prompt stringlengths 41 2k | solution stringlengths 1 22 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | content listlengths 1 1 |
|---|---|---|---|---|---|---|---|
As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$ and $AB > 14$, compute the minimal area of $\triangle AEG$. | 98 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAs in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ ... | MATH | {
"ground_truth": "98",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49d10c1c-56bc-40cd-9791-3d416872aa6d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAs in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ ... |
What is the sum of the prime factors of 85,085? | 53 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of the prime factors of 85,085?\n\nRemember to put your answer on its own line after \"Answer:\".",
... | MATH | {
"ground_truth": "53",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ae1814f9-8b02-4c23-8e49-5266d4d6562b"
} | [
{
"content": "Solve the following math problem step by step. 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 sum of the prime factors of 85,085?\n\nRemember to put your answer on its own line after \"Answer:\".",
... |
Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has three of their symbols in a... | 148 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAzar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then ... | MATH | {
"ground_truth": "148",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b70fc505-0800-49f0-99de-1710e7004315"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAzar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then ... |
Suppose that $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of $g$ is $[-2,1]$. The range of $f(x) \cdot g(x)$ is $[a,b]$. What is the largest possible value of $b$? | 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\nSuppose that $f(x)$ and $g(x)$ are functions on $\\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b18cd1af-d89d-45ae-9476-2c648b3ed5af"
} | [
{
"content": "Solve the following math problem step by step. 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 $f(x)$ and $g(x)$ are functions on $\\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of... |
Max has a spinner that lands on 1 with a probability of $\frac{1}{2}$, lands on 2 with a probability of $\frac{1}{4}$, lands on 3 with a probability of $\frac{1}{6}$, and lands on 4 with a probability of $\frac{1}{12}$. If Max spins the spinner, and then Zack spins the spinner, then what is the probability that Max get... | 191 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMax has a spinner that lands on 1 with a probability of $\\frac{1}{2}$, lands on 2 with a probability of $\\frac{1}{4}... | MATH | {
"ground_truth": "191",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "18ebac67-d1f5-4d6b-9da5-7bbc0ae51078"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMax has a spinner that lands on 1 with a probability of $\\frac{1}{2}$, lands on 2 with a probability of $\\frac{1}{4}... |
Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every ... | 1151 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a ga... | MATH | {
"ground_truth": "1151",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6c2e988d-7f11-4536-bac8-fd781c21eae1"
} | [
{
"content": "Solve the following math problem step by step. 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 uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a ga... |
Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \] | 215 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate \\[ \\frac{1}{729} \\sum_{a=1}^{9} \\sum_{b=1}^9 \\sum_{c=1}^9 \\left( abc+ab+bc+ca+a+b+c \\right). \\]\n\nRe... | MATH | {
"ground_truth": "215",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b338b15b-7802-4c7b-a90c-d0f697238038"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate \\[ \\frac{1}{729} \\sum_{a=1}^{9} \\sum_{b=1}^9 \\sum_{c=1}^9 \\left( abc+ab+bc+ca+a+b+c \\right). \\]\n\nRe... |
A right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of its area to its perimeter is a perfect square. Find the minimum possible area of this triangle. | 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\nA right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "108893cd-6138-40ad-9196-47a297b86eb6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA right triangle has the property that its sides are pairwise relatively prime positive integers and that the ratio of... |
Suppose \(a\) is a real number such that the equation
\[ a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)} \]
has more than one solution in the interval \((0, \pi)\). The set of all such \(a\) can be written in the form \((p,q) \cup (q,r),\) where \(p, q,\) and \(r\) are real numbers with \(p < q < r\). What is the value of \(p... | -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\nSuppose \\(a\\) is a real number such that the equation \n\\[ a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)} \\]\nhas mor... | MATH | {
"ground_truth": "-4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "87436320-bdf0-44d2-aa25-ea41448b999e"
} | [
{
"content": "Solve the following math problem step by step. 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\\) is a real number such that the equation \n\\[ a\\cdot(\\sin{x}+\\sin{(2x)}) = \\sin{(3x)} \\]\nhas mor... |
As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other p... | 40 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAs a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to d... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3bdf1bcd-52f4-43cb-a3aa-4221285c7959"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAs a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to d... |
A man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and str... | 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\nA man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ae939b57-6735-445a-a623-0abfad69b8ea"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA man with mass $m$ jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum... |
Let $a$ and $b$ be positive integers such that $1 \leq a < b \leq 100$. If there exists a positive integer $k$ such that $ab \mid a^k + b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs. | 96 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive integers such that $1 \\leq a < b \\leq 100$. If there exists a positive integer $k$ such ... | MATH | {
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "efde2a96-7019-476a-8368-60fb939e98df"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive integers such that $1 \\leq a < b \\leq 100$. If there exists a positive integer $k$ such ... |
For some positive integers $m > n$, the quantities $a = \text{lcm}(m, n)$ and $b = \gcd(m, n)$ satisfy $a = 30b$. If $m - n$ divides $a$, then what is the value of $\frac{m + n}{b}$? | 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\nFor some positive integers $m > n$, the quantities $a = \\text{lcm}(m, n)$ and $b = \\gcd(m, n)$ satisfy $a = 30b$. If... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1cd86045-bfd8-4525-99c6-bc88dd21265d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor some positive integers $m > n$, the quantities $a = \\text{lcm}(m, n)$ and $b = \\gcd(m, n)$ satisfy $a = 30b$. If... |
Suppose $PABCD$ is a right square pyramid with apex $P$ and base $ABCD$. If $PBD$ is an equilateral triangle with side length 6, then what is the volume of $PABCD$?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 21 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose $PABCD$ is a right square pyramid with apex $P$ and base $ABCD$. If $PBD$ is an equilateral triangle with sid... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ffffecb7-706a-4a94-ad1c-a5e60f8cd5de"
} | [
{
"content": "Solve the following math problem step by step. 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 $PABCD$ is a right square pyramid with apex $P$ and base $ABCD$. If $PBD$ is an equilateral triangle with sid... |
A long string is folded in half eight times, then cut in the middle. How many pieces are obtained? | 257 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 long string is folded in half eight times, then cut in the middle. How many pieces are obtained?\n\nRemember to put ... | MATH | {
"ground_truth": "257",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a8324853-2563-405c-8dc9-ef223b3c6c7b"
} | [
{
"content": "Solve the following math problem step by step. 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 long string is folded in half eight times, then cut in the middle. How many pieces are obtained?\n\nRemember to put ... |
How many ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer. | 3456 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer.... | MATH | {
"ground_truth": "3456",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6de1a817-d83d-4520-918e-0f1cb94e65c7"
} | [
{
"content": "Solve the following math problem step by step. 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 ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer.... |
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resultin... | 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\nSteve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ ... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "71f8c643-b424-4936-9884-83e5e9b522a7"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSteve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ ... |
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the th... | 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 data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third qua... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d5c592fd-4cd7-4244-bf56-f1fdf61ed717"
} | [
{
"content": "Solve the following math problem step by step. 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 data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third qua... |
A circle of radius \(1\) is internally tangent to two circles of radius \(2\) at points \(A\) and \(B\), where \(AB\) is a diameter of the smaller circle. Find the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles. The original answer is in the form ... | 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\nA circle of radius \\(1\\) is internally tangent to two circles of radius \\(2\\) at points \\(A\\) and \\(B\\), where... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ca31f6e3-c4c4-464b-a70b-2cabc0e7e1db"
} | [
{
"content": "Solve the following math problem step by step. 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 circle of radius \\(1\\) is internally tangent to two circles of radius \\(2\\) at points \\(A\\) and \\(B\\), where... |
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\} = x - \lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying:
\[\{x\} = \frac{x + \lfloor x \rfloor + \lfloor x + \frac{1}{2} \rfloor}{20}\]
If $S$ is the sum of a... | 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\nFor any real number $x$, let $\\lfloor x \\rfloor$ denote the integer part of $x$; $\\{ x \\}$ be the fractional part ... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5be0dd35-1dff-48a4-93ae-a3ba9abb8819"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any real number $x$, let $\\lfloor x \\rfloor$ denote the integer part of $x$; $\\{ x \\}$ be the fractional part ... |
Find the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \ge 0.$ | 0 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \\ge 0.$\n\nRemem... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "88919916-8e9f-4d8b-b106-0c9671f7055a"
} | [
{
"content": "Solve the following math problem step by step. 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 squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \\ge 0.$\n\nRemem... |
Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the ... | 840 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRectangle $ABCD$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ i... | MATH | {
"ground_truth": "840",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "40c352c9-a122-423a-b73b-a3d8527167f5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRectangle $ABCD$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ i... |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is the value of $k$? | 991 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 product $(8)(888\\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$... | MATH | {
"ground_truth": "991",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "15593fc5-9d6b-4e52-94bd-d9dd6aef956c"
} | [
{
"content": "Solve the following math problem step by step. 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 product $(8)(888\\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$... |
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
| 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\nHow many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? F... | MATH | {
"ground_truth": "256",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4a9596ca-39d0-4baa-8bac-8cbebb406cac"
} | [
{
"content": "Solve the following math problem step by step. 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 subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? F... |
Find the smallest number $n$ such that any set of $n$ points in a Cartesian plane, all of them with integer coordinates, contains two points such that the square of their mutual distance is a multiple of $2016$. | 14113 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 number $n$ such that any set of $n$ points in a Cartesian plane, all of them with integer coordinate... | MATH | {
"ground_truth": "14113",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f814d4c0-4ac0-4e22-b7d3-75ff6cbd27ad"
} | [
{
"content": "Solve the following math problem step by step. 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 number $n$ such that any set of $n$ points in a Cartesian plane, all of them with integer coordinate... |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. Find its area in the form of $x\sqrt{3}$. What is the value of $x$? | 25 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRegular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. Find its area in the form of $... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7c6e7419-b429-4c76-9acf-0326d7ef3530"
} | [
{
"content": "Solve the following math problem step by step. 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. Find its area in the form of $... |
The following line is parameterized, so that its direction vector is of the form $\begin{pmatrix} 2 \\ b \end{pmatrix}.$ Find $b.$
[asy]
unitsize(0.4 cm);
pair A, B, L, R;
int i, n;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
draw((-8,i)--(8,i),gray(0.7));
}
draw((-8,0)--(8,0),Arrows(6));
draw((... | 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\nThe following line is parameterized, so that its direction vector is of the form $\\begin{pmatrix} 2 \\\\ b \\end{pmat... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0f64e9ec-bc40-4ca8-b80f-3ae2b0dd8d41"
} | [
{
"content": "Solve the following math problem step by step. 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 following line is parameterized, so that its direction vector is of the form $\\begin{pmatrix} 2 \\\\ b \\end{pmat... |
George is about to get a certain amount of change less than one dollar from the cash register. If he gets the most quarters possible and the rest in pennies, he would need to receive 3 pennies to meet the amount. If he gets the most dimes possible and the rest in pennies, he would need to receive 8 pennies to meet the ... | 106 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGeorge is about to get a certain amount of change less than one dollar from the cash register. If he gets the most qua... | MATH | {
"ground_truth": "106",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "24ec4c12-04ca-4aab-80ee-4ddac37e60ef"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGeorge is about to get a certain amount of change less than one dollar from the cash register. If he gets the most qua... |
Let \( f \) be a function from \( \{1, 2, \ldots, 22\} \) to the positive integers such that \( mn \mid f(m) + f(n) \) for all \( m, n \in \{1, 2, \ldots, 22\} \). If \( d \) is the number of positive divisors of \( f(20) \), compute the minimum possible value of \( d \). | 2016 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 a function from \\( \\{1, 2, \\ldots, 22\\} \\) to the positive integers such that \\( mn \\mid f(m) ... | MATH | {
"ground_truth": "2016",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "415e84d1-1eb4-44db-87ba-a28a72e6eba7"
} | [
{
"content": "Solve the following math problem step by step. 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 a function from \\( \\{1, 2, \\ldots, 22\\} \\) to the positive integers such that \\( mn \\mid f(m) ... |
1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of match... | 94605 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of ... | MATH | {
"ground_truth": "94605",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3966717d-21d2-46b3-8785-33f79186dae9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\n1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of ... |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers whose sum is 15. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions. | 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\nCall a fraction $\\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers w... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1f9fba0f-c79b-4b28-b7b9-1edadaf848cf"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCall a fraction $\\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers w... |
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | 58 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?\n\nRemember to put your answer on its own line a... | MATH | {
"ground_truth": "58",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4b7cbc8d-26fc-44c5-9cfd-e3c9aaae335d"
} | [
{
"content": "Solve the following math problem step by step. 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 perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?\n\nRemember to put your answer on its own line a... |
Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today’s handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor’s. If the freshmen are distinguishable but the hando... | 125 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only... | MATH | {
"ground_truth": "125",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0053bfd4-fe2e-4cfe-80d9-eaf5091727fe"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only... |
How many rearrangements of the letters of "HMMTHMMT" do not contain the substring "HMMT"? For instance, one such arrangement is $HMMHMTMT$. | 361 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 rearrangements of the letters of \"HMMTHMMT\" do not contain the substring \"HMMT\"? For instance, one such a... | MATH | {
"ground_truth": "361",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b4c6874-7a38-4e5c-9ec5-337d6cf492d0"
} | [
{
"content": "Solve the following math problem step by step. 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 rearrangements of the letters of \"HMMTHMMT\" do not contain the substring \"HMMT\"? For instance, one such a... |
Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10$, inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260$, and the average of these favorite numbers is at most $5$. Find the sum of the four largest numbers. | 27 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10$, inclusive. The smallest number t... | MATH | {
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "97222636-4937-4130-81de-31872e4e1597"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10$, inclusive. The smallest number t... |
The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $... | 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\nThe integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the number... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7d15809c-8dca-4c4b-a1ca-6fe1ae8911b3"
} | [
{
"content": "Solve the following math problem step by step. 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 integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the number... |
Find the smallest perfect square whose last digit is not $0$, such that when its last two digits are removed, the resulting number is also a perfect square. | 121 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 perfect square whose last digit is not $0$, such that when its last two digits are removed, the resu... | MATH | {
"ground_truth": "121",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc553792-c2c5-4cba-8e95-61f461960e51"
} | [
{
"content": "Solve the following math problem step by step. 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 perfect square whose last digit is not $0$, such that when its last two digits are removed, the resu... |
A boat is traveling upstream at \(5\) mph relative to the current flowing against it at \(1\) mph. If a tree branch \(10\) miles upstream from the boat falls into the current of the river, how many hours does it take to reach the boat? | 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 boat is traveling upstream at \\(5\\) mph relative to the current flowing against it at \\(1\\) mph. If a tree branc... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5416acae-d6a6-4a5c-b22a-6d65512d8171"
} | [
{
"content": "Solve the following math problem step by step. 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 boat is traveling upstream at \\(5\\) mph relative to the current flowing against it at \\(1\\) mph. If a tree branc... |
There are two values of \(a\) for which the equation \(4x^2 + ax + 8x + 9 = 0\) has only one solution for \(x\). The discriminant of the quadratic equation can be expressed in the form \(m + n = 0\). Please find the sum of these values of \(m\) and \(n\). | -16 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are two values of \\(a\\) for which the equation \\(4x^2 + ax + 8x + 9 = 0\\) has only one solution for \\(x\\).... | MATH | {
"ground_truth": "-16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c5eca0f6-4491-4aee-91b9-5737656762c5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are two values of \\(a\\) for which the equation \\(4x^2 + ax + 8x + 9 = 0\\) has only one solution for \\(x\\).... |
Find the number of triples of sets $(A, B, C)$ such that:
(a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$.
(b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$.
(c) $|A| = |B| = |C| = 4$.
Here, $|S|$ denotes the number of elements in the set $S$. | 45360 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 triples of sets $(A, B, C)$ such that:\n\n(a) $A, B, C \\subseteq \\{1, 2, 3, \\dots , 8 \\}$.\n(b)... | MATH | {
"ground_truth": "45360",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e2a4d295-a35b-4ae7-80f1-39ff84c0935d"
} | [
{
"content": "Solve the following math problem step by step. 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 triples of sets $(A, B, C)$ such that:\n\n(a) $A, B, C \\subseteq \\{1, 2, 3, \\dots , 8 \\}$.\n(b)... |
How many ordered pairs of integers \((x, y)\) are there such that
\[ 0 < \left\vert xy \right\vert < 36? \] | 524 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 of integers \\((x, y)\\) are there such that\n\\[ 0 < \\left\\vert xy \\right\\vert < 36? \\]\n... | MATH | {
"ground_truth": "524",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0bd6d0f7-20bd-4c3d-8f0d-580e45954098"
} | [
{
"content": "Solve the following math problem step by step. 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 of integers \\((x, y)\\) are there such that\n\\[ 0 < \\left\\vert xy \\right\\vert < 36? \\]\n... |
I am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \le n \le 25$, I take one step forward if $n$ is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to my original starting point. How many steps long is my walk back? | 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\nI am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \\le n \\le 25$, I take on... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e273c186-7a3d-4df1-8d90-6b4012675f81"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nI am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \\le n \\le 25$, I take on... |
Sam spends his days walking around the following $2 \times 2$ grid of squares:
\[
\begin{array}{|c|c|}
\hline
1 & 2 \\
\hline
4 & 3 \\
\hline
\end{array}
\]
Two squares are considered adjacent if they share a side. Sam starts at the square labeled $1$ and every second, he walks to an adjacent square. How many paths can... | 167 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSam spends his days walking around the following $2 \\times 2$ grid of squares:\n\\[\n\\begin{array}{|c|c|}\n\\hline\n... | MATH | {
"ground_truth": "167",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b2ef73d0-2462-4e1b-b18e-8e3ba46b3a99"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSam spends his days walking around the following $2 \\times 2$ grid of squares:\n\\[\n\\begin{array}{|c|c|}\n\\hline\n... |
Find the smallest positive integer $n$ such that for every prime number $p$, the expression $p^2 + n$ is never prime. | 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 smallest positive integer $n$ such that for every prime number $p$, the expression $p^2 + n$ is never prime.\... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "09d81427-3fa8-4b38-857f-5e8f2681f033"
} | [
{
"content": "Solve the following math problem step by step. 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 every prime number $p$, the expression $p^2 + n$ is never prime.\... |
Nondegenerate $\triangle ABC$ has integer side lengths, $\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. What is the smallest possible value of the perimeter of $\triangle ABC$? | 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\nNondegenerate $\\triangle ABC$ has integer side lengths, $\\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. ... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "555f9954-7cef-42a0-be33-862e1fdb9fff"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nNondegenerate $\\triangle ABC$ has integer side lengths, $\\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. ... |
How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an integer. | 801 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an int... | MATH | {
"ground_truth": "801",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c598c6c9-e258-4abc-b2f9-f7262a7d055a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? Provide your answer as an int... |
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form... | 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\nPeter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers st... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b7042c20-6caa-4bef-baf4-1de16fa5c364"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPeter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers st... |
Find all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number. | 2 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number.\n\nRemember to put your ans... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4450ddbe-f0de-409d-938b-fa0fb735a624"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number.\n\nRemember to put your ans... |
Let $m, n, p$ be fixed positive real numbers which satisfy $m n p=8$. Depending on these constants, find the minimum of
$$
x^{2}+y^{2}+z^{2}+m x y+n x z+p y z,
$$
where $x, y, z$ are arbitrary positive real numbers satisfying $x y z=8$. When is the equality attained?
Solve the problem for: $m=n=p=2$ | 36 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $m, n, p$ be fixed positive real numbers which satisfy $m n p=8$. Depending on these constants, find the minimum o... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "aa0a6b6a-c2c3-48c7-9829-a696712a71d4"
} | [
{
"content": "Solve the following math problem step by step. 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 $m, n, p$ be fixed positive real numbers which satisfy $m n p=8$. Depending on these constants, find the minimum o... |
The smallest three positive proper divisors of an integer $n$ are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$. | 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\nThe smallest three positive proper divisors of an integer $n$ are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 ... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f8a283fb-8bc4-4c12-987f-bb550fc7cb9a"
} | [
{
"content": "Solve the following math problem step by step. 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 smallest three positive proper divisors of an integer $n$ are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 ... |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | 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\nFind the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$\n\nRemember to ... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de162efe-23bd-42b3-9a94-39abccaabd77"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$\n\nRemember to ... |
Following the directions of the arrows, how many different paths are there from $A$ to $C$?
[asy]
pair A,B,C;
A=(0,0);
B=(5,0);
C=(10,0);
dot(A);
dot(B);
dot(C);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,S);
draw((0,0)--(2.5,1)--(5,0),Arrow);
draw((0,0)--(2.5,-1)--(5,0),Arrow);
draw(B--(7.5,1)--C,Arrow);
draw(... | 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\nFollowing the directions of the arrows, how many different paths are there from $A$ to $C$?\n\n[asy]\npair A,B,C;\nA=(... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ed3ecd2f-2a53-4e10-9f6e-10672c7b6ef2"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFollowing the directions of the arrows, how many different paths are there from $A$ to $C$?\n\n[asy]\npair A,B,C;\nA=(... |
If \(\log (xy^3) = 1\) and \(\log (x^2y) = 1\), find the value of \(\log (xy)\). The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in its simplest form. Please provide the value of \(k + m\). | 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\nIf \\(\\log (xy^3) = 1\\) and \\(\\log (x^2y) = 1\\), find the value of \\(\\log (xy)\\). The answer should be in the ... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0fef2ea2-38fe-44d1-ba38-32f6a2f15835"
} | [
{
"content": "Solve the following math problem step by step. 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 \\(\\log (xy^3) = 1\\) and \\(\\log (x^2y) = 1\\), find the value of \\(\\log (xy)\\). The answer should be in the ... |
Find the number of real solutions to $\sin 6 \pi x = x.$ | 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\nFind the number of real solutions to $\\sin 6 \\pi x = x.$\n\nRemember to put your answer on its own line after \"Answ... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "51252aa5-9997-4bf7-8043-24e10ed65237"
} | [
{
"content": "Solve the following math problem step by step. 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 real solutions to $\\sin 6 \\pi x = x.$\n\nRemember to put your answer on its own line after \"Answ... |
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.) | 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\nHow many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6ca60d5a-cb5f-4446-bfe3-86503f4ebc62"
} | [
{
"content": "Solve the following math problem step by step. 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 distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring... |
Let $ABCD$ be a unit square. A circle with radius $\frac{32}{49}$ passes through point $D$ and is tangent to side $AB$ at point $E$. Then $DE = \frac{m}{n}$, where $m$, $n$ are positive integers and gcd $(m, n) = 1$. Find $100m + n$. | 807 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a unit square. A circle with radius $\\frac{32}{49}$ passes through point $D$ and is tangent to side $AB... | MATH | {
"ground_truth": "807",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9a2a52e9-6a23-4971-824c-22e74e359445"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a unit square. A circle with radius $\\frac{32}{49}$ passes through point $D$ and is tangent to side $AB... |
Frieda the frog begins a sequence of hops on a \(3 \times 3\) grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite ed... | 57 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFrieda the frog begins a sequence of hops on a \\(3 \\times 3\\) grid of squares, moving one square on each hop and ch... | MATH | {
"ground_truth": "57",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cefbcc14-0f6b-465a-bdcd-5fb3589b2445"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFrieda the frog begins a sequence of hops on a \\(3 \\times 3\\) grid of squares, moving one square on each hop and ch... |
How many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C$ in degrees are positive integers and satisfy the equation:
$$\cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1$$
for some positive integer $k$, where $kC$ does not exceed $360^{\circ}$? | 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\nHow many distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "03ed8eff-dfe8-44f6-8a97-60b289362d2b"
} | [
{
"content": "Solve the following math problem step by step. 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 distinct triangles $ABC$ are there, up to similarity, such that the magnitudes of the angles $A$, $B$, and $C... |
Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at le... | 12 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSeven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ fe... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8335d254-0d03-46f7-8ba8-a9aa34fe1405"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSeven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ fe... |
Suppose that $\sec x+\tan x=\frac{22}7.$ Find $\csc x+\cot x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 44 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 $\\sec x+\\tan x=\\frac{22}7.$ Find $\\csc x+\\cot x.$The answer is in the form \frac{m}{n}, where gcd(m... | MATH | {
"ground_truth": "44",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "20e65dcc-f760-4620-9f0a-b18682afbdce"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\sec x+\\tan x=\\frac{22}7.$ Find $\\csc x+\\cot x.$The answer is in the form \frac{m}{n}, where gcd(m... |
Find the number of integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323. | 323 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323.\n\nRemember to put ... | MATH | {
"ground_truth": "323",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "85ff6f63-6ff4-4bc0-82f7-584e3d458529"
} | [
{
"content": "Solve the following math problem step by step. 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 integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323.\n\nRemember to put ... |
In the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the points $W$, $X$, $Y$, and $Z$ is a vertex of one of the small squares. Square $ABCD$ can be constructed with sides passing through $W$, $X$, $Y$, and $Z$. What is the maximum possible distance from $A$ to... | 6 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the poi... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "60263102-bf85-4fc6-94cc-170c4023ff00"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram, four squares of side length 2 are placed in the corners of a square of side length 6. Each of the poi... |
In \(\triangle ABC\) shown in the figure, \(AB=7\), \(BC=8\), \(CA=9\), and \(\overline{AH}\) is an altitude. Points \(D\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, so that \(\overline{BD}\) and \(\overline{CE}\) are angle bisectors, intersecting \(\overline{AH}\) at \(Q\) and \(P\),... | 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\nIn \\(\\triangle ABC\\) shown in the figure, \\(AB=7\\), \\(BC=8\\), \\(CA=9\\), and \\(\\overline{AH}\\) is an altitu... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a055098b-d4aa-4dd6-b509-29e9b3c6adbe"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn \\(\\triangle ABC\\) shown in the figure, \\(AB=7\\), \\(BC=8\\), \\(CA=9\\), and \\(\\overline{AH}\\) is an altitu... |
Evaluate $\sqrt{5\cdot10\cdot14\cdot21\cdot15\cdot20}$. | 2100 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate $\\sqrt{5\\cdot10\\cdot14\\cdot21\\cdot15\\cdot20}$.\n\nRemember to put your answer on its own line after \"A... | MATH | {
"ground_truth": "2100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b1ef12da-d684-48f2-bb5f-a161c223c789"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate $\\sqrt{5\\cdot10\\cdot14\\cdot21\\cdot15\\cdot20}$.\n\nRemember to put your answer on its own line after \"A... |
Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\... | 96 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 a common chord between two intersecting circles to be the line segment connecting their two intersection points... | MATH | {
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "484d1c02-363c-4ada-9044-3cf78f75c1bc"
} | [
{
"content": "Solve the following math problem step by step. 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 a common chord between two intersecting circles to be the line segment connecting their two intersection points... |
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semic... | 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\nThree semicircles of radius $1$ are constructed on diameter $\\overline{AB}$ of a semicircle of radius $2$. The center... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a1e8378a-8c59-4d44-a3e0-f420698e785b"
} | [
{
"content": "Solve the following math problem step by step. 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 semicircles of radius $1$ are constructed on diameter $\\overline{AB}$ of a semicircle of radius $2$. The center... |
Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started? | 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\nMeredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 mile... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9597b9b2-c092-41a6-9952-d964c8fb00fe"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMeredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 mile... |
For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$. | 15 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor an odd number $n$, we define $n!! = n \\cdot (n-2) \\cdot (n-4) \\cdots 3 \\cdot 1$. Determine how many different ... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d5034984-b567-4eb9-b901-82a1ebdd2929"
} | [
{
"content": "Solve the following math problem step by step. 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 an odd number $n$, we define $n!! = n \\cdot (n-2) \\cdot (n-4) \\cdots 3 \\cdot 1$. Determine how many different ... |
Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$? | 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\nLet $f(x) = \\log_{10} \\left(\\sin(\\pi x) \\cdot \\sin(2 \\pi x) \\cdot \\sin (3 \\pi x) \\cdots \\sin(8 \\pi x)\\ri... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "684ee8ad-c7f4-4c1c-b2a9-9306fe6e90e4"
} | [
{
"content": "Solve the following math problem step by step. 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) = \\log_{10} \\left(\\sin(\\pi x) \\cdot \\sin(2 \\pi x) \\cdot \\sin (3 \\pi x) \\cdots \\sin(8 \\pi x)\\ri... |
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$. | 99 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(x)=x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cdots}}}$. Find $f(99)f^\\prime (99)$.\n\nRemember to put you... | MATH | {
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "59ed37da-fefb-49e8-bebc-f69057da6ce5"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $f(x)=x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cfrac{1}{2x+\\cdots}}}$. Find $f(99)f^\\prime (99)$.\n\nRemember to put you... |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | 103 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAlex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in retu... | MATH | {
"ground_truth": "103",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2abdd0ea-cbd7-42f8-96a2-e002f5773164"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAlex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in retu... |
What is the number of square units in the area of the hexagon below?
[asy]
unitsize(0.5cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;
int i,j;
for(i=0;i<=4;++i)
{
for(j=-3;j<=3;++j)
{
dot((i,j));
}
}
for(i=1;i<=4;++i)
{
draw((i,-1/3)--(i,1/3));
}
for(j=1;j<=3;++j)
{
draw((-1/3,j)--(1/3,j));
... | 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\nWhat is the number of square units in the area of the hexagon below?\n\n[asy]\nunitsize(0.5cm);\ndefaultpen(linewidth(... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9fda62be-8b90-4ad4-b581-6136e17b785e"
} | [
{
"content": "Solve the following math problem step by step. 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 number of square units in the area of the hexagon below?\n\n[asy]\nunitsize(0.5cm);\ndefaultpen(linewidth(... |
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute \[ \sum_{i=1}^{1000} g(i). \] | 8000 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Co... | MATH | {
"ground_truth": "8000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e43aae6f-f591-4e3e-8481-2c5235c21dd5"
} | [
{
"content": "Solve the following math problem step by step. 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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Co... |
When Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$-digit number. What is Lisa's favorite $2$-digit number? | 27 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of h... | MATH | {
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d62d665c-3c90-4aeb-b2b9-6764bbfc27cc"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen Lisa squares her favorite $2$-digit number, she gets the same result as when she cubes the sum of the digits of h... |
Consider the six by six grid of unit squares below. How many rectangles of area 8 square units can be formed using only the line segments of the grid as the sides of the rectangles?
[asy]unitsize(0.2inch);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((1,0)--(1,6));
draw((2,0)--(2,6));
draw((3,0)--(3,6));
draw((4,0)--... | 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\nConsider the six by six grid of unit squares below. How many rectangles of area 8 square units can be formed using onl... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2cb1703d-a00c-420f-9f60-41f8195a1566"
} | [
{
"content": "Solve the following math problem step by step. 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 six by six grid of unit squares below. How many rectangles of area 8 square units can be formed using onl... |
If $n$ is a real number, then the simultaneous system of equations:
\[
\begin{align*}
nx + y &= 1 \\
ny + z &= 1 \\
x + nz &= 1
\end{align*}
\]
has no solution if and only if $n$ is equal to $x$. Please provide the absolute value of $x$ as your final answer. | 1 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIf $n$ is a real number, then the simultaneous system of equations:\n\\[\n\\begin{align*}\n nx + y &= 1 \\\\\n ny + ... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c0c3c3b9-624f-4844-85d1-a1723d3cea26"
} | [
{
"content": "Solve the following math problem step by step. 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 $n$ is a real number, then the simultaneous system of equations:\n\\[\n\\begin{align*}\n nx + y &= 1 \\\\\n ny + ... |
The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree,... | 30202 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 set $\\{(x, y) \\in R^2| \\lfloor x + y\\rfloor \\cdot \\lceil x + y\\rceil = (\\lfloor x\\rfloor + \\lceil y \\rc... | MATH | {
"ground_truth": "30202",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "057aa9a6-77fb-46e7-be98-9fa2c376a21f"
} | [
{
"content": "Solve the following math problem step by step. 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 set $\\{(x, y) \\in R^2| \\lfloor x + y\\rfloor \\cdot \\lceil x + y\\rceil = (\\lfloor x\\rfloor + \\lceil y \\rc... |
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 6.\]Find the minimum value of $x^3 y^2 z.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 109 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ and $z$ be positive real numbers such that\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 6.\\]Find th... | MATH | {
"ground_truth": "109",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c805b069-5cf8-45f3-8c57-5f1952ce2991"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ and $z$ be positive real numbers such that\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 6.\\]Find th... |
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star-polygon? | 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\nA star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2b5ab6a1-962b-4121-a0a1-14d1a5157576"
} | [
{
"content": "Solve the following math problem step by step. 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 star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from... |
Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer.
How many triples $(a, b, c)$ satisfy the conditions? | 16 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "63c67dd3-935d-4af6-850e-dd6231eedf58"
} | [
{
"content": "Solve the following math problem step by step. 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 triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:... |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$. | 19 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ and let $P$ be the intersection of its diagonals $AC$... | MATH | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "63890aa8-e27f-42fd-812f-7e4d1506872c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ and let $P$ be the intersection of its diagonals $AC$... |
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations? | 683 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the ... | MATH | {
"ground_truth": "683",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3c4c54e9-4e21-4ce2-b3b1-6151be87012e"
} | [
{
"content": "Solve the following math problem step by step. 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 two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the ... |
For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime? | 22 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime?\n\... | MATH | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "40b709bb-d2c1-441c-b4cc-7b4a8eae38e1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime?\n\... |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, \ldots, x_N$ such that $x_i^2 - x_ix_j$ is not divisible by $1111$ for any $i \neq j$. | 1000 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the greatest positive integer $N$ with the following property: there exist integers $x_1, \\ldots, x_N$ such that... | MATH | {
"ground_truth": "1000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "edfd98d0-18f7-443d-b578-9b8eea4a9a79"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the greatest positive integer $N$ with the following property: there exist integers $x_1, \\ldots, x_N$ such that... |
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\fra... | 25 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.... | MATH | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6fd64891-07f8-4b62-a811-47aabe8be82c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.... |
The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. Find the net change over the four years, to the nearest percent. If the net change is $x\%$, what is the value of $x$? | -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\nThe yearly changes in the population census of a town for four consecutive years are, respectively, $25\\%$ increase, ... | MATH | {
"ground_truth": "-12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cbd3fcd8-7e26-4ddb-b1d9-1136931305b6"
} | [
{
"content": "Solve the following math problem step by step. 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 yearly changes in the population census of a town for four consecutive years are, respectively, $25\\%$ increase, ... |
If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 971 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the re... | MATH | {
"ground_truth": "971",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "89911306-0edf-446b-85e7-3d60f68d5da2"
} | [
{
"content": "Solve the following math problem step by step. 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 I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the re... |
Find the first three figures after the decimal point in the decimal expression of the number \(\frac{0.123456789101112\ldots495051}{0.515049\ldots121110987654321}\). Please provide the value of the first three digits after the decimal point as a single integer. | 239 | math_dapo | [
{
"content": "Solve the following math problem step by step. 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 first three figures after the decimal point in the decimal expression of the number \\(\\frac{0.1234567891011... | MATH | {
"ground_truth": "239",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "66268b00-e929-4d87-bed0-17153d28a3c7"
} | [
{
"content": "Solve the following math problem step by step. 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 first three figures after the decimal point in the decimal expression of the number \\(\\frac{0.1234567891011... |
The points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each other. The line segment $MN$ is parallel to $AB$ and intersects the radius $OA$ at point $P$. Find the radius of the circle given that $|MP| = 12$ and $|PN| = 2\sqrt{14}$. | 10 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each o... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1518901b-1f3f-4da1-b20b-f60965944376"
} | [
{
"content": "Solve the following math problem step by step. 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 points $A$, $B$, $M$, and $N$ lie on a circle with center $O$. The radii $OA$ and $OB$ are perpendicular to each o... |
Let $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x$, where $\lfloor a \rfloor$ is the greatest integer not exceeding $a$. | 652400 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\\lfloor x \\rfloor y ... | MATH | {
"ground_truth": "652400",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "03a0565a-02ae-458d-a568-9fb9a46fd4ea"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\\lfloor x \\rfloor y ... |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | 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\nSally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards s... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "380b2b8d-178c-46d1-b593-0cfdc80d8af9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards s... |
Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after hims... | 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\nDone with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit a... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4c23076f-5d82-4796-8fde-df4c68c91edd"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDone with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit a... |
A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. Find the number of possible values of $n$. | 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\nA fifth number, $n$, is added to the set $\\{ 3,6,9,10 \\}$ to make the mean of the set of five numbers equal to its m... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ad1cd92e-6908-4acc-a9b4-fe3eecef05de"
} | [
{
"content": "Solve the following math problem step by step. 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 fifth number, $n$, is added to the set $\\{ 3,6,9,10 \\}$ to make the mean of the set of five numbers equal to its m... |
Suppose that \( P = 2^m \) and \( Q = 3^n \). Find the expression that is equal to \( 12^{mn} \) for every pair of integers \( (m, n) \). The original answer is in the form \( P^{a}Q^{b} \), please give the value of \( a + b + c \). | 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\nSuppose that \\( P = 2^m \\) and \\( Q = 3^n \\). Find the expression that is equal to \\( 12^{mn} \\) for every pair ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5ae3e585-ce15-4218-9b01-b4d730f1d050"
} | [
{
"content": "Solve the following math problem step by step. 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 \\( P = 2^m \\) and \\( Q = 3^n \\). Find the expression that is equal to \\( 12^{mn} \\) for every pair ... |
Find the positive solution to
$$\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0$$ | 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\nFind the positive solution to \n$$\\frac 1{x^2-10x-29}+\\frac1{x^2-10x-45}-\\frac 2{x^2-10x-69}=0$$\n\nRemember to pu... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f01a5b88-86d2-4445-9bfe-c255beab7b22"
} | [
{
"content": "Solve the following math problem step by step. 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 positive solution to \n$$\\frac 1{x^2-10x-29}+\\frac1{x^2-10x-45}-\\frac 2{x^2-10x-69}=0$$\n\nRemember to pu... |
On the $xy$-plane, the origin is labeled with an $M$. The points $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$ are labeled with $A$'s. The points $(2,0)$, $(1,1)$, $(0,2)$, $(-1, 1)$, $(-2, 0)$, $(-1, -1)$, $(0, -2)$, and $(1, -1)$ are labeled with $T$'s. The points $(3,0)$, $(2,1)$, $(1,2)$, $(0, 3)$, $(-1, 2)$, $(-2, 1)$,... | 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\nOn the $xy$-plane, the origin is labeled with an $M$. The points $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$ are labeled ... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b4acd3bf-688d-4724-9b02-b3b05cc242f4"
} | [
{
"content": "Solve the following math problem step by step. 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 $xy$-plane, the origin is labeled with an $M$. The points $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$ are labeled ... |
In rectangle $ABCD$, $AD=1$, $P$ is on $\overline{AB}$, and $\overline{DB}$ and $\overline{DP}$ trisect $\angle ADC$. Write the perimeter of $\triangle BDP$ in simplest form as: $w + \frac{x \cdot \sqrt{y}}{z}$, where $w, x, y, z$ are nonnegative integers. What is $w + x + y + z$?
[asy]
pair A,B,C,D,P;
A=(0,5);
B=(8... | 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\nIn rectangle $ABCD$, $AD=1$, $P$ is on $\\overline{AB}$, and $\\overline{DB}$ and $\\overline{DP}$ trisect $\\angle AD... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "04178b4b-96f6-43f5-af7d-e6b0fe425236"
} | [
{
"content": "Solve the following math problem step by step. 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 rectangle $ABCD$, $AD=1$, $P$ is on $\\overline{AB}$, and $\\overline{DB}$ and $\\overline{DP}$ trisect $\\angle AD... |
Horses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3\text{-to-}1$ and the odds against Y winning are $2\text{-to-}3$, what are the odds against Z winning? Express your answer in the form $k\text{-to-}m$. Please provide the value of k + m. | 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\nHorses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cf6f7053-3799-492e-b790-7f25187a3a3f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHorses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are... |
Dave's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths contained cayenne, and one-sixths contained salted soy nuts. What is the largest possible number of pies that had none of these ingredients? | 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\nDave's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "39150900-786e-4338-ba8c-3475a0f657d4"
} | [
{
"content": "Solve the following math problem step by step. 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's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths... |
In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark.
```latex
\begin{asy}
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$2$",(... | 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\nIn the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two ... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "568e7a80-3824-4372-ba25-8df9950e1dd3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two ... |
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