | [
{
"content": "Solve the following math problem step by step. 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 fractions in the form $\\frac{n}{99}$, with $0<n<99$, are in lowest terms?\n\nRemember to put your answer on ... |
$2017$ voters vote by submitting a ranking of the integers $\{1, 2, ..., 38\}$ from favorite (a vote for that value in $1$st place) to least favorite (a vote for that value in $38$th/last place). Let $a_k$ be the integer that received the most $k$th place votes (the smallest such integer if there is a tie). Find the maximum possible value of $\Sigma_{k=1}^{38}
a_k$.
| 1442 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$2017$ voters vote by submitting a ranking of the integers $\\{1, 2, ..., 38\\}$ from favorite (a vote for that value ... | MATH | {
"ground_truth": "1442",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4e8b29fe-5220-44f1-b8a0-d8cd7446ebc6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n$2017$ voters vote by submitting a ranking of the integers $\\{1, 2, ..., 38\\}$ from favorite (a vote for that value ... |
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there? | 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\nTogether, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de157406-4d69-4f1a-b7fb-cee8dd75b420"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTogether, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is... |
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether? | 1990 | 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 numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are w... | MATH | {
"ground_truth": "1990",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f44fc70c-16ba-4fe5-922d-76219f4490b3"
} | [
{
"content": "Solve the following math problem step by step. 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 numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are w... |
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