data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. The vectors $\\overrightarrow{m} = (a-b,\\sin A+\\sin C)$ and $\\overrightarrow{n} = (a-c, \\sin(A+C))$ are collinear. \n(1) Find the value of angle $C$; \n(2) If $\\overr... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{6}"
} | {
"split": "train",
"index": 33649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with side lengths as labeled. What is the area of the shaded $\\text L$-shaped region? [asy]\n/* AMC8 2000 #6 Problem */\ndraw((0,0)--(5,0)--(5,5)--(0,5)--cycle);\ndraw((1,5)--(1,1)--(5,1));\ndraw((0,... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 33650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a square of side length 2. Let points \\(X, Y\\), and \\(Z\\) be constructed inside \\(ABCD\\) such that \\(ABX, BCY\\), and \\(CDZ\\) are equilateral triangles. Let point \\(W\\) be outside \\(ABCD\\) such that triangle \\(DAW\\) is equilateral. Let the area ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33651,
"solution": "\\(WXYZ\\) is a kite with diagonals \\(XZ\\) and \\(WY\\), which have lengths \\(2 \\sqrt{3}-2\\) and 2, so the area is \\(2 \\sqrt{3}-2=\\sqrt{12}-2\\)."
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a_{1}, a_{2}, \\ldots\\) be an infinite sequence of integers such that \\(a_{i}\\) divides \\(a_{i+1}\\) for all \\(i \\geq 1\\), and let \\(b_{i}\\) be the remainder when \\(a_{i}\\) is divided by 210. What is the maximal number of distinct terms in the sequence \\(b_{1... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 33652,
"solution": "It is clear that the sequence \\(\\{a_{i}\\}\\) will be a concatenation of sequences of the form \\(\\{v_{i}\\}_{i=1}^{N_{0}},\\{w_{i} \\cdot p_{1}\\}_{i=1}^{N_{1}},\\{x_{i} \\cdot p_{1} p_{2}\\}_{i=1}^{N_{2}},\\{y_{i} \\cdot p_{1} p_{2} p_{3}\\}_{i=1}^{N_{3}}\\), ... |
deepscaler | [
{
"role": "user",
"content": "If a worker receives a $20$% cut in wages, calculate the exact percentage or dollar amount of a raise needed for the worker to regain their original pay. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 33653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers are between $\\sqrt[3]{10}$ and $\\sqrt[3]{200}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parallelogram $ABCD$ has a base $\\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by p... | math | {
"style": "rule",
"ground_truth": "18 + 3\\sqrt{5}"
} | {
"split": "train",
"index": 33655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(a) In a tennis tournament with 64 players, how many matches are played?\n\n(b) In a tournament with 2011 players, how many matches are played? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 33656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the enclosed shape formed by the line $y=x-2$ and the curve $y^2=x$ can be calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 33657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ s... | math | {
"style": "rule",
"ground_truth": "1, 5, 9"
} | {
"split": "train",
"index": 33658,
"solution": "\n\nLet \\( A_1B_1C_1D_1 \\) be an arbitrary convex quadrilateral. \\( P \\) is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point \\( P \\) is acute. We recursivel... |
deepscaler | [
{
"role": "user",
"content": "The sum of the first and the third of three consecutive odd integers is 152. What is the value of the second integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 33659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $3 \times 3$ grid shown, the central square contains the integer 5. The remaining eight squares contain $a, b, c, d, e, f, g, h$, which are each to be replaced with an integer from 1 to 9, inclusive. Integers can be repeated. There are $N$ ways to complete the grid so th... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 33660,
"solution": "Consider the grid as laid out in the problem: \n\\begin{tabular}{|l|l|l|}\n\\hline$a$ & $b$ & $c$ \\\\\n\\hline$d$ & 5 & $e$ \\\\\n\\hline$f$ & $g$ & $h$ \\\\\n\\hline\n\\end{tabular}\nWe know that the sums of the integers along each row, along each column, and alo... |
deepscaler | [
{
"role": "user",
"content": "A set $\\mathcal{T}$ of distinct positive integers has the following property: for every integer $y$ in $\\mathcal{T},$ the arithmetic mean of the set of values obtained by deleting $y$ from $\\mathcal{T}$ is an integer. Given that 1 belongs to $\\mathcal{T}$ and that 1764 is t... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 33661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{2}"
} | {
"split": "train",
"index": 33662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(g(x)\\) is a function such that\n\\[g(xy) + 2x = xg(y) + g(x)\\] for all real numbers \\(x\\) and \\(y.\\) If \\(g(-1) = 3\\) and \\(g(1) = 1\\), then compute \\(g(-101).\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 33663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5880"
} | {
"split": "train",
"index": 33664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the angle between the vectors $\\begin{pmatrix} 2 \\\\ -1 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} -1 \\\\ 1 \\\\ 0 \\end{pmatrix},$ in degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"split": "train",
"index": 33665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $\\angle Y = 90^\\circ$, $YZ = 4$, and $XY = \\sqrt{34}$. What is $\\tan X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 33666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $q(x) = 2x^6 - 3x^4 + Dx^2 + 6$ be a polynomial. When $q(x)$ is divided by $x - 2$, the remainder is 14. Find the remainder when $q(x)$ is divided by $x + 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 33667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33668,
"solution": "To solve this problem, we need to determine how many days of the week could be the first day of a month with 31 days such that the number of Mondays and Wednesdays in that month are equal.\n\n1. **Understanding the Weekday Distribution:**\n A month with 31 days s... |
deepscaler | [
{
"role": "user",
"content": "Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Sup... | math | {
"style": "rule",
"ground_truth": "25517"
} | {
"split": "train",
"index": 33669,
"solution": "There are a total of $16^{5}$ ways for the people to collectively ace the test. Consider groups of people who share the same problems that they got incorrect. We either have a group of 2 and a group of 3 , or a group 5 . In the first case, we can pick the group of ... |
deepscaler | [
{
"role": "user",
"content": "The function $y=f(x)$ is an even function with the smallest positive period of 4, and when $x \\in [-2, 0]$, $f(x) = 2x + 1$. If there exist $x_1, x_2, \\ldots, x_n$ satisfying $0 \\leq x_1 < x_2 < \\ldots < x_n$, and $|f(x_1) - f(x_2)| + |f(x_2) - f(x_3)| + \\ldots + |f(x_{n-1... | math | {
"style": "rule",
"ground_truth": "1513"
} | {
"split": "train",
"index": 33670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can two rooks be arranged on a chessboard such that one cannot capture the other? (A rook can capture another if it is on the same row or column of the chessboard). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3136"
} | {
"split": "train",
"index": 33671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this figure $\\angle RFS = \\angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\dfrac{1}{2}"
} | {
"split": "train",
"index": 33672,
"solution": "1. **Identify Given Angles and Sides**: We are given that $\\angle RFS = \\angle FDR = \\theta$, and the side lengths $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, and $FS = 7\\frac{1}{2}$ inches.\n\n2. **Apply the Law of Cosines in $\\triangle FDR$**:\n \\[... |
deepscaler | [
{
"role": "user",
"content": "The legs of a right triangle have lengths $\\log_4 27$ and $\\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 33673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin ^{8} x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35\\pi}{8}"
} | {
"split": "train",
"index": 33674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $1 \\leq k \\leq 2013$ does the decimal representation of $k^{k}$ end with a 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 33675,
"solution": "We claim that this is only possible if $k$ has a units digit of 1. Clearly, it is true in these cases. Additionally, $k^{k}$ cannot have a units digit of 1 when $k$ has a units digit of $2,4,5,6$, or 8. If $k$ has a units digit of 3 or 7, then $k^{k}$ has a units d... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = |\\log_2 x|$, let $m$ and $n$ be positive real numbers such that $m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ on the interval $[m^2, n]$ is $2$, find the value of $n+m$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 33676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 1235 numbers written on a board. One of them appears more frequently than the others - 10 times. What is the smallest possible number of different numbers that can be written on the board? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 33677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral \\( \\square ABCD \\), point \\( M \\) lies on diagonal \\( BD \\) with \\( MD = 3BM \\). Let \\( AM \\) intersect \\( BC \\) at point \\( N \\). Find the value of \\( \\frac{S_{\\triangle MND}}{S_{\\square ABCD}} \\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 33678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. The area of \\(... | math | {
"style": "rule",
"ground_truth": "2535"
} | {
"split": "train",
"index": 33679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular tetrahedron with four vertices \\(A, B, C, D\\) and each edge of length 1 meter, a bug starts at point \\(A\\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability a... | math | {
"style": "rule",
"ground_truth": "\\frac{182}{729}"
} | {
"split": "train",
"index": 33680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 33681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the two distinct square roots of a positive number $x$ are $a+3$ and $2a-15$, and $\\sqrt [3] {x+y-2}=4$. Find the value of $x-2y+2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with a slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$? Let's think st... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ n$ be a positive integer and $ [ \\ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:\r\n\r $ (1)$ $ n$ is not a perfect square;\r\n\r $ (2)$ $ a^{3}$ divides $ n^{2}$ . Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 33685,
"solution": "To solve this problem, we need to count the number of distinct intersection points formed by the diagonals of a regular octagon. We will use combinatorial geometry to determine this count.\n\n#### Step 1: Counting the Total Number of Diagonals\nIn an $n$-sided poly... |
deepscaler | [
{
"role": "user",
"content": "Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\\frac{3}... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 33686,
"solution": "1. **Identify the given information:**\n - Mean score of the morning class, $M = 84$.\n - Mean score of the afternoon class, $A = 70$.\n - Ratio of the number of students in the morning class to the afternoon class, $\\frac{m}{a} = \\frac{3}{4}$.\n\n2. **Expr... |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the shape bounded by the lines given by the equations:\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array}{l}\nx=t-\\sin t \\\\\ny=1-\\cos t\n\\end{array}\\right. \\\\\n& y=1 \\quad (0<x<2\\pi, \\, y \\geq 1)\n\\end{aligned}\n$$ Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} + 2"
} | {
"split": "train",
"index": 33687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let event $A$ be \"Point $M(x,y)$ satisfies $x^{2}+y^{2}\\leqslant a(a > 0)$\", and event $B$ be \"Point $M(x,y)$ satisfies $\\begin{cases} & x-y+1\\geqslant 0 \\\\ & 5x-2y-4\\leqslant 0 \\\\ & 2x+y+2\\geqslant 0 \\end{cases}$. If $P(B|A)=1$, then find the maximum value of the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 33688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a certain unfair die is rolled, an even number is $5$ times as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 33689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all non-empty subsets of the set \\( S = \\{1, 2, \\cdots, 10\\} \\). A subset is called a \"good subset\" if the number of even numbers in the subset is not less than the number of odd numbers. How many \"good subsets\" are there? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "637"
} | {
"split": "train",
"index": 33690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( x_{1}, x_{2}, \\ldots, x_{49} \\) are real numbers such that\n\n\\[ x_{1}^{2} + 2 x_{2}^{2} + \\cdots + 49 x_{49}^{2} = 1. \\]\n\nFind the maximum value of \\( x_{1} + 2 x_{2} + \\cdots + 49 x_{49} \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x + \\frac{1}{x} = 5,$ then compute the value of\n\\[(x - 2)^2 + \\frac{25}{(x - 2)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. Afte... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"split": "train",
"index": 33693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ in the equation $\\frac{1}{x} + \\frac{2}{x} \\div \\frac{4}{x} = 0.75$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the d... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 33695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone with a base radius of $15$ cm and a height of $30$ cm has a sphere inscribed within it. The radius of the sphere can be expressed in the form $b\\sqrt{d} - b$ cm. Determine the values of $b$ and $d$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 33696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Teacher Tan awarded a stack of exercise books to the students who were named \"Outstanding Students\" in the math Olympiad class. If each student is awarded 3 books, there are 7 books left over; if each student is awarded 5 books, there are 9 books short. How many students rece... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 33697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\\mathbb Z\\rightarrow\\mathbb Z$ and $g:\\mathbb Z\\rightarrow\\mathbb Z$ satisfying \\[f(g(x))=x+a\\quad\\text{and}\\quad g(f(x))=x+b\\] for all integers $x$ . L... | math | {
"style": "rule",
"ground_truth": "|a| = |b|"
} | {
"split": "train",
"index": 33698,
"solution": "We claim that the answer is $|a|=|b|$ .\nProof: $f$ and $g$ are surjective because $x+a$ and $x+b$ can take on any integral value, and by evaluating the parentheses in different order, we find $f(g(f(x)))=f(x+b)=f(x)+a$ and $g(f(g(x)))=g(x+a)=g(x)+b$ . We see that ... |
deepscaler | [
{
"role": "user",
"content": "The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "5\\pi"
} | {
"split": "train",
"index": 33699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit negative integer congruent to $1 \\pmod{17}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1002"
} | {
"split": "train",
"index": 33700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach? Express your answer in terms of $\\pi$. Let's think step... | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 33701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest sum that nine consecutive natural numbers can have if this sum ends in 2050306? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22050306"
} | {
"split": "train",
"index": 33702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob are playing in the forest. They have six sticks of length $1,2,3,4,5,6$ inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon. Let's t... | math | {
"style": "rule",
"ground_truth": "33 \\sqrt{3}"
} | {
"split": "train",
"index": 33704,
"solution": "Let the side lengths, in counterclockwise order, be $a, b, c, d, e, f$. Place the hexagon on the coordinate plane with edge $a$ parallel to the $x$-axis and the intersection between edge $a$ and edge $f$ at the origin (oriented so that edge $b$ lies in the first qu... |
deepscaler | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $2\\cos C(a\\cos C+c\\cos A)+b=0$.\n\n$(1)$ Find the magnitude of angle $C$;\n\n$(2)$ If $b=2$ and $c=2\\sqrt{3}$, find the area of $\\triangle ABC... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\( A \\) divided by \\( 3! \\) gives a result where the number of factors is \\(\\frac{1}{3}\\) of the original number of factors. What is the smallest such \\( A \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?\n Le... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 33707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33708,
"solution": "1. **Identify the formula for the sum of an arithmetic progression (AP)**:\n The sum $S_n$ of the first $n$ terms of an AP where the first term is $a$ and the common difference is $d$ is given by:\n \\[\n S_n = \\frac{n}{2} \\left(2a + (n-1)d\\right)\n \\]\... |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C:x^{2}-\\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6\\sqrt{6})$, the minimum perimeter of $\\triangle APF$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 33709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A'Niu is riding a horse to cross a river. There are four horses named A, B, C, and D. It takes 2 minutes for horse A to cross the river, 3 minutes for horse B, 7 minutes for horse C, and 6 minutes for horse D. Only two horses can be driven across the river at a time. The questi... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $b$ such that the roots of the polynomial\n$$x^3 - 9x^2 + 39x + b$$ form an arithmetic progression, and one of the roots is real while the other two are complex conjugates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"split": "train",
"index": 33711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A novel is to be recorded onto compact discs and takes 528 minutes to read aloud. Each disc can now hold up to 45 minutes of reading. Assuming that the smallest possible number of discs is used and that each disc contains the same length of reading, calculate the number of minu... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 33712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number \\( x \\) is \\( x \\) times as likely as rolling a 1. For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rollin... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 33713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common multiple of 8 and 12 that is less than 90? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 33714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$ . $z$ is the maximum real number such that [ $\\frac{5}{z}$ ] + [ $\\frac{6}{z}$ ] = 7. Find the value of $ 20z$ . Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 33715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.\nThese checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside dow... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 33716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=3$, what is the value of $-(5x - 6x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33717,
"solution": "When $x=3$, we have $-(5x - 6x) = -(-x) = x = 3$. Alternatively, when $x=3$, we have $-(5x - 6x) = -(15 - 18) = -(-3) = 3$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $231_5 + 414_5 + 123_5$. Express your answer in base $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1323_5"
} | {
"split": "train",
"index": 33718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the quotient $2033_4 \\div 22_4$ in base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11_4"
} | {
"split": "train",
"index": 33719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jessica's six assignment scores are 87, 94, 85, 92, 90, and 88. What is the arithmetic mean of these six scores? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89.3"
} | {
"split": "train",
"index": 33720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $3 \\times (7 - 5) - 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 33721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $A$ which can be represented in the form: \\[ A = \\left ( m - \\dfrac 1n \\right) \\left( n - \\dfrac 1p \\right) \\left( p - \\dfrac 1m \\right) \\]\r\nwhere $m\\geq n\\geq p \\geq 1$ are integer numbers. \r\n\r\n*Ioan Bogdan* Let's think step b... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 33722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-negative real numbers \\( a, b, c, x, y, z \\) that satisfy \\( a + b + c = x + y + z = 1 \\), find the minimum value of \\( \\left(a - x^{2}\\right)\\left(b - y^{2}\\right)\\left(c - z^{2}\\right) \\). Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"split": "train",
"index": 33723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 33724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{123}{40}"
} | {
"split": "train",
"index": 33725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the sep... | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 33726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces ... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 33727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega \\in \\mathbb{C}$ , and $\\left | \\omega \\right | = 1$ . Find the maximum length of $z = \\left( \\omega + 2 \\right) ^3 \\left( \\omega - 3 \\right)^2$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 33728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per m... | math | {
"style": "rule",
"ground_truth": "23.5"
} | {
"split": "train",
"index": 33729,
"solution": "Amy and Ben will continuously eat carrots, then stop (not necessarily at the same time), and continuously eat muffins until no food is left. Suppose that Amy and Ben finish eating the carrots in $T_{1}$ minutes and the muffins $T_{2}$ minutes later; we wish to find... |
deepscaler | [
{
"role": "user",
"content": "For the power of _n_ of a natural number _m_ greater than or equal to 2, the following decomposition formula is given: \n2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7… \n2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11… \n2<sup>4</sup> = 7 ... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 33730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is moved from one vertex of the triangle to ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{64}"
} | {
"split": "train",
"index": 33731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of $\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33732,
"solution": "1. **Understanding the operation $[a,b,c]$:** \n The operation $[a,b,c]$ is defined as $\\frac{a+b}{c}$, where $c \\neq 0$.\n\n2. **Simplifying each component:** \n - For $[60,30,90]$, we calculate:\n \\[\n [60,30,90] = \\frac{60+30}{90} = \\frac{90}{... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC, AB=32, AC=35$, and $BC=x$. What is the smallest positive integer $x$ such that $1+\\cos^{2}A, \\cos^{2}B$, and $\\cos^{2}C$ form the sides of a non-degenerate triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 33733,
"solution": "By the triangle inequality, we wish $\\cos^{2}B+\\cos^{2}C>1+\\cos^{2}A$. The other two inequalities are always satisfied, since $1+\\cos^{2}A \\geq 1 \\geq \\cos^{2}B, \\cos^{2}C$. Rewrite the above as $$2-\\sin^{2}B-\\sin^{2}C>2-\\sin^{2}A$$ so it is equivalent t... |
deepscaler | [
{
"role": "user",
"content": "If \\( 50\\% \\) of \\( N \\) is 16, what is \\( 75\\% \\) of \\( N \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33734,
"solution": "The percentage \\( 50\\% \\) is equivalent to the fraction \\( \\frac{1}{2} \\), while \\( 75\\% \\) is equivalent to \\( \\frac{3}{4} \\). Since \\( 50\\% \\) of \\( N \\) is 16, then \\( \\frac{1}{2}N=16 \\) or \\( N=32 \\). Therefore, \\( 75\\% \\) of \\( N \\) ... |
deepscaler | [
{
"role": "user",
"content": "A regular octahedron has a side length of 1. What is the distance between two opposite faces? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6} / 3"
} | {
"split": "train",
"index": 33735,
"solution": "Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the oc... |
deepscaler | [
{
"role": "user",
"content": "Given a frustum of a cone with the areas of its two bases being $4\\pi$ and $25\\pi$ respectively, and the height of the frustum is 4, find the volume and the lateral surface area of the frustum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35\\pi"
} | {
"split": "train",
"index": 33736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a = \\pi/4032$. Find the smallest positive integer $n$ such that \\[2[\\cos(a)\\sin(a) + \\cos(9a)\\sin(3a) + \\cos(25a)\\sin(5a) + \\cdots + \\cos(n^2a)\\sin(na)]\\] is an integer, where $n$ is odd. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4031"
} | {
"split": "train",
"index": 33737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer greater than $-\\frac{17}{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 33738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In plane $\\alpha$, there are four points, and in plane $\\beta$, there are five points. From these nine points, the maximum number of planes that can be determined by any three points is ; the maximum number of tetrahedrons that can be determined by any four points is ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 33739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a notebook, there is a grid rectangle of size $3 \\times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \\times 5$ grid ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 33740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of sequences of integers $(a_{1}, \\ldots, a_{200})$ such that the following conditions hold. - $0 \\leq a_{1}<a_{2}<\\cdots<a_{200} \\leq 202$. - There exists a positive integer $N$ with the following property: for every index $i \\in\\{1, \\ldots, 200\\}$ t... | math | {
"style": "rule",
"ground_truth": "20503"
} | {
"split": "train",
"index": 33741,
"solution": "Let $m:=203$ be an integer not divisible by 3. We'll show the answer for general such $m$ is $m\\left\\lceil\\frac{m-1}{2}\\right\\rceil$. Let $x, y, z$ be the three excluded residues. Then $N$ works if and only if $\\{x, y, z\\} \\equiv\\{N-x, N-y, N-z\\} (\\bmod ... |
deepscaler | [
{
"role": "user",
"content": "When the base-12 integer $1531_{12}$ is divided by $8$, what is the remainder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of $\\frac{(3^t-2t)t}{9^t}$ for real values of $t$?\nA) $\\frac{1}{10}$\nB) $\\frac{1}{12}$\nC) $\\frac{1}{8}$\nD) $\\frac{1}{6}$\nE) $\\frac{1}{4}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 33743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane Cartesian coordinate system \\( xO y \\), the circle \\( \\Omega \\) and the parabola \\( \\Gamma: y^{2} = 4x \\) share exactly one common point, and the circle \\( \\Omega \\) is tangent to the x-axis at the focus \\( F \\) of \\( \\Gamma \\). Find the radius of t... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{9}"
} | {
"split": "train",
"index": 33744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of a rectangle which remains unchanged when it is made $3 \\frac{1}{2}$ inches longer and $1 \\frac{1}{3}$ inch narrower, or when it is made $3 \\frac{1}{2}$ inches shorter and $1 \\frac{2}{3}$ inch wider. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the ellipse $C\\_1$: $\\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\\_2$: $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\\_2$ has the coordinates $(\\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\\_2$ at points $A$ ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 33746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 33747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\\triangle ACD$. Find $s.$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 33748,
"solution": "[asy] size(220); pointpen = black; pathpen = black + linewidth(0.7); pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8); D(D(MP(\"A\",A))--D(MP(\"C\",C))--D(MP(\"D\",D,NE))--cycle); D(D(MP(\"B\",B))--D); D((0,-4)--(0,12),linetype(\"4 4\")+linewidth(0.7)); MP(\"6\",B/... |
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