data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5\\sqrt{13}",
"style": "rule"
} | {
"index": "36900",
"split": "train"
} |
math | [
{
"content": "In an $8 \\times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in t... | {
"ground_truth": "234",
"style": "rule"
} | {
"index": "36901",
"split": "train"
} |
math | [
{
"content": "In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces, there are a total o... | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "36902",
"split": "train"
} |
math | [
{
"content": "A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, if two bulbs are lit,... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "36903",
"split": "train"
} |
math | [
{
"content": "I ponder some numbers in bed, all products of three primes I've said, apply $\\phi$ they're still fun: $$n=37^{2} \\cdot 3 \\ldots \\phi(n)= 11^{3}+1 ?$$ now Elev'n cubed plus one. What numbers could be in my head? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2007, 2738, 3122",
"style": "rule"
} | {
"index": "36904",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos α=-\\dfrac{4}{5}\\left(\\dfrac{π}{2}<α<π\\right)$, find $\\cos\\left(\\dfrac{π}{6}-α\\right)$ and $\\cos\\left(\\dfrac{π}{6}+α\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{3+4\\sqrt{3}}{10}",
"style": "rule"
} | {
"index": "36905",
"split": "train"
} |
math | [
{
"content": "Let $x$, $y$, $z$, $u$, and $v$ be positive integers with $x+y+z+u+v=2505$. Let $N$ be the largest of the sums $x+y$, $y+z$, $z+u$, and $u+v$. Determine the smallest possible value of $N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1253",
"style": "rule"
} | {
"index": "36906",
"split": "train"
} |
math | [
{
"content": "Determine the number of four-digit integers $n$ such that $n$ and $2n$ are both palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "36907",
"split": "train"
} |
math | [
{
"content": "Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute $$\\frac{1}{255} \\sum_{0 \\leq n<16} 2^{n}(-1)^{s(n)}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "36908",
"split": "train"
} |
math | [
{
"content": "Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "36909",
"split": "train"
} |
math | [
{
"content": "In a triangle with sides of lengths $a,$ $b,$ and $c,$\n\\[(a + b + c)(a + b - c) = 3ab.\\]Find the angle opposite the side of length $c,$ in degrees. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "36910",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ is a non-zero arithmetic sequence, $S_n$ denotes the sum of the first $n$ terms, and $S_{2n-1} = a_n^2$ for any $n \\in \\mathbb{N^*}$. If the inequality $\\dfrac{1}{a_1a_2} + \\dfrac{1}{a_2a_3} + \\ldots + \\dfrac{1}{a_na_{n+1}} \\leqslant n\\log_{\\frac{1}{8}}\\lam... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "36911",
"split": "train"
} |
math | [
{
"content": "Given the function $g(x) = \\frac{6x^2 + 11x + 17}{7(2 + x)}$, find the minimum value of $g(x)$ for $x \\ge 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{127}{24}",
"style": "rule"
} | {
"index": "36912",
"split": "train"
} |
math | [
{
"content": "A grocery store manager decides to design a more compact pyramid-like stack of apples with a rectangular base of 4 apples by 6 apples. Each apple above the first level still rests in a pocket formed by four apples below, and the stack is completed with a double row of apples on top. Determine the ... | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "36913",
"split": "train"
} |
math | [
{
"content": "Let point P be the intersection point in the first quadrant of the hyperbola $\\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. F\\1 and F\\2 are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Find the eccentricity... | {
"ground_truth": "\\frac{\\sqrt{10}}{2}",
"style": "rule"
} | {
"index": "36914",
"split": "train"
} |
math | [
{
"content": "Randomly select a number $x$ in the interval $[-1, 1]$. The probability that the value of $\\cos \\frac{\\pi x}{2}$ falls between 0 and $\\frac{1}{2}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "36915",
"split": "train"
} |
math | [
{
"content": "Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance. Please output the final answer within \\boxed{}."... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "36916",
"split": "train"
} |
math | [
{
"content": "On the base \\(AC\\) of an isosceles triangle \\(ABC\\), a point \\(E\\) is taken, and on the sides \\(AB\\) and \\(BC\\), points \\(K\\) and \\(M\\) are taken such that \\(KE \\parallel BC\\) and \\(EM \\parallel AB\\). What fraction of the area of triangle \\(\\mathrm{ABC}\\) is occupied by the ... | {
"ground_truth": "6/25",
"style": "rule"
} | {
"index": "36917",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) belongs to the edge \\( CD \\) of the parallelepiped \\( ABCDA_1B_1C_1D_1 \\), where \\( CM: MD = 1:2 \\). Construct the section of the parallelepiped with a plane passing through point \\( M \\) parallel to the lines \\( DB \\) and \\( AC_1 \\). In what ratio does this plane divid... | {
"ground_truth": "1 : 11",
"style": "rule"
} | {
"index": "36918",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive number \\( c \\) with the following property: For any integer \\( n \\geqslant 4 \\) and any set \\( A \\subseteq \\{1, 2, \\ldots, n\\} \\), if \\( |A| > c n \\), then there exists a function \\( f: A \\rightarrow \\{1, -1\\} \\) such that \\( \\left|\\sum_{a \\in A} f(... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "36919",
"split": "train"
} |
math | [
{
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "36920",
"split": "train"
} |
math | [
{
"content": "Find all the values of $m$ for which the zeros of $2 x^{2}-m x-8$ differ by $m-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6,-\\frac{10}{3}",
"style": "rule"
} | {
"index": "36921",
"split": "train"
} |
math | [
{
"content": "There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "36922",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:\n\\[\n\\frac{1}{b^2 + c^2 - 3a^2} + \\frac{1}{a^2 + c^2 - 3b^2} + \\frac{1}{a^2 + b^2 - 3c^2}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "36923",
"split": "train"
} |
math | [
{
"content": "Given the equation $x^{2}+4ax+3a+1=0 (a > 1)$, whose two roots are $\\tan \\alpha$ and $\\tan \\beta$, with $\\alpha, \\beta \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$, find $\\tan \\frac{\\alpha + \\beta}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "36924",
"split": "train"
} |
math | [
{
"content": "The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \\sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{128\\pi}{3}",
"style": "rule"
} | {
"index": "36925",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$, $\\sin A = \\frac{8}{17}$ and $\\sin B = 1$. Find $\\sin C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{15}{17}",
"style": "rule"
} | {
"index": "36926",
"split": "train"
} |
math | [
{
"content": "Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\\sin \\angle FAP$. Please output the final answer... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "36927",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \\le 2y \\le 60$ and $y \\le 2x \\le 60$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "480",
"style": "rule"
} | {
"index": "36928",
"split": "train"
} |
math | [
{
"content": "The product of $7d^2-3d+g$ and $3d^2+hd-8$ is $21d^4-44d^3-35d^2+14d-16$. What is $g+h$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "36929",
"split": "train"
} |
math | [
{
"content": "Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "36930",
"split": "train"
} |
math | [
{
"content": "A flea is jumping on the vertices of square \\(ABCD\\), starting from vertex \\(A\\). With each jump, it moves to an adjacent vertex with a probability of \\(\\frac{1}{2}\\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be t... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "36931",
"split": "train"
} |
math | [
{
"content": "The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a \"wealth number.\" Among the numbers from 1 to 100, how many \"wealth numbers\" ar... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "36932",
"split": "train"
} |
math | [
{
"content": "A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\nHow many different paths can the particle take from $(0,0)$ to $(5,5... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "36933",
"split": "train"
} |
math | [
{
"content": "The roots of the equation $2\\sqrt{x} + 2x^{-\\frac{1}{2}} = 5$ can be found by solving: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4x^2-17x+4 = 0",
"style": "rule"
} | {
"index": "36934",
"split": "train"
} |
math | [
{
"content": "Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \\frac{1}{2 + \\frac{1}{2 + \\frac{1}{x}}}.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{9}{10}",
"style": "rule"
} | {
"index": "36935",
"split": "train"
} |
math | [
{
"content": "Consider the function $f(x) = x^2 +2\\sqrt{x}$. Evaluate $2f(2) - f(8)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-56",
"style": "rule"
} | {
"index": "36936",
"split": "train"
} |
math | [
{
"content": "A hiker is exploring a trail. The trail has three sections: the first $25 \\%$ of the trail is along a river, the next $\\frac{5}{8}$ of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "24 \\text{ km}",
"style": "rule"
} | {
"index": "36937",
"split": "train"
} |
math | [
{
"content": "Sara conducted a survey among a group to find out how many people were aware that bats can transmit diseases. She found out that $75.3\\%$ believed bats could transmit diseases. Among those who believed this, $60.2\\%$ incorrectly thought that all bats transmit Zika virus, which amounted to 37 peo... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "36938",
"split": "train"
} |
math | [
{
"content": "Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "36939",
"split": "train"
} |
math | [
{
"content": "The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "36940",
"split": "train"
} |
math | [
{
"content": "The projection of $\\begin{pmatrix} 0 \\\\ 3 \\end{pmatrix}$ onto a certain vector $\\mathbf{w}$ is $\\begin{pmatrix} -9/10 \\\\ 3/10 \\end{pmatrix}.$ Find the projection of $\\begin{pmatrix} 4 \\\\ 1 \\end{pmatrix}$ onto $\\mathbf{w}.$ Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\begin{pmatrix} 33/10 \\\\ -11/10 \\end{pmatrix}",
"style": "rule"
} | {
"index": "36941",
"split": "train"
} |
math | [
{
"content": "How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "36942",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\log_{m}(m - x)$, if the maximum value in the interval $[3, 5]$ is 1 greater than the minimum value, determine the real number $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + \\sqrt{6}",
"style": "rule"
} | {
"index": "36943",
"split": "train"
} |
math | [
{
"content": "Calculate \\(\\int_{0}^{1} e^{-x^{2}} \\, dx\\) to an accuracy of 0.001. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.747",
"style": "rule"
} | {
"index": "36944",
"split": "train"
} |
math | [
{
"content": "If 8 is added to the square of 5, the result is divisible by: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "36945",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\\frac{{{x}^{2}}}{{{a}^{2}}}-\\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \\perp x$-axis, find the eccentricity of the hyperbola. ... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "36946",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $5^{137}$ is divided by 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "36947",
"split": "train"
} |
math | [
{
"content": "Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of 5 or 7 or both? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{25}",
"style": "rule"
} | {
"index": "36948",
"split": "train"
} |
math | [
{
"content": "Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.\nA) $54 + 18\\sqrt{2} - 9\\pi$\nB... | {
"ground_truth": "54 + 36\\sqrt{2} - 9\\pi",
"style": "rule"
} | {
"index": "36949",
"split": "train"
} |
math | [
{
"content": "When every vector on the line $y = \\frac{5}{2} x + 4$ is projected onto a certain vector $\\mathbf{w},$ the result is always the vector $\\mathbf{p}.$ Find the vector $\\mathbf{p}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -40/29 \\\\ 16/29 \\end{pmatrix}",
"style": "rule"
} | {
"index": "36950",
"split": "train"
} |
math | [
{
"content": "Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "36951",
"split": "train"
} |
math | [
{
"content": "If the product $\\frac{3}{2} \\cdot \\frac{4}{3} \\cdot \\frac{5}{4} \\cdot \\frac{6}{5} \\cdot \\ldots \\cdot \\frac{a}{b} = 9$, what is the sum of $a$ and $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "36952",
"split": "train"
} |
math | [
{
"content": "Appending three digits at the end of 2007, one obtains an integer \\(N\\) of seven digits. In order to get \\(N\\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "075",
"style": "rule"
} | {
"index": "36953",
"split": "train"
} |
math | [
{
"content": "Given regular hexagon $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$ , where $P$ is the midpoint of $AB$ , $Q$ is the midpoint of $CD$ , and $R$ is the midpoint of $EF$ . Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "36954",
"split": "train"
} |
math | [
{
"content": "A research study group is investigating the traffic volume at a certain intersection near the school during the rush hour from 8:00 to 10:00. After long-term observation and statistics, they have established a simple function model between traffic volume and average vehicle speed. The model is as ... | {
"ground_truth": "\\frac{25}{3}",
"style": "rule"
} | {
"index": "36955",
"split": "train"
} |
math | [
{
"content": "How many ways are there for Nick to travel from $(0,0)$ to $(16,16)$ in the coordinate plane by moving one unit in the positive $x$ or $y$ direction at a time, such that Nick changes direction an odd number of times? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\cdot\\binom{30}{15} = 310235040",
"style": "rule"
} | {
"index": "36956",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = 4x + c$ and $g(x) = cx + 2.$ If $f(g(x)) = 12x + d,$ then find $d.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "36957",
"split": "train"
} |
math | [
{
"content": "Evaluate: $6 - 8\\left(9 - 4^2\\right) \\div 2 - 3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "36958",
"split": "train"
} |
math | [
{
"content": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = a \\cdot d - b \\cdot c$, what is the value of $\\begin{vmatrix} 3 & 4 \\\\ 1 & 2 \\end{vmatrix}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$2$",
"style": "rule"
} | {
"index": "36959",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \\frac{\\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.\n\n(1) Find the equation of the ellipse $(C)$;\n(... | {
"ground_truth": "\\frac{3 \\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "36960",
"split": "train"
} |
math | [
{
"content": "Solve in integers the system of equations: $$x^2-y^2=z$$\n $$3xy+(x-y)z=z^2$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "{(2, 1, 3), (1, 2, -3), (1, 0, 1), (0, 1, -1), (0, 0, 0)}",
"style": "rule"
} | {
"index": "36961",
"split": "train"
} |
math | [
{
"content": "If the odds for pulling a prize out of the box are $3:4$, what is the probability of not pulling the prize out of the box? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{7}",
"style": "rule"
} | {
"index": "36962",
"split": "train"
} |
math | [
{
"content": "A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \\(\\frac{1}{4}\\) inch. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\%",
"style": "rule"
} | {
"index": "36963",
"split": "train"
} |
math | [
{
"content": "In a store, we paid with a 1000 forint bill. On the receipt, the amount to be paid and the change were composed of the same digits but in a different order. What is the sum of the digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "36964",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a\\_n\\}$ is a geometric sequence, $a\\_2=2$, $a\\_6=162$, find $a\\_{10}$ = $\\_\\_\\_\\_\\_\\_$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13122",
"style": "rule"
} | {
"index": "36965",
"split": "train"
} |
math | [
{
"content": "In square ABCD, point E is on AB and point F is on CD such that AE = 3EB and CF = 3FD. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{32}",
"style": "rule"
} | {
"index": "36966",
"split": "train"
} |
math | [
{
"content": "How many cubic centimeters are in the volume of a cone having a diameter of 10cm and a height of 6cm? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50 \\pi",
"style": "rule"
} | {
"index": "36967",
"split": "train"
} |
math | [
{
"content": "Define a new function $\\$N$ such that $\\$N = 0.75N + 2$. Calculate $\\$(\\$(\\$30))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17.28125",
"style": "rule"
} | {
"index": "36968",
"split": "train"
} |
math | [
{
"content": "A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new to... | {
"ground_truth": "927",
"style": "rule"
} | {
"index": "36969",
"split": "train"
} |
math | [
{
"content": "In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of ... | {
"ground_truth": "\\frac{1513}{2017}",
"style": "rule"
} | {
"index": "36970",
"split": "train"
} |
math | [
{
"content": "A basketball championship has been played in a round-robin format, with each pair of teams playing twice and no ties (overtime is played until one team wins). The winner of a match receives 2 points, and the loser receives 1 point. At the end of the championship, the sum of the points obtained by ... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "36971",
"split": "train"
} |
math | [
{
"content": "How many even numbers are greater than 202 and less than 405? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "36972",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) \\) be defined on \\( \\mathbb{R} \\), and for any \\( x \\), the condition \\( f(x+2) + f(x) = x \\) holds. It is also known that \\( f(x) = x^3 \\) on the interval \\( (-2, 0] \\). Find \\( f(2012) \\). Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "1006",
"style": "rule"
} | {
"index": "36973",
"split": "train"
} |
math | [
{
"content": "A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "36974",
"split": "train"
} |
math | [
{
"content": "There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them? Please output the final answer wi... | {
"ground_truth": "41/56",
"style": "rule"
} | {
"index": "36975",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\\mathcal{... | {
"ground_truth": "1247",
"style": "rule"
} | {
"index": "36976",
"split": "train"
} |
math | [
{
"content": "Liu Yulei bought 4 packs of yogurt and 4 packs of fresh milk at the supermarket, paying a total of 14 yuan. Later, she returned 2 packs of yogurt and bought 4 more packs of fresh milk, and the cashier gave her 1 yuan back. The price of each pack of yogurt is ____ yuan. Please output the final answ... | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "36977",
"split": "train"
} |
math | [
{
"content": "For a given integer $n\\ge 2$, let $a_0,a_1,\\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\\{ a_i+a_j \\mid 0\\le i \\le j \\le n \\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3n",
"style": "rule"
} | {
"index": "36978",
"split": "train"
} |
math | [
{
"content": "If a positive four-digit number's thousand digit \\\\(a\\\\), hundred digit \\\\(b\\\\), ten digit \\\\(c\\\\), and unit digit \\\\(d\\\\) satisfy the relation \\\\((a-b)(c-d) < 0\\\\), then it is called a \"Rainbow Four-Digit Number\", for example, \\\\(2012\\\\) is a \"Rainbow Four-Digit Number\... | {
"ground_truth": "3645",
"style": "rule"
} | {
"index": "36979",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of kings that can be placed on a chessboard such that no two of them attack each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "36980",
"split": "train"
} |
math | [
{
"content": "In the figure, $\\angle A$, $\\angle B$, and $\\angle C$ are right angles. If $\\angle AEB = 40^\\circ$ and $\\angle BED = \\angle BDE$, then $\\angle CDE =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95^\\circ",
"style": "rule"
} | {
"index": "36981",
"split": "train"
} |
math | [
{
"content": "The polynomial $ax^4 + bx^3 + 32x^2 - 16x + 6$ has a factor of $3x^2 - 2x + 1.$ Find the ordered pair $(a,b).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(18,-24)",
"style": "rule"
} | {
"index": "36982",
"split": "train"
} |
math | [
{
"content": "Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \\div 24$ and $X$ is an integer, what is the smallest possible value of $X$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4625",
"style": "rule"
} | {
"index": "36983",
"split": "train"
} |
math | [
{
"content": "Find the number of complex numbers $z$ satisfying $|z| = 1$ and\n\\[\\left| \\frac{z}{\\overline{z}} + \\frac{\\overline{z}}{z} \\right| = 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "36984",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the area of the shaded triangle to the area of the square? The square is divided into a 5x5 grid of smaller, equal-sized squares. A triangle is shaded such that it covers half of a square at the center of the grid and three full squares adjacent to this half-covered square. Th... | {
"ground_truth": "\\frac{7}{50}",
"style": "rule"
} | {
"index": "36985",
"split": "train"
} |
math | [
{
"content": "James borrows $2000$ dollars from Alice, who charges an interest of $3\\%$ per month (which compounds monthly). What is the least integer number of months after which James will owe more than three times as much as he borrowed? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "36986",
"split": "train"
} |
math | [
{
"content": "A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "245 yd.",
"style": "rule"
} | {
"index": "36987",
"split": "train"
} |
math | [
{
"content": "Find all real values of $x$ that satisfy $x + \\frac{45}{x-4} = -10.$\n\nEnter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1, \\; -5",
"style": "rule"
} | {
"index": "36988",
"split": "train"
} |
math | [
{
"content": "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$. What is the probability that it rains on at most 2 days in June? Express your answer as a decimal to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.411",
"style": "rule"
} | {
"index": "36989",
"split": "train"
} |
math | [
{
"content": "If $e^{i \\alpha} = \\frac{3}{5} +\\frac{4}{5} i$ and $e^{i \\beta} = -\\frac{12}{13} + \\frac{5}{13} i,$ then find $\\sin (\\alpha + \\beta).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{33}{65}",
"style": "rule"
} | {
"index": "36990",
"split": "train"
} |
math | [
{
"content": "If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "36991",
"split": "train"
} |
math | [
{
"content": "For a certain complex number $c$, the polynomial\n\\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\\]has exactly 4 distinct roots. What is $|c|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{10}",
"style": "rule"
} | {
"index": "36992",
"split": "train"
} |
math | [
{
"content": "In the isosceles right triangle \\(ABC\\) with \\(\\angle A = 90^\\circ\\) and \\(AB = AC = 1\\), a rectangle \\(EHGF\\) is inscribed such that \\(G\\) and \\(H\\) lie on the side \\(BC\\). Find the maximum area of the rectangle \\(EHGF\\). Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1/4",
"style": "rule"
} | {
"index": "36993",
"split": "train"
} |
math | [
{
"content": "What is the largest four-digit number whose digits add up to 23? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9950",
"style": "rule"
} | {
"index": "36994",
"split": "train"
} |
math | [
{
"content": "Compute $(4+8)^2+(4^2+8^2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "224",
"style": "rule"
} | {
"index": "36995",
"split": "train"
} |
math | [
{
"content": "Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "36996",
"split": "train"
} |
math | [
{
"content": "Rotate an equilateral triangle with side length $2$ around one of its sides to form a solid of revolution. The surface area of this solid is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{3}\\pi",
"style": "rule"
} | {
"index": "36997",
"split": "train"
} |
math | [
{
"content": "A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers. Please output the final answer within \\bo... | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "36998",
"split": "train"
} |
math | [
{
"content": "Point $O$ is the center of an ellipse with major axis $\\overline{AB}$ and minor axis $\\overline{CD}.$ Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is 2, compute the product $(AB)(CD).$ Please output the final answer within \\boxe... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "36999",
"split": "train"
} |
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