data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In an isosceles triangle, one of the angles opposite an equal side is $40^{\\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label(\"$\\backslash$\",(1.5,1)); label(\"{/}\",(4.5,1));\nlabel(\"$40^{\\circ}$\",(.5,0),dir(45));\n[/a... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3600",
"split": "train"
} |
math | [
{
"content": "Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3601",
"split": "train"
} |
math | [
{
"content": "The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "3602",
"split": "train"
} |
math | [
{
"content": "The numbers $4x, 2x-3, 4x-3$ are three consecutive terms in an arithmetic sequence. What is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{4}",
"style": "rule"
} | {
"index": "3603",
"split": "train"
} |
math | [
{
"content": "Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \\geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "3604",
"split": "train"
} |
math | [
{
"content": "Three cars leave city $A$ at the same time and travel along a closed path composed of three straight segments $AB, BC$, and $CA$. The speeds of the first car on these segments are 12, 10, and 15 kilometers per hour, respectively. The speeds of the second car are 15, 15, and 10 kilometers per hour,... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "3605",
"split": "train"
} |
math | [
{
"content": "In the triangle $\\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\\alpha$ and $\\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\\beta = 2 \\tan^{-1} (1/3)$. Find $\\alpha$... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "3606",
"split": "train"
} |
math | [
{
"content": "4.\tFind the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:\nIf $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1944",
"style": "rule"
} | {
"index": "3607",
"split": "train"
} |
math | [
{
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ denote the sequence of real numbers such that $a_{0}=2$ and $a_{n+1}=\\frac{a_{n}}{1+a_{n}}$ for $n \\geq 0$. Compute $a_{2012}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{4025}",
"style": "rule"
} | {
"index": "3608",
"split": "train"
} |
math | [
{
"content": "If $5(\\cos a + \\cos b) + 4(\\cos a \\cos b + 1) = 0,$ then find all possible values of\n\\[\\tan \\frac{a}{2} \\tan \\frac{b}{2}.\\]Enter all the possible values, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3,-3",
"style": "rule"
} | {
"index": "3609",
"split": "train"
} |
math | [
{
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 54, and passes through the points $(-2,0)$ and $(4,0).$ Find $a + b + c.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "3610",
"split": "train"
} |
math | [
{
"content": "Arthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he walk? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3611",
"split": "train"
} |
math | [
{
"content": "Murtha has decided to start a pebble collection. She collects one pebble the first day and two pebbles on the second day. On each subsequent day she collects one more pebble than the previous day. How many pebbles will she have collected at the end of the twelfth day? Please output the final answe... | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "3612",
"split": "train"
} |
math | [
{
"content": "Find the sum of the areas of all distinct rectangles that can be formed from 9 squares (not necessarily all), if the side of each square is $1 \\text{ cm}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "3613",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3614",
"split": "train"
} |
math | [
{
"content": "Consider the set $\\{2, 7, 12, 17, 22, 27, 32\\}$. Calculate the number of different integers that can be expressed as the sum of three distinct members of this set. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3615",
"split": "train"
} |
math | [
{
"content": "Given the values 1432, 3214, 2143, and 4321, calculate their sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11110",
"style": "rule"
} | {
"index": "3616",
"split": "train"
} |
math | [
{
"content": "Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.\n ~ pi_is_3.14 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "3617",
"split": "train"
} |
math | [
{
"content": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3618",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=x^{3}+3x^{2}+6x$, $f(a)=1$, $f(b)=-9$, the value of $a+b$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "3619",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{vmatrix} x & \\frac {1}{x} \\\\ -2 & 1\\end{vmatrix}$ ($x>0$), its range is set $A$, \n(1) If the universal set $U=\\mathbb{R}$, find $C_UA$; \n(2) For any $x \\in (0, \\frac {1}{2}]$, the inequality $f(x)+a \\geq 0$ always holds, find the range of the real numb... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "3620",
"split": "train"
} |
math | [
{
"content": "Evaluate the polynomial \\[ x^3 - 3x^2 - 12x + 9, \\] where \\(x\\) is the positive number such that \\(x^2 - 3x - 12 = 0\\) and \\(x \\neq -2\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-23",
"style": "rule"
} | {
"index": "3621",
"split": "train"
} |
math | [
{
"content": "For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive integer value of $n$?... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "3622",
"split": "train"
} |
math | [
{
"content": "Shelly writes down a vector $v=(a, b, c, d)$, where $0<a<b<c<d$ are integers. Let $\\sigma(v)$ denote the set of 24 vectors whose coordinates are $a, b, c$, and $d$ in some order. For instance, $\\sigma(v)$ contains $(b, c, d, a)$. Shelly notes that there are 3 vectors in $\\sigma(v)$ whose sum is... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3623",
"split": "train"
} |
math | [
{
"content": "Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "3624",
"split": "train"
} |
math | [
{
"content": "Evaluate \n\\[\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3} + ... + \\frac{1}{t_{2003}}\\]\nwhere \\( t_n = \\frac{n(n+1)}{2} \\) is the $n$th triangular number.\nA) $\\frac{2005}{1002}$ \nB) $\\frac{4006}{2003}$ \nC) $\\frac{2003}{1001}$ \nD) $\\frac{2003}{1002}$ \nE) 2 Please output the f... | {
"ground_truth": "\\frac{2003}{1002}",
"style": "rule"
} | {
"index": "3625",
"split": "train"
} |
math | [
{
"content": "At a national competition, 31 participants are accommodated in a hotel where each participant gets his/her own room. The rooms are numbered from 1 to 31. When all participants have arrived, except those assigned to rooms 15, 16, and 17, what is the median room number of the remaining 28 participan... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3626",
"split": "train"
} |
math | [
{
"content": "Given the line $y = \\frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y =\\frac{3}{4}x+1",
"style": "rule"
} | {
"index": "3627",
"split": "train"
} |
math | [
{
"content": "$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\\angle A P B=90^{\\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find $A F$. Please output... | {
"ground_truth": "499 / 500",
"style": "rule"
} | {
"index": "3628",
"split": "train"
} |
math | [
{
"content": "How many ways are there to choose 6 different numbers from $1, 2, \\cdots, 49$, where at least two numbers are consecutive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\binom{49}{6} - \\binom{44}{6}",
"style": "rule"
} | {
"index": "3629",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs $(s, d)$ of positive integers with $4 \\leq s \\leq d \\leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\\frac{1}{2}$ ? Please output the fin... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "3630",
"split": "train"
} |
math | [
{
"content": "The product of the first three terms of a geometric sequence is 2, the product of the last three terms is 4, and the product of all terms is 64. Calculate the number of terms in this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3631",
"split": "train"
} |
math | [
{
"content": "On a Cartesian coordinate plane, points \\((4,-1)\\) and \\((-1, 3)\\) are adjacent corners on a square. Calculate the area of this square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "3632",
"split": "train"
} |
math | [
{
"content": "Carl drove continuously from 7:30 a.m. until 2:15 p.m. of the same day and covered a distance of 234 miles. What was his average speed in miles per hour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{936}{27}",
"style": "rule"
} | {
"index": "3633",
"split": "train"
} |
math | [
{
"content": "The average of 10 numbers is 85. If the numbers 70 and 76 are removed from the set of numbers, what is the average of the remaining numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "3634",
"split": "train"
} |
math | [
{
"content": "Given that point P is any point on the graph of the function $f(x) = 2\\sqrt{2x}$, and a tangent line is drawn from point P to circle D: $x^2 + y^2 - 4x + 3 = 0$, with the points of tangency being A and B, find the minimum value of the area of quadrilateral PADB. Please output the final answer wit... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "3635",
"split": "train"
} |
math | [
{
"content": "Given that $\\triangle ABC$ is an acute triangle, vector $\\overrightarrow{m}=(\\cos (A+ \\frac{\\pi}{3}),\\sin (A+ \\frac{\\pi}{3}))$, $\\overrightarrow{n}=(\\cos B,\\sin B)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n(I) Find the value of $A-B$;\n(II) If $\\cos B= \\frac{3}{5}$, and... | {
"ground_truth": "4\\sqrt{3}+3",
"style": "rule"
} | {
"index": "3636",
"split": "train"
} |
math | [
{
"content": "The product of positive integers $a$, $b$, and $c$ equals 2450. What is the minimum possible value of the sum $a + b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "3637",
"split": "train"
} |
math | [
{
"content": "Read the material first, then answer the question. \n$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\\sqrt{5-2\\sqrt{6}}$. \nAfter thinking about it, Xiao Zhang's process of solving this problem is as follows: \n$\\sqrt{5-2\\sqrt{6}}=\\sqrt{2-2\\sqrt{2\\t... | {
"ground_truth": "\\sqrt{6}+\\sqrt{2}",
"style": "rule"
} | {
"index": "3638",
"split": "train"
} |
math | [
{
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, C, D, E;\nA=(0,20.785);\nB=(0,0);\nC=(9,-5.196);\nD=(13.5,-2.598);\nE=(12,0);\ndraw(A--B--C--D--E--A);\ndraw(B--E);\ndr... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3639",
"split": "train"
} |
math | [
{
"content": "A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, ski... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3640",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ.$ What is the measure of $\\angle R?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"index": "3641",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=|\\log_{x}|$, $a > b > 0$, $f(a)=f(b)$, calculate the minimum value of $\\frac{a^2+b^2}{a-b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "3642",
"split": "train"
} |
math | [
{
"content": "How many cubic centimeters are in the volume of a cone having a diameter of 12 cm and a slant height of 10 cm? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96 \\pi",
"style": "rule"
} | {
"index": "3643",
"split": "train"
} |
math | [
{
"content": "Simplify $({\\frac{3}{{a+1}}-a+1})÷\\frac{{{a^2}-4}}{{{a^2}+2a+1}}$, then choose a suitable number from $-1$, $2$, $3$ to substitute and evaluate. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "3644",
"split": "train"
} |
math | [
{
"content": "Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster than Fluffball, and se... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "3645",
"split": "train"
} |
math | [
{
"content": "Given that the sides of triangle $\\triangle ABC$ opposite to angles $A$, $B$, and $C$ are in arithmetic sequence and $C=2\\left(A+B\\right)$, calculate the ratio $\\frac{b}{a}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "3646",
"split": "train"
} |
math | [
{
"content": "For $k\\ge 1$ , define $a_k=2^k$ . Let $$ S=\\sum_{k=1}^{\\infty}\\cos^{-1}\\left(\\frac{2a_k^2-6a_k+5}{\\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\\right). $$ Compute $\\lfloor 100S\\rfloor$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "157",
"style": "rule"
} | {
"index": "3647",
"split": "train"
} |
math | [
{
"content": "Compute $\\tan (-3645^\\circ)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "3648",
"split": "train"
} |
math | [
{
"content": "What is the mean of the set $\\{m, m + 6, m + 8, m + 11, m + 18, m + 20\\}$ if the median of the set is 19? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "3649",
"split": "train"
} |
math | [
{
"content": "If \\(\\frac{\\left(\\frac{a}{c}+\\frac{a}{b}+1\\right)}{\\left(\\frac{b}{a}+\\frac{b}{c}+1\\right)}=11\\), and \\(a, b\\), and \\(c\\) are positive integers, then the number of ordered triples \\((a, b, c)\\), such that \\(a+2b+c \\leq 40\\), is: Please output the final answer within \\boxed{}.",... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "3650",
"split": "train"
} |
math | [
{
"content": "Let the function $y=f(x)$ have the domain $D$. If for any $x_{1}, x_{2} \\in D$, when $x_{1}+x_{2}=2a$, it always holds that $f(x_{1})+f(x_{2})=2b$, then the point $(a,b)$ is called the symmetry center of the graph of the function $y=f(x)$. Study a symmetry point of the graph of the function $f(x)... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "3651",
"split": "train"
} |
math | [
{
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $\\overrightarrow{a}\\cdot (\\overrightarrow{b}-\\overrightarrow{a})=0$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "3652",
"split": "train"
} |
math | [
{
"content": "On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them and said: \"Now togeth... | {
"ground_truth": "1346",
"style": "rule"
} | {
"index": "3653",
"split": "train"
} |
math | [
{
"content": "The opposite number of $2- \\sqrt{3}$ is ______, and its absolute value is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2- \\sqrt{3}",
"style": "rule"
} | {
"index": "3654",
"split": "train"
} |
math | [
{
"content": "Given a tetrahedron \\(ABCD\\). Points \\(M\\), \\(N\\), and \\(K\\) lie on edges \\(AD\\), \\(BC\\), and \\(DC\\) respectively, such that \\(AM:MD = 1:3\\), \\(BN:NC = 1:1\\), and \\(CK:KD = 1:2\\). Construct the section of the tetrahedron with the plane \\(MNK\\). In what ratio does this plane d... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "3655",
"split": "train"
} |
math | [
{
"content": "Given the following propositions:\n\n(1) The graph of the function $y=3^{x} (x \\in \\mathbb{R})$ is symmetric to the graph of the function $y=\\log_{3}x (x > 0)$ with respect to the line $y=x$;\n\n(2) The smallest positive period of the function $y=|\\sin x|$ is $2\\pi$;\n\n(3) The graph of the f... | {
"ground_truth": "(1)(3)(4)",
"style": "rule"
} | {
"index": "3656",
"split": "train"
} |
math | [
{
"content": "Three people, Jia, Yi, and Bing, participated in a competition and they took the top 3 places (with no ties). Jia said: \"I am first\", Yi said: \"I am not first\", and Bing said: \"I am not third\". Only one of them is telling the truth. If the rankings of Jia, Yi, and Bing are respectively $A, B... | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "3657",
"split": "train"
} |
math | [
{
"content": "Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to the volume of Bonnie... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "3658",
"split": "train"
} |
math | [
{
"content": "Which one of the following points is not on the graph of $y=\\dfrac{x}{x+1}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-1,1)",
"style": "rule"
} | {
"index": "3659",
"split": "train"
} |
math | [
{
"content": "What is the value of $x$ if $|x-1| = |x-2|$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "3660",
"split": "train"
} |
math | [
{
"content": "Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3661",
"split": "train"
} |
math | [
{
"content": "When 5 integers are arranged in ascending order, the median is 6. If the only mode of this data set is 8, determine the possible maximum sum of these 5 integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "3662",
"split": "train"
} |
math | [
{
"content": "Let \\( a_1, a_2, \\dots \\) be a sequence of positive real numbers such that\n\\[ a_n = 7a_{n-1} - n \\]for all \\( n > 1 \\). Find the smallest possible value of \\( a_1 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{36}",
"style": "rule"
} | {
"index": "3663",
"split": "train"
} |
math | [
{
"content": "Given that $(a + \\frac{1}{a})^3 = 3$, find the value of $a^4 + \\frac{1}{a^4}$.\nA) $9^{1/3} - 4 \\cdot 3^{1/3} + 2$\nB) $9^{1/3} - 2 \\cdot 3^{1/3} + 2$\nC) $9^{1/3} + 4 \\cdot 3^{1/3} + 2$\nD) $4 \\cdot 3^{1/3} - 9^{1/3}$ Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "9^{1/3} - 4 \\cdot 3^{1/3} + 2",
"style": "rule"
} | {
"index": "3664",
"split": "train"
} |
math | [
{
"content": "Cars A and B simultaneously depart from locations $A$ and $B$, and travel uniformly towards each other. When car A has traveled 12 km past the midpoint of $A$ and $B$, the two cars meet. If car A departs 10 minutes later than car B, they meet exactly at the midpoint of $A$ and $B$. When car A reac... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3665",
"split": "train"
} |
math | [
{
"content": "Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming ... | {
"ground_truth": "136",
"style": "rule"
} | {
"index": "3666",
"split": "train"
} |
math | [
{
"content": "Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed a... | {
"ground_truth": "290",
"style": "rule"
} | {
"index": "3667",
"split": "train"
} |
math | [
{
"content": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.\n[... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "3668",
"split": "train"
} |
math | [
{
"content": "Mrs. Thompson teaches science to 20 students. She noticed that when she had recorded the grades for 19 of these students, their average was 76. After including Oliver's grade, the new average grade for all 20 students became 78. Calculate Oliver's score on the test. Please output the final answer ... | {
"ground_truth": "116",
"style": "rule"
} | {
"index": "3669",
"split": "train"
} |
math | [
{
"content": "In the star shown, the sum of the four integers along each straight line is to be the same. Five numbers have been entered. The five missing numbers are 19, 21, 23, 25, and 27. Which number is represented by \\( q \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "3670",
"split": "train"
} |
math | [
{
"content": "Find the fraction \\(\\frac{p}{q}\\) with the smallest possible natural denominator for which \\(\\frac{1}{2014} < \\frac{p}{q} < \\frac{1}{2013}\\). Enter the denominator of this fraction in the provided field. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4027",
"style": "rule"
} | {
"index": "3671",
"split": "train"
} |
math | [
{
"content": "Cyclic pentagon \\( A B C D E \\) has a right angle \\( \\angle A B C = 90^\\circ \\) and side lengths \\( A B = 15 \\) and \\( B C = 20 \\). Supposing that \\( A B = D E = E A \\), find \\( C D \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "3672",
"split": "train"
} |
math | [
{
"content": "A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-2k-24\\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "3673",
"split": "train"
} |
math | [
{
"content": "In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ , the product of $2$ of its roots is $- 32$ . Find $k$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ k = 86 \\]",
"style": "rule"
} | {
"index": "3674",
"split": "train"
} |
math | [
{
"content": "Let $r$, $s$, and $t$ be the three roots of the equation \\[8x^3 + 1001x + 2008 = 0.\\] Find $(r + s)^3 + (s + t)^3 + (t + r)^3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "753",
"style": "rule"
} | {
"index": "3675",
"split": "train"
} |
math | [
{
"content": "Find the curve defined by the equation\n\\[r = 2.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(B)}",
"style": "rule"
} | {
"index": "3676",
"split": "train"
} |
math | [
{
"content": "What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$ ?\n\n*2016 CCA Math Bonanza Lightning #2.4* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3677",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) represent the lengths of the sides of a triangle, with \\(\\alpha\\), \\(\\beta\\), and \\(\\gamma\\) being the angles opposite these sides respectively. Given that \\(a^2 + b^2 = 2023c^2\\), determine the value of \n\\[\n\\frac{\\cot \\alpha}{\\cot \\beta + \\cot... | {
"ground_truth": "1011",
"style": "rule"
} | {
"index": "3678",
"split": "train"
} |
math | [
{
"content": "For how many non-negative real values of $x$ is $\\sqrt{169-\\sqrt[4]{x}}$ an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3679",
"split": "train"
} |
math | [
{
"content": "100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''\n\nHow many of those... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "3680",
"split": "train"
} |
math | [
{
"content": "For each positive integer $p$, let $c(p)$ denote the unique positive integer $k$ such that $|k - \\sqrt[3]{p}| < \\frac{1}{2}$. For example, $c(8)=2$ and $c(27)=3$. Find $T = \\sum_{p=1}^{1728} c(p)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18252",
"style": "rule"
} | {
"index": "3681",
"split": "train"
} |
math | [
{
"content": "Determine the sum of the real numbers \\( x \\) for which \\(\\frac{2 x}{x^{2}+5 x+3}+\\frac{3 x}{x^{2}+x+3}=1\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "3682",
"split": "train"
} |
math | [
{
"content": "Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "3683",
"split": "train"
} |
math | [
{
"content": "The number 3003 is the only number known to appear eight times in Pascal's triangle, at positions $\\binom{3003}{1},\\binom{3003}{3002},\\binom{a}{2},\\binom{a}{a-2},\\binom{15}{b},\\binom{15}{15-b},\\binom{14}{6},\\binom{14}{8}$. Compute $a+b(15-b)$. Please output the final answer within \\boxed{... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "3684",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=\\sin\\left(2x+\\frac{\\pi}{3}\\right)+\\sqrt{3}\\sin^2x-\\sqrt{3}\\cos^2x-\\frac{1}{2}$. \n$(1)$ Find the smallest positive period and the interval of monotonicity of $f(x)$; \n$(2)$ If $x_0\\in\\left[\\frac{5\\pi}{12},\\frac{2\\pi}{3}\\right]$ and $f(x_{0})=\\frac{\\sqrt{3}}{3}-\\fra... | {
"ground_truth": "-\\frac{3+\\sqrt{6}}{6}",
"style": "rule"
} | {
"index": "3685",
"split": "train"
} |
math | [
{
"content": "A polynomial with integer coefficients is of the form\n\\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 18.\\]You are told that the integer $r$ is a double root of this polynomial. (In other words, the polynomial is divisible by $(x - r)^2.$) Enter all the possible values of $r,$ separated by commas. Please... | {
"ground_truth": "-3,-1,1,3",
"style": "rule"
} | {
"index": "3686",
"split": "train"
} |
math | [
{
"content": "Given three points in space \\\\(A(0,2,3)\\\\), \\\\(B(-2,1,6)\\\\), and \\\\(C(1,-1,5)\\\\):\n\\\\((1)\\\\) Find \\\\(\\cos < \\overrightarrow{AB}, \\overrightarrow{AC} >\\\\).\n\\\\((2)\\\\) Find the area of the parallelogram with sides \\\\(AB\\\\) and \\\\(AC\\\\). Please output the final answ... | {
"ground_truth": "7\\sqrt{3}",
"style": "rule"
} | {
"index": "3687",
"split": "train"
} |
math | [
{
"content": "There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3688",
"split": "train"
} |
math | [
{
"content": "Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each ju... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3689",
"split": "train"
} |
math | [
{
"content": "A triangle with integral sides has perimeter $8$. The area of the triangle is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "3690",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\frac{(2112-2021)^2}{169}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "3691",
"split": "train"
} |
math | [
{
"content": "A conical flask has a base radius of $15$ cm and a height of $30$ cm. The flask is filled with a liquid to a height of $10$ cm from the base, and a sphere is inscribed such that it just touches the liquid surface. What is the radius of the sphere? Express your answer in cm. Please output the final... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3692",
"split": "train"
} |
math | [
{
"content": "A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, expressed as the rat... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "3693",
"split": "train"
} |
math | [
{
"content": "At the end of $1997$, the desert area in a certain region was $9\\times 10^{5}hm^{2}$ (note: $hm^{2}$ is the unit of area, representing hectares). Geologists started continuous observations from $1998$ to understand the changes in the desert area of this region. The observation results at the end ... | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "3694",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ for which the matrix\n\\[\\begin{pmatrix} 1 + x & 7 \\\\ 3 - x & 8 \\end{pmatrix}\\]is not invertible. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{15}",
"style": "rule"
} | {
"index": "3695",
"split": "train"
} |
math | [
{
"content": "How many positive whole numbers have cube roots that are less than $8$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "3696",
"split": "train"
} |
math | [
{
"content": "In the diagram, what is the value of $x?$ [asy]\ndraw(Arc((0,0),1,-120,42));\n\ndraw((-2,-2*sqrt(3))--(3,3*sqrt(3))--(3,-2*sqrt(3))--(-2*sqrt(3)/.9,-2*sqrt(3))--(3,2.7));\ndraw((2.7,-2*sqrt(3))--(2.7,-2*sqrt(3)+.3)--(3,-2*sqrt(3)+.3));\nlabel(\"$48^\\circ$\",(3,1.5),W);\nlabel(\"$60^\\circ$\",(-1.... | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "3697",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a rectangle such that $A B=20$ and $A D=24$. Point $P$ lies inside $A B C D$ such that triangles $P A C$ and $P B D$ have areas 20 and 24, respectively. Compute all possible areas of triangle $P A B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98, 118, 122, 142",
"style": "rule"
} | {
"index": "3698",
"split": "train"
} |
math | [
{
"content": "The distance between locations A and B is 291 kilometers. Persons A and B depart simultaneously from location A and travel to location B at a constant speed, while person C departs from location B and heads towards location A at a constant speed. When person B has traveled \\( p \\) kilometers and... | {
"ground_truth": "221",
"style": "rule"
} | {
"index": "3699",
"split": "train"
} |
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