data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=0$. At some point during a given week, a law enforcement officer had issued $T+2$ traffic warnings, 20 tickets, and had made $T+5$ arrests. How many more tickets must the officer issue in order for the combined number of tickets and arrests to be 20 times the number of w... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2721,
"split": "test",
"question": "Let $T=0$. At some point during a given week, a law enforcement officer had issued $T+2$ traffic warnings, 20 tickets, and had made $T+5$ arrests. How many more tickets must the officer issue in order for the combined number of tickets and arrests to be 20 times the ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=15$. In parallelogram $A R M L$, points $P$ and $Q$ trisect $\\overline{A R}$ and points $W, X, Y, Z$ divide $\\overline{M L}$ into fifths (where $W$ is closest to $M$, and points $X$ and $Y$ are both between $W$ and $Z$ ). If $[A R M L]=T$, compute $[P Q W Z]$.\n\nLet's... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2722,
"split": "test",
"question": "Let $T=15$. In parallelogram $A R M L$, points $P$ and $Q$ trisect $\\overline{A R}$ and points $W, X, Y, Z$ divide $\\overline{M L}$ into fifths (where $W$ is closest to $M$, and points $X$ and $Y$ are both between $W$ and $Z$ ). If $[A R M L]=T$, compute $[P Q W Z]... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=7$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2723,
"split": "test",
"question": "Let $T=7$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=36$. The graph of $y=x^{2}+2 x-T$ intersects the $x$-axis at points $A$ and $M$, which are diagonally opposite vertices of square $A R M L$. Compute $[A R M L]$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 2724,
"split": "test",
"question": "Let $T=36$. The graph of $y=x^{2}+2 x-T$ intersects the $x$-axis at points $A$ and $M$, which are diagonally opposite vertices of square $A R M L$. Compute $[A R M L]$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $S$ be the set of prime factors of the numbers you receive from positions 7 and 9 , and let $p$ and $q$ be the two least distinct elements of $S$, with $p<q$. Hexagon HEXAGO is inscribed in circle $\\omega$, and every angle of $H E X A G O$ is $120^{\\circ}$. If $H E=X A=G ... | math | {
"style": "rule",
"ground_truth": "$\\frac{67 \\pi}{3}$"
} | {
"index": 2725,
"split": "test",
"question": "Let $S$ be the set of prime factors of the numbers you receive from positions 7 and 9 , and let $p$ and $q$ be the two least distinct elements of $S$, with $p<q$. Hexagon HEXAGO is inscribed in circle $\\omega$, and every angle of $H E X A G O$ is $120^{\\circ}$. If ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting t... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2726,
"split": "test",
"question": "Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=8$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2 T)^{2}+b$. Compute $b$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "184"
} | {
"index": 2727,
"split": "test",
"question": "Let $T=8$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2 T)^{2}+b$. Compute $b$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=19$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the sequence.\n\nL... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2728,
"split": "test",
"question": "Let $T=19$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the seq... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=17$. A regular $n$-gon has exactly $T$ more diagonals than a regular $(n-1)$-gon. Compute the value of $n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2729,
"split": "test",
"question": "Let $T=17$. A regular $n$-gon has exactly $T$ more diagonals than a regular $(n-1)$-gon. Compute the value of $n$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=27/2$. The sequence $a_{1}, a_{2}, a_{3}, \\ldots$, is arithmetic with $a_{16}=13$ and $a_{30}=20$. Compute the value of $k$ for which $a_{k}=T$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 2730,
"split": "test",
"question": "Let $T=27/2$. The sequence $a_{1}, a_{2}, a_{3}, \\ldots$, is arithmetic with $a_{16}=13$ and $a_{30}=20$. Compute the value of $k$ for which $a_{k}=T$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=114$. A rectangular prism has a length of 1 , a width of 3 , a height of $h$, and has a total surface area of $T$. Compute the value of $h$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{27}{2}$"
} | {
"index": 2731,
"split": "test",
"question": "Let $T=114$. A rectangular prism has a length of 1 , a width of 3 , a height of $h$, and has a total surface area of $T$. Compute the value of $h$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The zeros of $x^{2}+b x+93$ are $r$ and $s$. If the zeros of $x^{2}-22 x+c$ are $r+1$ and $s+1$, compute $c$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"index": 2732,
"split": "test",
"question": "The zeros of $x^{2}+b x+93$ are $r$ and $s$. If the zeros of $x^{2}-22 x+c$ are $r+1$ and $s+1$, compute $c$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $N=888,888 \\times 9,999,999$. Compute the sum of the digits of $N$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 2733,
"split": "test",
"question": "Let $N=888,888 \\times 9,999,999$. Compute the sum of the digits of $N$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Five equilateral triangles are drawn in the plane so that no two sides of any of the triangles are parallel. Compute the maximum number of points of intersection among all five triangles.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 2734,
"split": "test",
"question": "Five equilateral triangles are drawn in the plane so that no two sides of any of the triangles are parallel. Compute the maximum number of points of intersection among all five triangles."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Let $S$ be the set of four-digit positive integers for which the sum of the squares of their digits is 17 . For example, $2023 \\in S$ because $2^{2}+0^{2}+2^{2}+3^{2}=17$. Compute the median of $S$.\n\nLet's think step by step and output the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "2302"
} | {
"index": 2735,
"split": "test",
"question": "$\\quad$ Let $S$ be the set of four-digit positive integers for which the sum of the squares of their digits is 17 . For example, $2023 \\in S$ because $2^{2}+0^{2}+2^{2}+3^{2}=17$. Compute the median of $S$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $E U C L I D$ be a hexagon inscribed in a circle of radius 5 . Given that $E U=U C=L I=I D=6$, and $C L=D E$, compute $C L$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{14}{5}$"
} | {
"index": 2736,
"split": "test",
"question": "Let $E U C L I D$ be a hexagon inscribed in a circle of radius 5 . Given that $E U=U C=L I=I D=6$, and $C L=D E$, compute $C L$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The ARMLLexicon consists of 10 letters: $\\{A, R, M, L, e, x, i, c, o, n\\}$. A palindrome is an ordered list of letters that read the same backwards and forwards; for example, MALAM, n, oncecno, and MoM are palindromes. Compute the number of 15-letter palindromes that can be s... | math | {
"style": "rule",
"ground_truth": "99956"
} | {
"index": 2737,
"split": "test",
"question": "The ARMLLexicon consists of 10 letters: $\\{A, R, M, L, e, x, i, c, o, n\\}$. A palindrome is an ordered list of letters that read the same backwards and forwards; for example, MALAM, n, oncecno, and MoM are palindromes. Compute the number of 15-letter palindromes th... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $10^{y}$ be the product of all real numbers $x$ such that $\\log x=\\frac{3+\\left\\lfloor(\\log x)^{2}\\right\\rfloor}{4}$. Compute $y$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2738,
"split": "test",
"question": "Let $10^{y}$ be the product of all real numbers $x$ such that $\\log x=\\frac{3+\\left\\lfloor(\\log x)^{2}\\right\\rfloor}{4}$. Compute $y$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute\n\n$$\n\\frac{r_{1}}{\\sqrt[3]{r_{2}}}+\\frac{r_{2}}{\\sqrt[3]{r_{1}}}\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "508"
} | {
"index": 2739,
"split": "test",
"question": "The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute\n\n$$\n\\frac{r_{1}}{\\sqrt[3]{r_{2}}}+\\frac{r_{2}}{\\sqrt[3]{r_{1}}}\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Circle $\\omega$ is tangent to parallel lines $\\ell_{1}$ and $\\ell_{2}$ at $A$ and $B$ respectively. Circle $\\omega_{1}$ is tangent to $\\ell_{1}$ at $C$ and to $\\omega$ externally at $P$. Circle $\\omega_{2}$ is tangent to $\\ell_{2}$ at $D$ and to $\\omega$ externally at ... | math | {
"style": "rule",
"ground_truth": "$5 \\sqrt{10}$"
} | {
"index": 2740,
"split": "test",
"question": "Circle $\\omega$ is tangent to parallel lines $\\ell_{1}$ and $\\ell_{2}$ at $A$ and $B$ respectively. Circle $\\omega_{1}$ is tangent to $\\ell_{1}$ at $C$ and to $\\omega$ externally at $P$. Circle $\\omega_{2}$ is tangent to $\\ell_{2}$ at $D$ and to $\\omega$ ext... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given quadrilateral $A R M L$ with $A R=20, R M=23, M L=25$, and $A M=32$, compute the number of different integers that could be the perimeter of $A R M L$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 2741,
"split": "test",
"question": "Given quadrilateral $A R M L$ with $A R=20, R M=23, M L=25$, and $A M=32$, compute the number of different integers that could be the perimeter of $A R M L$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ denote the set of all real polynomials $A(x)$ with leading coefficient 1 such that there exists a real polynomial $B(x)$ that satisfies\n\n$$\n\\frac{1}{A(x)}+\\frac{1}{B(x)}+\\frac{1}{x+10}=\\frac{1}{x}\n$$\n\nfor all real numbers $x$ for which $A(x) \\neq 0... | math | {
"style": "rule",
"ground_truth": "46760"
} | {
"index": 2742,
"split": "test",
"question": "Let $\\mathcal{S}$ denote the set of all real polynomials $A(x)$ with leading coefficient 1 such that there exists a real polynomial $B(x)$ that satisfies\n\n$$\n\\frac{1}{A(x)}+\\frac{1}{B(x)}+\\frac{1}{x+10}=\\frac{1}{x}\n$$\n\nfor all real numbers $x$ for which $A... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greatest digit in $T$. In square $N O R M, N O=b$, and points $P_{1}$ and $P_{2}$ lie on $\\overline{N O}$ and $\\overline{O R}$, respectively, so that $O P_{1}=O P_{2}=a$. A circle centered at $O$ has r... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2744,
"split": "test",
"question": "Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greatest digit in $T$. In square $N O R M, N O=b$, and points $P_{1}$ and $P_{2}$ lie on $\\overline{N O}$ and $\\overline{O R}$, respectively, so that $O P_{1}=O P_{2}=a$. A circle centered a... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=36$. Square $A B C D$ has area $T$. Points $M, N, O$, and $P$ lie on $\\overline{A B}$, $\\overline{B C}, \\overline{C D}$, and $\\overline{D A}$, respectively, so that quadrilateral $M N O P$ is a rectangle with $M P=2$. Compute $M N$.\n\nLet's think step by step and ou... | math | {
"style": "rule",
"ground_truth": "$6 \\sqrt{2}-2$"
} | {
"index": 2745,
"split": "test",
"question": "Let $T=36$. Square $A B C D$ has area $T$. Points $M, N, O$, and $P$ lie on $\\overline{A B}$, $\\overline{B C}, \\overline{C D}$, and $\\overline{D A}$, respectively, so that quadrilateral $M N O P$ is a rectangle with $M P=2$. Compute $M N$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In a game, a player chooses 2 of the 13 letters from the first half of the alphabet (i.e., A-M) and 2 of the 13 letters from the second half of the alphabet (i.e., N-Z). Aditya plays the game, and then Ayesha plays the game. Compute the probability that Aditya and Ayesha choose... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{6084}$"
} | {
"index": 2746,
"split": "test",
"question": "In a game, a player chooses 2 of the 13 letters from the first half of the alphabet (i.e., A-M) and 2 of the 13 letters from the second half of the alphabet (i.e., N-Z). Aditya plays the game, and then Ayesha plays the game. Compute the probability that Aditya and Ay... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=\\frac{1}{6084}$. Compute the least positive integer $n$ such that when a fair coin is flipped $n$ times, the probability of it landing heads on all $n$ flips is less than $T$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2747,
"split": "test",
"question": "Let $T=\\frac{1}{6084}$. Compute the least positive integer $n$ such that when a fair coin is flipped $n$ times, the probability of it landing heads on all $n$ flips is less than $T$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=13$. Compute the least integer $n>2023$ such that the equation $x^{2}-T x-n=0$ has integer solutions.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"index": 2748,
"split": "test",
"question": "Let $T=13$. Compute the least integer $n>2023$ such that the equation $x^{2}-T x-n=0$ has integer solutions."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relat... | math | {
"style": "rule",
"ground_truth": "1,2"
} | {
"index": 2753,
"split": "test",
"question": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integer... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relat... | math | {
"style": "rule",
"ground_truth": "1,3"
} | {
"index": 2754,
"split": "test",
"question": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integer... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For an integer $n \\geq 4$, define $a_{n}$ to be the product of all real numbers that are roots to at least one quadratic polynomial whose coefficients are positive integers that sum to $n$. Compute\n\n$$\n\\frac{a_{4}}{a_{5}}+\\frac{a_{5}}{a_{6}}+\\frac{a_{6}}{a_{7}}+\\cdots+\... | math | {
"style": "rule",
"ground_truth": "-2019"
} | {
"index": 2759,
"split": "test",
"question": "For an integer $n \\geq 4$, define $a_{n}$ to be the product of all real numbers that are roots to at least one quadratic polynomial whose coefficients are positive integers that sum to $n$. Compute\n\n$$\n\\frac{a_{4}}{a_{5}}+\\frac{a_{5}}{a_{6}}+\\frac{a_{6}}{a_{7}... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Suppose that $u$ and $v$ are distinct numbers chosen at random from the set $\\{1,2,3, \\ldots, 30\\}$. Compute the probability that the roots of the polynomial $(x+u)(x+v)+4$ are integers.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{17}{145}$"
} | {
"index": 2760,
"split": "test",
"question": "Suppose that $u$ and $v$ are distinct numbers chosen at random from the set $\\{1,2,3, \\ldots, 30\\}$. Compute the probability that the roots of the polynomial $(x+u)(x+v)+4$ are integers."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The degree-measures of the interior angles of convex hexagon TIEBRK are all integers in arithmetic progression. Compute the least possible degree-measure of the smallest interior angle in hexagon TIEBRK.\n\nLet's think step by step and output the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 2761,
"split": "test",
"question": "The degree-measures of the interior angles of convex hexagon TIEBRK are all integers in arithmetic progression. Compute the least possible degree-measure of the smallest interior angle in hexagon TIEBRK."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A six-digit natural number is \"sort-of-decreasing\" if its first three digits are in strictly decreasing order and its last three digits are in strictly decreasing order. For example, 821950 and 631631 are sort-of-decreasing but 853791 and 911411 are not. Compute the number of... | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"index": 2762,
"split": "test",
"question": "A six-digit natural number is \"sort-of-decreasing\" if its first three digits are in strictly decreasing order and its last three digits are in strictly decreasing order. For example, 821950 and 631631 are sort-of-decreasing but 853791 and 911411 are not. Compute th... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For each positive integer $N$, let $P(N)$ denote the product of the digits of $N$. For example, $P(8)=8$, $P(451)=20$, and $P(2023)=0$. Compute the least positive integer $n$ such that $P(n+23)=P(n)+23$.\n\nLet's think step by step and output the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 2763,
"split": "test",
"question": "For each positive integer $N$, let $P(N)$ denote the product of the digits of $N$. For example, $P(8)=8$, $P(451)=20$, and $P(2023)=0$. Compute the least positive integer $n$ such that $P(n+23)=P(n)+23$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the least integer value of the function\n\n$$\nf(x)=\\frac{x^{4}-6 x^{3}+2 x^{2}-6 x+2}{x^{2}+1}\n$$\n\nwhose domain is the set of all real numbers.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 2764,
"split": "test",
"question": "Compute the least integer value of the function\n\n$$\nf(x)=\\frac{x^{4}-6 x^{3}+2 x^{2}-6 x+2}{x^{2}+1}\n$$\n\nwhose domain is the set of all real numbers."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Suppose that noncongruent triangles $A B C$ and $X Y Z$ are given such that $A B=X Y=10, B C=$ $Y Z=9$, and $\\mathrm{m} \\angle C A B=\\mathrm{m} \\angle Z X Y=30^{\\circ}$. Compute $[A B C]+[X Y Z]$.\n\nLet's think step by step and output the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "$25 \\sqrt{3}$"
} | {
"index": 2765,
"split": "test",
"question": "Suppose that noncongruent triangles $A B C$ and $X Y Z$ are given such that $A B=X Y=10, B C=$ $Y Z=9$, and $\\mathrm{m} \\angle C A B=\\mathrm{m} \\angle Z X Y=30^{\\circ}$. Compute $[A B C]+[X Y Z]$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The mean, median, and unique mode of a list of positive integers are three consecutive integers in some order. Compute the least possible sum of the integers in the original list.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2766,
"split": "test",
"question": "The mean, median, and unique mode of a list of positive integers are three consecutive integers in some order. Compute the least possible sum of the integers in the original list."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the other. For exa... | math | {
"style": "rule",
"ground_truth": "7711"
} | {
"index": 2767,
"split": "test",
"question": "David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the oth... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In quadrilateral $A B C D, \\mathrm{~m} \\angle B+\\mathrm{m} \\angle D=270^{\\circ}$. The circumcircle of $\\triangle A B D$ intersects $\\overline{C D}$ at point $E$, distinct from $D$. Given that $B C=4, C E=5$, and $D E=7$, compute the diameter of the circumcircle of $\\tri... | math | {
"style": "rule",
"ground_truth": "$\\sqrt{130}$"
} | {
"index": 2768,
"split": "test",
"question": "In quadrilateral $A B C D, \\mathrm{~m} \\angle B+\\mathrm{m} \\angle D=270^{\\circ}$. The circumcircle of $\\triangle A B D$ intersects $\\overline{C D}$ at point $E$, distinct from $D$. Given that $B C=4, C E=5$, and $D E=7$, compute the diameter of the circumcircl... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $i=\\sqrt{-1}$. The complex number $z=-142+333 \\sqrt{5} i$ can be expressed as a product of two complex numbers in multiple different ways, two of which are $(57-8 \\sqrt{5} i)(-6+5 \\sqrt{5} i)$ and $(24+\\sqrt{5} i)(-3+14 \\sqrt{5} i)$. Given that $z=-142+333 \\sqrt{5} i... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 2770,
"split": "test",
"question": "Let $i=\\sqrt{-1}$. The complex number $z=-142+333 \\sqrt{5} i$ can be expressed as a product of two complex numbers in multiple different ways, two of which are $(57-8 \\sqrt{5} i)(-6+5 \\sqrt{5} i)$ and $(24+\\sqrt{5} i)(-3+14 \\sqrt{5} i)$. Given that $z=-142+333 ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Parallelogram $A B C D$ is rotated about $A$ in the plane, resulting in $A B^{\\prime} C^{\\prime} D^{\\prime}$, with $D$ on $\\overline{A B^{\\prime}}$. Suppose that $\\left[B^{\\prime} C D\\right]=\\left[A B D^{\\prime}\\right]=\\left[B C C^{\\prime}\\right]$. Compute $\\tan ... | math | {
"style": "rule",
"ground_truth": "$\\sqrt{2}-1$,$\\frac{3-\\sqrt{2}}{7}$"
} | {
"index": 2771,
"split": "test",
"question": "Parallelogram $A B C D$ is rotated about $A$ in the plane, resulting in $A B^{\\prime} C^{\\prime} D^{\\prime}$, with $D$ on $\\overline{A B^{\\prime}}$. Suppose that $\\left[B^{\\prime} C D\\right]=\\left[A B D^{\\prime}\\right]=\\left[B C C^{\\prime}\\right]$. Comp... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the least integer greater than 2023 , the sum of whose digits is 17 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2069"
} | {
"index": 2772,
"split": "test",
"question": "Compute the least integer greater than 2023 , the sum of whose digits is 17 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T$ = 2069, and let $K$ be the sum of the digits of $T$. Let $r$ and $s$ be the two roots of the polynomial $x^{2}-18 x+K$. Compute $|r-s|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2773,
"split": "test",
"question": "Let $T$ = 2069, and let $K$ be the sum of the digits of $T$. Let $r$ and $s$ be the two roots of the polynomial $x^{2}-18 x+K$. Compute $|r-s|$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 7, and let $K=9 T$. Let $A_{1}=2$, and for $n \\geq 2$, let\n\n$$\nA_{n}= \\begin{cases}A_{n-1}+1 & \\text { if } n \\text { is not a perfect square } \\\\ \\sqrt{n} & \\text { if } n \\text { is a perfect square. }\\end{cases}\n$$\n\nCompute $A_{K}$.\n\nLet's think st... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 2775,
"split": "test",
"question": "Let $T=$ 7, and let $K=9 T$. Let $A_{1}=2$, and for $n \\geq 2$, let\n\n$$\nA_{n}= \\begin{cases}A_{n-1}+1 & \\text { if } n \\text { is not a perfect square } \\\\ \\sqrt{n} & \\text { if } n \\text { is a perfect square. }\\end{cases}\n$$\n\nCompute $A_{K}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 21. The number $20^{T} \\cdot 23^{T}$ has $K$ positive divisors. Compute the greatest prime factor of $K$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 2776,
"split": "test",
"question": "Let $T=$ 21. The number $20^{T} \\cdot 23^{T}$ has $K$ positive divisors. Compute the greatest prime factor of $K$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=43$. Compute the positive integer $n \\neq 17$ for which $\\left(\\begin{array}{c}T-3 \\\\ 17\\end{array}\\right)=\\left(\\begin{array}{c}T-3 \\\\ n\\end{array}\\right)$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 2777,
"split": "test",
"question": "Let $T=43$. Compute the positive integer $n \\neq 17$ for which $\\left(\\begin{array}{c}T-3 \\\\ 17\\end{array}\\right)=\\left(\\begin{array}{c}T-3 \\\\ n\\end{array}\\right)$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=23$ . Compute the units digit of $T^{2023}+T^{20}-T^{23}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 2778,
"split": "test",
"question": "Let $T=23$ . Compute the units digit of $T^{2023}+T^{20}-T^{23}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 3. Suppose that $T$ fair coins are flipped. Compute the probability that at least one tails is flipped.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{7}{8}$"
} | {
"index": 2780,
"split": "test",
"question": "Let $T=$ 3. Suppose that $T$ fair coins are flipped. Compute the probability that at least one tails is flipped."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ $\\frac{7}{8}$. The number $T$ can be expressed as a reduced fraction $\\frac{m}{n}$, where $m$ and $n$ are positive integers whose greatest common divisor is 1 . The equation $x^{2}+(m+n) x+m n=0$ has two distinct real solutions. Compute the lesser of these two soluti... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 2781,
"split": "test",
"question": "Let $T=$ $\\frac{7}{8}$. The number $T$ can be expressed as a reduced fraction $\\frac{m}{n}$, where $m$ and $n$ are positive integers whose greatest common divisor is 1 . The equation $x^{2}+(m+n) x+m n=0$ has two distinct real solutions. Compute the lesser of these... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ -8, and let $i=\\sqrt{-1}$. Compute the positive integer $k$ for which $(-1+i)^{k}=\\frac{1}{2^{T}}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2782,
"split": "test",
"question": "Let $T=$ -8, and let $i=\\sqrt{-1}$. Compute the positive integer $k$ for which $(-1+i)^{k}=\\frac{1}{2^{T}}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 16. Compute the value of $x$ that satisfies $\\log _{4} T=\\log _{2} x$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2783,
"split": "test",
"question": "Let $T=$ 16. Compute the value of $x$ that satisfies $\\log _{4} T=\\log _{2} x$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, whose area is $T$. Given that $L E=5 \\sqrt{2}$, compute $[L E O]$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2784,
"split": "test",
"question": "Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, whose area is $T$. Given that $L E=5 \\sqrt{2}$, compute $[L E O]$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=$ 7. Compute the units digit of $T^{2023}+(T-2)^{20}-(T+10)^{23}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2785,
"split": "test",
"question": "Let $T=$ 7. Compute the units digit of $T^{2023}+(T-2)^{20}-(T+10)^{23}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\\overline{B P}$ is tangent to the smaller circle. Compute $B P$.\n\nLet's th... | math | {
"style": "rule",
"ground_truth": "$\\sqrt{15}$"
} | {
"index": 2786,
"split": "test",
"question": "Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\\overline{B P}$ is tangent to the smaller circle. Compute $B P$."... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the largest prime divisor of $15 !-13$ !.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2787,
"split": "test",
"question": "Compute the largest prime divisor of $15 !-13$ !."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Three non-overlapping squares of positive integer side lengths each have one vertex at the origin and sides parallel to the coordinate axes. Together, the three squares enclose a region whose area is 41 . Compute the largest possible perimeter of the region.\n\nLet's think step... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2788,
"split": "test",
"question": "Three non-overlapping squares of positive integer side lengths each have one vertex at the origin and sides parallel to the coordinate axes. Together, the three squares enclose a region whose area is 41 . Compute the largest possible perimeter of the region."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A circle with center $O$ and radius 1 contains chord $\\overline{A B}$ of length 1 , and point $M$ is the midpoint of $\\overline{A B}$. If the perpendicular to $\\overline{A O}$ through $M$ intersects $\\overline{A O}$ at $P$, compute $[M A P]$.\n\nLet's think step by step and... | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{3}}{32}$"
} | {
"index": 2789,
"split": "test",
"question": "A circle with center $O$ and radius 1 contains chord $\\overline{A B}$ of length 1 , and point $M$ is the midpoint of $\\overline{A B}$. If the perpendicular to $\\overline{A O}$ through $M$ intersects $\\overline{A O}$ at $P$, compute $[M A P]$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 2790,
"split": "test",
"question": "$\\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real numbers in arithmetic progression. Compute the value of $j$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-50"
} | {
"index": 2791,
"split": "test",
"question": "The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real numbers in arithmetic progression. Compute the value of $j$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that\n\n$$\nn,\\lfloor\\sqrt{n}\\rfloor,\\lfloor\\sqrt[3]{n}\\rfloor,\\lfloor\\sqrt[4]{n}\\rfloor,\\lfloor\\sqrt[5]{n}\\rfloor,\\lfloor\\sqrt[6]{n}\\rfloor,\\lfloor\\sqrt[7]{n}\\rfloor, \\text { and }\\lfloor\\sqrt[8]{n}\\rfloor\n$... | math | {
"style": "rule",
"ground_truth": "4096"
} | {
"index": 2792,
"split": "test",
"question": "Compute the smallest positive integer $n$ such that\n\n$$\nn,\\lfloor\\sqrt{n}\\rfloor,\\lfloor\\sqrt[3]{n}\\rfloor,\\lfloor\\sqrt[4]{n}\\rfloor,\\lfloor\\sqrt[5]{n}\\rfloor,\\lfloor\\sqrt[6]{n}\\rfloor,\\lfloor\\sqrt[7]{n}\\rfloor, \\text { and }\\lfloor\\sqrt[8]{n}... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "If $n$ is a positive integer, then $n$ !! is defined to be $n(n-2)(n-4) \\cdots 2$ if $n$ is even and $n(n-2)(n-4) \\cdots 1$ if $n$ is odd. For example, $8 ! !=8 \\cdot 6 \\cdot 4 \\cdot 2=384$ and $9 ! !=9 \\cdot 7 \\cdot 5 \\cdot 3 \\cdot 1=945$. Compute the number of positi... | math | {
"style": "rule",
"ground_truth": "1510"
} | {
"index": 2793,
"split": "test",
"question": "If $n$ is a positive integer, then $n$ !! is defined to be $n(n-2)(n-4) \\cdots 2$ if $n$ is even and $n(n-2)(n-4) \\cdots 1$ if $n$ is odd. For example, $8 ! !=8 \\cdot 6 \\cdot 4 \\cdot 2=384$ and $9 ! !=9 \\cdot 7 \\cdot 5 \\cdot 3 \\cdot 1=945$. Compute the numbe... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "On the complex plane, the parallelogram formed by the points $0, z, \\frac{1}{z}$, and $z+\\frac{1}{z}$ has area $\\frac{35}{37}$, and the real part of $z$ is positive. If $d$ is the smallest possible value of $\\left|z+\\frac{1}{z}\\right|$, compute $d^{2}$.\n\nLet's think ste... | math | {
"style": "rule",
"ground_truth": "$\\frac{50}{37}$"
} | {
"index": 2794,
"split": "test",
"question": "On the complex plane, the parallelogram formed by the points $0, z, \\frac{1}{z}$, and $z+\\frac{1}{z}$ has area $\\frac{35}{37}$, and the real part of $z$ is positive. If $d$ is the smallest possible value of $\\left|z+\\frac{1}{z}\\right|$, compute $d^{2}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "One face of a $2 \\times 2 \\times 2$ cube is painted (not the entire cube), and the cube is cut into eight $1 \\times 1 \\times 1$ cubes. The small cubes are reassembled randomly into a $2 \\times 2 \\times 2$ cube. Compute the probability that no paint is showing.\n\nLet's th... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{16}$"
} | {
"index": 2795,
"split": "test",
"question": "One face of a $2 \\times 2 \\times 2$ cube is painted (not the entire cube), and the cube is cut into eight $1 \\times 1 \\times 1$ cubes. The small cubes are reassembled randomly into a $2 \\times 2 \\times 2$ cube. Compute the probability that no paint is showing."... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In triangle $A B C, A B=B C$. A trisector of $\\angle B$ intersects $\\overline{A C}$ at $D$. If $A B, A C$, and $B D$ are integers and $A B-B D=7$, compute $A C$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 2796,
"split": "test",
"question": "In triangle $A B C, A B=B C$. A trisector of $\\angle B$ intersects $\\overline{A C}$ at $D$. If $A B, A C$, and $B D$ are integers and $A B-B D=7$, compute $A C$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The rational number $r$ is the largest number less than 1 whose base-7 expansion consists of two distinct repeating digits, $r=0 . \\underline{A} \\underline{B} \\underline{A} \\underline{B} \\underline{A} \\underline{B} \\ldots$ Written as a reduced fraction, $r=\\frac{p}{q}$.... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 2797,
"split": "test",
"question": "The rational number $r$ is the largest number less than 1 whose base-7 expansion consists of two distinct repeating digits, $r=0 . \\underline{A} \\underline{B} \\underline{A} \\underline{B} \\underline{A} \\underline{B} \\ldots$ Written as a reduced fraction, $r=\\f... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=95$. Triangle $A B C$ has $A B=A C$. Points $M$ and $N$ lie on $\\overline{B C}$ such that $\\overline{A M}$ and $\\overline{A N}$ trisect $\\angle B A C$, with $M$ closer to $C$. If $\\mathrm{m} \\angle A M C=T^{\\circ}$, then $\\mathrm{m} \\angle A C B=U^{\\circ}$. Com... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 2798,
"split": "test",
"question": "Let $T=95$. Triangle $A B C$ has $A B=A C$. Points $M$ and $N$ lie on $\\overline{B C}$ such that $\\overline{A M}$ and $\\overline{A N}$ trisect $\\angle B A C$, with $M$ closer to $C$. If $\\mathrm{m} \\angle A M C=T^{\\circ}$, then $\\mathrm{m} \\angle A C B=U^{\\... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=75$. At Wash College of Higher Education (Wash Ed.), the entering class has $n$ students. Each day, two of these students are selected to oil the slide rules. If the entering class had two more students, there would be $T$ more ways of selecting the two slide rule oilers... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 2799,
"split": "test",
"question": "Let $T=75$. At Wash College of Higher Education (Wash Ed.), the entering class has $n$ students. Each day, two of these students are selected to oil the slide rules. If the entering class had two more students, there would be $T$ more ways of selecting the two slide ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the least positive integer $n$ such that the set of angles\n\n$$\n\\left\\{123^{\\circ}, 246^{\\circ}, \\ldots, n \\cdot 123^{\\circ}\\right\\}\n$$\n\ncontains at least one angle in each of the four quadrants.\n\nLet's think step by step and output the final answer with... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2800,
"split": "test",
"question": "Compute the least positive integer $n$ such that the set of angles\n\n$$\n\\left\\{123^{\\circ}, 246^{\\circ}, \\ldots, n \\cdot 123^{\\circ}\\right\\}\n$$\n\ncontains at least one angle in each of the four quadrants."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=11$. In ARMLvania, license plates use only the digits 1-9, and each license plate contains exactly $T-3$ digits. On each plate, all digits are distinct, and for all $k \\leq T-3$, the $k^{\\text {th }}$ digit is at least $k$. Compute the number of valid ARMLvanian licens... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 2801,
"split": "test",
"question": "Let $T=11$. In ARMLvania, license plates use only the digits 1-9, and each license plate contains exactly $T-3$ digits. On each plate, all digits are distinct, and for all $k \\leq T-3$, the $k^{\\text {th }}$ digit is at least $k$. Compute the number of valid ARMLva... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=256$. Let $\\mathcal{R}$ be the region in the plane defined by the inequalities $x^{2}+y^{2} \\geq T$ and $|x|+|y| \\leq \\sqrt{2 T}$. Compute the area of region $\\mathcal{R}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$1024-256 \\pi$"
} | {
"index": 2802,
"split": "test",
"question": "Let $T=256$. Let $\\mathcal{R}$ be the region in the plane defined by the inequalities $x^{2}+y^{2} \\geq T$ and $|x|+|y| \\leq \\sqrt{2 T}$. Compute the area of region $\\mathcal{R}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Triangle $A B C$ has $\\mathrm{m} \\angle A>\\mathrm{m} \\angle B>\\mathrm{m} \\angle C$. The angle between the altitude and the angle bisector at vertex $A$ is $6^{\\circ}$. The angle between the altitude and the angle bisector at vertex $B$ is $18^{\\circ}$. Compute the degre... | math | {
"style": "rule",
"ground_truth": "$44$"
} | {
"index": 2817,
"split": "test",
"question": "Triangle $A B C$ has $\\mathrm{m} \\angle A>\\mathrm{m} \\angle B>\\mathrm{m} \\angle C$. The angle between the altitude and the angle bisector at vertex $A$ is $6^{\\circ}$. The angle between the altitude and the angle bisector at vertex $B$ is $18^{\\circ}$. Comput... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(b, c)$, with $-20 \\leq b \\leq 20,-20 \\leq c \\leq 20$, such that the equations $x^{2}+b x+c=0$ and $x^{2}+c x+b=0$ share at least one root.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 2818,
"split": "test",
"question": "Compute the number of ordered pairs of integers $(b, c)$, with $-20 \\leq b \\leq 20,-20 \\leq c \\leq 20$, such that the equations $x^{2}+b x+c=0$ and $x^{2}+c x+b=0$ share at least one root."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A seventeen-sided die has faces numbered 1 through 17, but it is not fair: 17 comes up with probability $1 / 2$, and each of the numbers 1 through 16 comes up with probability $1 / 32$. Compute the probability that the sum of two rolls is either 20 or 12.\n\nLet's think step by... | math | {
"style": "rule",
"ground_truth": "$\\frac{7}{128}$"
} | {
"index": 2819,
"split": "test",
"question": "A seventeen-sided die has faces numbered 1 through 17, but it is not fair: 17 comes up with probability $1 / 2$, and each of the numbers 1 through 16 comes up with probability $1 / 32$. Compute the probability that the sum of two rolls is either 20 or 12."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \\leq 50,1<b \\leq 50$, and $\\log _{b} a$ is rational.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 2821,
"split": "test",
"question": "Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \\leq 50,1<b \\leq 50$, and $\\log _{b} a$ is rational."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Suppose that 5-letter \"words\" are formed using only the letters A, R, M, and L. Each letter need not be used in a word, but each word must contain at least two distinct letters. Compute the number of such words that use the letter A more than any other letter.\n\nLet's think ... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 2822,
"split": "test",
"question": "Suppose that 5-letter \"words\" are formed using only the letters A, R, M, and L. Each letter need not be used in a word, but each word must contain at least two distinct letters. Compute the number of such words that use the letter A more than any other letter."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ form an arithmetic sequence. If $a_{1}=10$ and $a_{a_{2}}=100$, compute $a_{a_{a_{3}}}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "820"
} | {
"index": 2823,
"split": "test",
"question": "Positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ form an arithmetic sequence. If $a_{1}=10$ and $a_{a_{2}}=100$, compute $a_{a_{a_{3}}}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The graphs of $y=x^{2}-|x|-12$ and $y=|x|-k$ intersect at distinct points $A, B, C$, and $D$, in order of increasing $x$-coordinates. If $A B=B C=C D$, compute $k$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$10+2 \\sqrt{2}$"
} | {
"index": 2824,
"split": "test",
"question": "The graphs of $y=x^{2}-|x|-12$ and $y=|x|-k$ intersect at distinct points $A, B, C$, and $D$, in order of increasing $x$-coordinates. If $A B=B C=C D$, compute $k$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The zeros of $f(x)=x^{6}+2 x^{5}+3 x^{4}+5 x^{3}+8 x^{2}+13 x+21$ are distinct complex numbers. Compute the average value of $A+B C+D E F$ over all possible permutations $(A, B, C, D, E, F)$ of these six numbers.\n\nLet's think step by step and output the final answer within \\... | math | {
"style": "rule",
"ground_truth": "$-\\frac{23}{60}$"
} | {
"index": 2825,
"split": "test",
"question": "The zeros of $f(x)=x^{6}+2 x^{5}+3 x^{4}+5 x^{3}+8 x^{2}+13 x+21$ are distinct complex numbers. Compute the average value of $A+B C+D E F$ over all possible permutations $(A, B, C, D, E, F)$ of these six numbers."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $N=\\left\\lfloor(3+\\sqrt{5})^{34}\\right\\rfloor$. Compute the remainder when $N$ is divided by 100 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 2827,
"split": "test",
"question": "Let $N=\\left\\lfloor(3+\\sqrt{5})^{34}\\right\\rfloor$. Compute the remainder when $N$ is divided by 100 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $\\mathrm{m} \\angle B=\\mathrm{m} \\angle C=80^{\\circ}$. Compute the number of points $P$ in the plane such that triangles $P A B, P B C$, and $P C A$ are all isosceles and non-degenerate. Note: the approximation $\\cos 80^{\\circ} \\approx 0.17... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2828,
"split": "test",
"question": "Let $A B C$ be a triangle with $\\mathrm{m} \\angle B=\\mathrm{m} \\angle C=80^{\\circ}$. Compute the number of points $P$ in the plane such that triangles $P A B, P B C$, and $P C A$ are all isosceles and non-degenerate. Note: the approximation $\\cos 80^{\\circ} \\... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "If $\\lceil u\\rceil$ denotes the least integer greater than or equal to $u$, and $\\lfloor u\\rfloor$ denotes the greatest integer less than or equal to $u$, compute the largest solution $x$ to the equation\n\n$$\n\\left\\lfloor\\frac{x}{3}\\right\\rfloor+\\lceil 3 x\\rceil=\\... | math | {
"style": "rule",
"ground_truth": "$\\frac{189 \\sqrt{11}}{11}$"
} | {
"index": 2829,
"split": "test",
"question": "If $\\lceil u\\rceil$ denotes the least integer greater than or equal to $u$, and $\\lfloor u\\rfloor$ denotes the greatest integer less than or equal to $u$, compute the largest solution $x$ to the equation\n\n$$\n\\left\\lfloor\\frac{x}{3}\\right\\rfloor+\\lceil 3 ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "If $x, y$, and $z$ are positive integers such that $x y=20$ and $y z=12$, compute the smallest possible value of $x+z$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2830,
"split": "test",
"question": "If $x, y$, and $z$ are positive integers such that $x y=20$ and $y z=12$, compute the smallest possible value of $x+z$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=8$. Let $A=(1,5)$ and $B=(T-1,17)$. Compute the value of $x$ such that $(x, 3)$ lies on the perpendicular bisector of $\\overline{A B}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2831,
"split": "test",
"question": "Let $T=8$. Let $A=(1,5)$ and $B=(T-1,17)$. Compute the value of $x$ such that $(x, 3)$ lies on the perpendicular bisector of $\\overline{A B}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let T be a rational number. Let $N$ be the smallest positive $T$-digit number that is divisible by 33 . Compute the product of the last two digits of $N$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2832,
"split": "test",
"question": "Let T be a rational number. Let $N$ be the smallest positive $T$-digit number that is divisible by 33 . Compute the product of the last two digits of $N$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n}(z)=$ $f_{n-1}(\\bar{z})$. If $f_{1}(z)+2 f_{2}(z)+3 f_{3}(z)+4 f_{4}(z)+5 f_{5}(z)=T+T i$, compute $|z|$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{26}$"
} | {
"index": 2834,
"split": "test",
"question": "Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n}(z)=$ $f_{n-1}(\\bar{z})$. If $f_{1}(z)+2 f_{2}(z)+3 f_{3}(z)+4 f_{4}(z)+5 f_{5}(z)=T+T i$, compute $|z|$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=\\sqrt{26}$. Compute the number of ordered pairs of positive integers $(a, b)$ with the property that $a b=T^{20} \\cdot 210^{12}$, and the greatest common divisor of $a$ and $b$ is 1 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2835,
"split": "test",
"question": "Let $T=\\sqrt{26}$. Compute the number of ordered pairs of positive integers $(a, b)$ with the property that $a b=T^{20} \\cdot 210^{12}$, and the greatest common divisor of $a$ and $b$ is 1 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=32$. Given that $\\sin \\theta=\\frac{\\sqrt{T^{2}-64}}{T}$, compute the largest possible value of the infinite series $\\cos \\theta+\\cos ^{2} \\theta+\\cos ^{3} \\theta+\\ldots$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{3}$"
} | {
"index": 2836,
"split": "test",
"question": "Let $T=32$. Given that $\\sin \\theta=\\frac{\\sqrt{T^{2}-64}}{T}$, compute the largest possible value of the infinite series $\\cos \\theta+\\cos ^{2} \\theta+\\cos ^{3} \\theta+\\ldots$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=\\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of square $A B C... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 2838,
"split": "test",
"question": "Let $T=\\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of sq... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=-14$, and let $d=|T|$. A person whose birthday falls between July 23 and August 22 inclusive is called a Leo. A person born in July is randomly selected, and it is given that her birthday is before the $d^{\\text {th }}$ day of July. Another person who was also born in J... | math | {
"style": "rule",
"ground_truth": "$\\frac{9}{17}$"
} | {
"index": 2839,
"split": "test",
"question": "Let $T=-14$, and let $d=|T|$. A person whose birthday falls between July 23 and August 22 inclusive is called a Leo. A person born in July is randomly selected, and it is given that her birthday is before the $d^{\\text {th }}$ day of July. Another person who was als... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=-10$. Given that $\\log _{2} 4^{8 !}+\\log _{4} 2^{8 !}=6 ! \\cdot T \\cdot x$, compute $x$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 2840,
"split": "test",
"question": "Let $T=-10$. Given that $\\log _{2} 4^{8 !}+\\log _{4} 2^{8 !}=6 ! \\cdot T \\cdot x$, compute $x$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=20$. For some real constants $a$ and $b$, the solution sets of the equations $x^{2}+(5 b-T-a) x=T+1$ and $2 x^{2}+(T+8 a-2) x=-10 b$ are the same. Compute $a$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 2841,
"split": "test",
"question": "Let $T=20$. For some real constants $a$ and $b$, the solution sets of the equations $x^{2}+(5 b-T-a) x=T+1$ and $2 x^{2}+(T+8 a-2) x=-10 b$ are the same. Compute $a$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let T be a rational number, and let $K=T-2$. If $K$ workers can produce 9 widgets in 1 hour, compute the number of workers needed to produce $\\frac{720}{K}$ widgets in 4 hours.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2842,
"split": "test",
"question": "Let T be a rational number, and let $K=T-2$. If $K$ workers can produce 9 widgets in 1 hour, compute the number of workers needed to produce $\\frac{720}{K}$ widgets in 4 hours."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=2018$, and append the digits of $T$ to $\\underline{A} \\underline{A} \\underline{B}$ (for example, if $T=17$, then the result would be $\\underline{1} \\underline{\\underline{A}} \\underline{A} \\underline{B}$ ). If the resulting number is divisible by 11 , compute the ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2843,
"split": "test",
"question": "Let $T=2018$, and append the digits of $T$ to $\\underline{A} \\underline{A} \\underline{B}$ (for example, if $T=17$, then the result would be $\\underline{1} \\underline{\\underline{A}} \\underline{A} \\underline{B}$ ). If the resulting number is divisible by 11 , c... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given that April $1^{\\text {st }}, 2012$ fell on a Sunday, what is the next year in which April $1^{\\text {st }}$ will fall on a Sunday?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"index": 2844,
"split": "test",
"question": "Given that April $1^{\\text {st }}, 2012$ fell on a Sunday, what is the next year in which April $1^{\\text {st }}$ will fall on a Sunday?"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $p$ be a prime number. If $p$ years ago, the ages of three children formed a geometric sequence with a sum of $p$ and a common ratio of 2 , compute the sum of the children's current ages.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2845,
"split": "test",
"question": "Let $p$ be a prime number. If $p$ years ago, the ages of three children formed a geometric sequence with a sum of $p$ and a common ratio of 2 , compute the sum of the children's current ages."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Define a reverse prime to be a positive integer $N$ such that when the digits of $N$ are read in reverse order, the resulting number is a prime. For example, the numbers 5, 16, and 110 are all reverse primes. Compute the largest two-digit integer $N$ such that the numbers $N, 4... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 2846,
"split": "test",
"question": "Define a reverse prime to be a positive integer $N$ such that when the digits of $N$ are read in reverse order, the resulting number is a prime. For example, the numbers 5, 16, and 110 are all reverse primes. Compute the largest two-digit integer $N$ such that the nu... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Some students in a gym class are wearing blue jerseys, and the rest are wearing red jerseys. There are exactly 25 ways to pick a team of three players that includes at least one player wearing each color. Compute the number of students in the class.\n\nLet's think step by step ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2847,
"split": "test",
"question": "Some students in a gym class are wearing blue jerseys, and the rest are wearing red jerseys. There are exactly 25 ways to pick a team of three players that includes at least one player wearing each color. Compute the number of students in the class."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Point $P$ is on the hypotenuse $\\overline{E N}$ of right triangle $B E N$ such that $\\overline{B P}$ bisects $\\angle E B N$. Perpendiculars $\\overline{P R}$ and $\\overline{P S}$ are drawn to sides $\\overline{B E}$ and $\\overline{B N}$, respectively. If $E N=221$ and $P R... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{60}$"
} | {
"index": 2848,
"split": "test",
"question": "Point $P$ is on the hypotenuse $\\overline{E N}$ of right triangle $B E N$ such that $\\overline{B P}$ bisects $\\angle E B N$. Perpendiculars $\\overline{P R}$ and $\\overline{P S}$ are drawn to sides $\\overline{B E}$ and $\\overline{B N}$, respectively. If $E N=22... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Compute all real values of $x$ such that $\\log _{2}\\left(\\log _{2} x\\right)=\\log _{4}\\left(\\log _{4} x\\right)$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{2}$"
} | {
"index": 2849,
"split": "test",
"question": "$\\quad$ Compute all real values of $x$ such that $\\log _{2}\\left(\\log _{2} x\\right)=\\log _{4}\\left(\\log _{4} x\\right)$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $k$ be the least common multiple of the numbers in the set $\\mathcal{S}=\\{1,2, \\ldots, 30\\}$. Determine the number of positive integer divisors of $k$ that are divisible by exactly 28 of the numbers in the set $\\mathcal{S}$.\n\nLet's think step by step and output the f... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 2850,
"split": "test",
"question": "Let $k$ be the least common multiple of the numbers in the set $\\mathcal{S}=\\{1,2, \\ldots, 30\\}$. Determine the number of positive integer divisors of $k$ that are divisible by exactly 28 of the numbers in the set $\\mathcal{S}$."
} |
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