data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In an isosceles trapezoid, the lengths of the bases are 9 cm and 21 cm, and the height is 8 cm. Find the radius of the circumscribed circle around the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10.625",
"style": "rule"
} | {
"index": "38800",
"split": "train"
} |
math | [
{
"content": "Define $a$ $\\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\\$$ $3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "38801",
"split": "train"
} |
math | [
{
"content": "A convex polyhedron \\( P \\) has 2021 edges. By cutting off a pyramid at each vertex, which uses one edge of \\( P \\) as a base edge, a new convex polyhedron \\( Q \\) is obtained. The planes of the bases of the pyramids do not intersect each other on or inside \\( P \\). How many edges does the... | {
"ground_truth": "6063",
"style": "rule"
} | {
"index": "38802",
"split": "train"
} |
math | [
{
"content": "Phil has 7 green marbles and 3 purple marbles in a bag. He removes a marble at random, records the color, puts it back, and then repeats this process until he has withdrawn 6 marbles. What is the probability that exactly three of the marbles that he removes are green? Express your answer as a deci... | {
"ground_truth": ".185",
"style": "rule"
} | {
"index": "38803",
"split": "train"
} |
math | [
{
"content": "How many of the first 1000 positive integers can be expressed in the form\n$\\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor + \\lfloor 6x \\rfloor + \\lfloor 8x \\rfloor$,\n where $x$ is a real number, and $\\lfloor z \\rfloor$ denotes the greatest integer less than or equal to $z$? Please output the ... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "38804",
"split": "train"
} |
math | [
{
"content": "For every $m$ and $k$ integers with $k$ odd, denote by $\\left[ \\frac{m}{k} \\right]$ the integer closest to $\\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that\n\\[\\left[ \\frac{n}{k} \\right] + \\left[ \\frac{100 - n}{k} \\right] = \\left[ \\frac{100}{k} \\right]\\]fo... | {
"ground_truth": "\\frac{34}{67}",
"style": "rule"
} | {
"index": "38805",
"split": "train"
} |
math | [
{
"content": "What is the largest integer \\( k \\) such that \\( k+1 \\) divides\n\\[ k^{2020} + 2k^{2019} + 3k^{2018} + \\cdots + 2020k + 2021? \\ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "38806",
"split": "train"
} |
math | [
{
"content": "Ivan and Peter are running in opposite directions on circular tracks with a common center, initially positioned at the minimal distance from each other. Ivan completes one full lap every 20 seconds, while Peter completes one full lap every 28 seconds. In the shortest possible time, when will they ... | {
"ground_truth": "35/6",
"style": "rule"
} | {
"index": "38807",
"split": "train"
} |
math | [
{
"content": "Given a \"double-single\" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "38808",
"split": "train"
} |
math | [
{
"content": "What is the matrix $\\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$? (In particular, $A$ goes to $A',$ and so on.)\n\n[asy]\n\nsize(200);\nimport graph;\n\npair Z=(0,0), A=(2,3), B=(-3,2), C=(-4,1), D=(-1,-4);\n\nLabel f; \nf.p=fontsize(6); \nxaxis(-1.... | {
"ground_truth": "\\begin{pmatrix} 1 & -1 \\\\ 1 & \\phantom -1 \\end{pmatrix}",
"style": "rule"
} | {
"index": "38809",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$ and $y$ satisfy that three of the four numbers $x+y$, $x-y$, $\\frac{x}{y}$, and $xy$ are equal, determine the value of $|y|-|x|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38810",
"split": "train"
} |
math | [
{
"content": "An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$? Please output the final answer within \\boxed{}... | {
"ground_truth": "9/35",
"style": "rule"
} | {
"index": "38811",
"split": "train"
} |
math | [
{
"content": "One of the asymptotes of a hyperbola has equation $y=3x.$ The foci of the hyperbola have the same $x-$coordinate, which is $5.$ Find the equation of the other asymptote of the hyperbola, giving your answer in the form \"$y = mx + b$\". Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "y = -3x + 30",
"style": "rule"
} | {
"index": "38812",
"split": "train"
} |
math | [
{
"content": "2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k} is equal to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2^{-(2k+1)}",
"style": "rule"
} | {
"index": "38813",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$ . Comp... | {
"ground_truth": "3\\sqrt{3} - 6",
"style": "rule"
} | {
"index": "38814",
"split": "train"
} |
math | [
{
"content": "Calculate the radius of the circle where all the complex roots of the equation $(z - 1)^6 = 64z^6$ lie when plotted in the complex plane. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "38815",
"split": "train"
} |
math | [
{
"content": "What is the least possible value of \\((x+1)(x+2)(x+3)(x+4)+2023\\) where \\(x\\) is a real number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "38816",
"split": "train"
} |
math | [
{
"content": "A rectangle has side lengths $6$ and $8$ . There are relatively prime positive integers $m$ and $n$ so that $\\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Fin... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38817",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer \\( n \\) such that for any given \\( n \\) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \\( R_{1}, R_{2}, R_{3} \\) such that \\( R_{1} \\) can be nested inside \\( R_{2} \\) and \\( R_{2} \\) can be nested inside \\( R_{3} \\)... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "38818",
"split": "train"
} |
math | [
{
"content": "How many of the integers $1,2, \\ldots, 2004$ can be represented as $(m n+1) /(m+n)$ for positive integers $m$ and $n$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "38819",
"split": "train"
} |
math | [
{
"content": "Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\\frac{{y^2}}{4}=1$, and they satisfy $\\overrightarrow{PA}+\\overrightarrow{PB}=2\\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the mini... | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "38820",
"split": "train"
} |
math | [
{
"content": "Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ whe... | {
"ground_truth": "489",
"style": "rule"
} | {
"index": "38821",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the left focus of the ellipse $C:\\frac{{x}^{2}}{3}+\\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 - 2\\sqrt{3}",
"style": "rule"
} | {
"index": "38822",
"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.\n How many different paths can the particle take from $(0,0)$ to $(5,... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "38823",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $C: \\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\\_1|=3|PF\\_2|$ and $\\overrightarrow{PF\\_1} \\cdot \\overrightarrow{PF\\_2}=a^{2}$,... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "38824",
"split": "train"
} |
math | [
{
"content": "The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?\n\nNote: We say that the fraction \\( p / q \\) is irreducible if the integers \\( p \\) and \\( q \\) do not have common prim... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "38825",
"split": "train"
} |
math | [
{
"content": "Given the vectors $\\overrightarrow{a} = (1, 1)$, $\\overrightarrow{b} = (2, 0)$, the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "38826",
"split": "train"
} |
math | [
{
"content": "What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\pi + 2",
"style": "rule"
} | {
"index": "38827",
"split": "train"
} |
math | [
{
"content": "Compute the integrals:\n1) \\(\\int_{0}^{5} \\frac{x \\, dx}{\\sqrt{1+3x}}\\)\n2) \\(\\int_{\\ln 2}^{\\ln 9} \\frac{dx}{e^{x}-e^{-x}}\\)\n3) \\(\\int_{1}^{\\sqrt{3}} \\frac{(x^{3}+1) \\, dx}{x^{2} \\sqrt{4-x^{2}}}\\)\n4) \\(\\int_{0}^{\\frac{\\pi}{2}} \\frac{dx}{2+\\cos x}\\) Please output the fin... | {
"ground_truth": "\\frac{\\pi}{3 \\sqrt{3}}",
"style": "rule"
} | {
"index": "38828",
"split": "train"
} |
math | [
{
"content": "A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\\text{th}}$ pirate to take a share takes $\\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement ... | {
"ground_truth": "1925",
"style": "rule"
} | {
"index": "38829",
"split": "train"
} |
math | [
{
"content": "ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7 and 8",
"style": "rule"
} | {
"index": "38830",
"split": "train"
} |
math | [
{
"content": "For all triples \\((x, y, z)\\) that satisfy the system\n\n$$\n\\left\\{\\begin{array}{l}\n2 \\cos x = \\operatorname{ctg} y \\\\\n2 \\sin y = \\operatorname{tg} z \\\\\n\\cos z = \\operatorname{ctg} x\n\\end{array}\\right.\n$$\n\nfind the minimum value of the expression \\(\\sin x + \\cos z\\). P... | {
"ground_truth": "-\\frac{5 \\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "38831",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "38832",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$, where $\\angle A = \\angle B$ and $AB = 10$. What is the area of $\\triangle ABC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "38833",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = ax+b$, where $a$ and $b$ are real constants, and $g(x) = 2x - 5$. Suppose that for all $x$, it is true that $g(f(x)) = 3x + 4$. What is $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38834",
"split": "train"
} |
math | [
{
"content": "Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then\n\\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.\\]\nFor every $m\\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "38835",
"split": "train"
} |
math | [
{
"content": "Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, …)? (Answer with a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "38836",
"split": "train"
} |
math | [
{
"content": "Given $\\sin\\theta= \\frac {m-3}{m+5}$ and $\\cos\\theta= \\frac {4-2m}{m+5}$ ($\\frac {\\pi}{2} < \\theta < \\pi$), calculate $\\tan\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac {5}{12}",
"style": "rule"
} | {
"index": "38837",
"split": "train"
} |
math | [
{
"content": "Given that $| \\overrightarrow{a}|=5$, $| \\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ on vector $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "38838",
"split": "train"
} |
math | [
{
"content": "Find the $r$ that satisfies $\\log_{16} (r+16) = \\frac{5}{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38839",
"split": "train"
} |
math | [
{
"content": "Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "B",
"style": "rule"
} | {
"index": "38840",
"split": "train"
} |
math | [
{
"content": "In a large square, points $P$, $Q$, $R$, and $S$ are midpoints of the sides. Inside the square, a triangle is formed by connecting point $P$ to the center of the square and point $Q$. If the area of the larger square is 80, what are the areas of the smaller square and the triangle formed? Please o... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38841",
"split": "train"
} |
math | [
{
"content": "The number 74 can be factored as 2(37), so 74 is said to have two distinct prime factors. How many distinct prime factors does 210 have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38842",
"split": "train"
} |
math | [
{
"content": "$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\\circ}$. Find the angular measure of arc $B D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "38843",
"split": "train"
} |
math | [
{
"content": "In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $-9$ and $-1$ for the roots. The correct ... | {
"ground_truth": "x^2-10x+9=0",
"style": "rule"
} | {
"index": "38844",
"split": "train"
} |
math | [
{
"content": "Three points, $A, B$, and $C$, are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of $\\angle A B C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60^{\\circ}",
"style": "rule"
} | {
"index": "38845",
"split": "train"
} |
math | [
{
"content": "What is the simplified value of the sum: $-1^{2004} + (-1)^{2005} + 1^{2006} -1^{2007}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "38846",
"split": "train"
} |
math | [
{
"content": "A positive integer \\( n \\) is said to be increasing if, by reversing the digits of \\( n \\), we get an integer larger than \\( n \\). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are incre... | {
"ground_truth": "4005",
"style": "rule"
} | {
"index": "38847",
"split": "train"
} |
math | [
{
"content": "Given the coordinates of points $A(3, 0)$, $B(0, -3)$, and $C(\\cos\\alpha, \\sin\\alpha)$, where $\\alpha \\in \\left(\\frac{\\pi}{2}, \\frac{3\\pi}{2}\\right)$. If $\\overrightarrow{OC}$ is parallel to $\\overrightarrow{AB}$ and $O$ is the origin, find the value of $\\alpha$. Please output the f... | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "38848",
"split": "train"
} |
math | [
{
"content": "How many miles can a car travel in 20 minutes if it travels $ \\, \\frac{3}{4} \\, $ as fast as a train going 80 miles per hour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20\\text{ miles}",
"style": "rule"
} | {
"index": "38849",
"split": "train"
} |
math | [
{
"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? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "138",
"style": "rule"
} | {
"index": "38850",
"split": "train"
} |
math | [
{
"content": "A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what perce... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "38851",
"split": "train"
} |
math | [
{
"content": "During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic,\\[x^2-sx+p,\\]with roots $r_1$ and $r_2$. He notices that\\[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\\cdots=r_1^{2007}+r_2^{2007}.\\]He w... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "38852",
"split": "train"
} |
math | [
{
"content": "Given that $0 < a < \\pi, \\tan a=-2$,\n\n(1) Find the value of $\\cos a$;\n\n(2) Find the value of $2\\sin^{2}a - \\sin a \\cos a + \\cos^{2}a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{5}",
"style": "rule"
} | {
"index": "38853",
"split": "train"
} |
math | [
{
"content": "The polynomial \\( f(x) \\) satisfies the equation \\( f(x) - f(x-2) = (2x-1)^{2} \\) for all \\( x \\). Find the sum of the coefficients of \\( x^{2} \\) and \\( x \\) in \\( f(x) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "38854",
"split": "train"
} |
math | [
{
"content": "Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$ . He knows the following three facts: $P(0) = -\\frac{1}{8}$ , the roots form a geometric progression in the order $a,b,c$ , and \\[ \\sum_{k=1}^{\\infty... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38855",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD$, $AB=2$, $BC=4$, and points $E$, $F$, and $G$ are located as follows: $E$ is the midpoint of $\\overline{BC}$, $F$ is the midpoint of $\\overline{CD}$, and $G$ is one fourth of the way down $\\overline{AD}$ from $A$. If point $H$ is the midpoint of $\\overline{GE}$, what is the... | {
"ground_truth": "\\dfrac{5}{4}",
"style": "rule"
} | {
"index": "38856",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\\{ A,B,C,D,M,N\\}$ . Find the least number $k$ such that $s(M,N... | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "38857",
"split": "train"
} |
math | [
{
"content": "The function $f(x)$ satisfies\n\\[f(x + y) = f(x) f(y)\\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$\n\nEnter all the possible values, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0,1",
"style": "rule"
} | {
"index": "38858",
"split": "train"
} |
math | [
{
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black... | {
"ground_truth": "\\dfrac14",
"style": "rule"
} | {
"index": "38859",
"split": "train"
} |
math | [
{
"content": "One mole of an ideal monatomic gas is first heated isobarically, performing 10 Joules of work. Then it is heated isothermally, receiving the same amount of heat as in the first case. How much work does the gas perform (in Joules) in the second case? Please output the final answer within \\boxed{}.... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "38860",
"split": "train"
} |
math | [
{
"content": "Point $P$ is on the ellipse $\\frac{x^{2}}{16}+ \\frac{y^{2}}{9}=1$. The maximum and minimum distances from point $P$ to the line $3x-4y=24$ are $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{12(2- \\sqrt{2})}{5}",
"style": "rule"
} | {
"index": "38861",
"split": "train"
} |
math | [
{
"content": "If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of \"quasi-geometric progression.\" Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these ... | {
"ground_truth": "\\frac{13}{35}",
"style": "rule"
} | {
"index": "38862",
"split": "train"
} |
math | [
{
"content": "Compute $\\dbinom{11}{9}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "38863",
"split": "train"
} |
math | [
{
"content": "Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ \ffind $n$ such that the sum of the \ffirst $n$ terms is $2008$ or $2009$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1026",
"style": "rule"
} | {
"index": "38864",
"split": "train"
} |
math | [
{
"content": "In a local election based on proportional representation, each voter is given a ballot with 10 candidate names. The voter must mark No. 1 against the candidate for whom they cast their first vote, and they may optionally fill out No. 2 through No. 10 for their subsequent preferences.\n\nHow many d... | {
"ground_truth": "3628800",
"style": "rule"
} | {
"index": "38865",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x+3$ and $g(x)=3x+5$. Find $f(g(4))-g(f(4))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "38866",
"split": "train"
} |
math | [
{
"content": "There are $12$ small balls in a bag, which are red, black, and yellow respectively (these balls are the same in other aspects except for color). The probability of getting a red ball when randomly drawing one ball is $\\frac{1}{3}$, and the probability of getting a black ball is $\\frac{1}{6}$ mor... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "38867",
"split": "train"
} |
math | [
{
"content": "A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside this cylinder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{61}",
"style": "rule"
} | {
"index": "38868",
"split": "train"
} |
math | [
{
"content": "A three-digit number is called a \"concave number\" if the digit in the tens place is smaller than both the digit in the hundreds place and the digit in the units place. For example, 504 and 746 are concave numbers. How many three-digit concave numbers are there if all the digits are distinct? Ple... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "38869",
"split": "train"
} |
math | [
{
"content": "Find the coefficient of the x term in the expansion of $(x^2-x-2)^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "38870",
"split": "train"
} |
math | [
{
"content": "Compute the value of the following expression:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2046",
"style": "rule"
} | {
"index": "38871",
"split": "train"
} |
math | [
{
"content": "Let set $A=\\{x \\mid |x-2| \\leq 2\\}$, and $B=\\{y \\mid y=-x^2, -1 \\leq x \\leq 2\\}$, then $A \\cap B=$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\{0\\}",
"style": "rule"
} | {
"index": "38872",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(1-\\frac{2}{{m+1}})\\div \\frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\\tan 60^{\\circ}-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3-\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "38873",
"split": "train"
} |
math | [
{
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu,20]=60$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "38874",
"split": "train"
} |
math | [
{
"content": "Evaluate: $\\frac{10^{-2}5^0}{10^{-3}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38875",
"split": "train"
} |
math | [
{
"content": "To make a rectangular frame with lengths of 3cm, 4cm, and 5cm, the total length of wire needed is cm. If paper is then glued around the outside (seams not considered), the total area of paper needed is cm<sup>2</sup>. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "38876",
"split": "train"
} |
math | [
{
"content": "Given the original spherical dome has a height of $55$ meters and can be represented as holding $250,000$ liters of air, and Emily's scale model can hold only $0.2$ liters of air, determine the height, in meters, of the spherical dome in Emily's model. Please output the final answer within \\boxed... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "38877",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(C\\) lies on \\(AE\\) and \\(AB=BC=CD\\). If \\(\\angle CDE=t^{\\circ}, \\angle DEC=(2t)^{\\circ}\\), and \\(\\angle BCA=\\angle BCD=x^{\\circ}\\), determine the measure of \\(\\angle ABC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "38878",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ that satisfies the equation $25^{-2} = \\frac{5^{48/x}}{5^{26/x} \\cdot 25^{17/x}}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38879",
"split": "train"
} |
math | [
{
"content": "Semicircles of diameter 2'' are lined up as shown. What is the area, in square inches, of the shaded region in a 1-foot length of this pattern? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\nsize(101);\n\npath tophalf = Arc((0,0),1,180,0) -- Arc((2,0),1,180,0) -- Arc((4,0),1,180,0)... | {
"ground_truth": "6\\pi",
"style": "rule"
} | {
"index": "38880",
"split": "train"
} |
math | [
{
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $\\overrightarrow{a}^{2}=(5 \\overrightarrow{a}-4 \\overrightarrow{b})\\cdot \\overrightarrow{b}$, find the minimum value of $\\cos < \\overrightarrow{a}, \\overrightarrow{b} >$. Please output the final answer within... | {
"ground_truth": "\\frac {4}{5}",
"style": "rule"
} | {
"index": "38881",
"split": "train"
} |
math | [
{
"content": "Whole numbers whose decimal representation reads the same from left to right as from right to left are called symmetrical. For example, the number 5134315 is symmetrical, while 5134415 is not. How many seven-digit symmetrical numbers exist such that adding 1100 to them leaves them unchanged as sym... | {
"ground_truth": "810",
"style": "rule"
} | {
"index": "38882",
"split": "train"
} |
math | [
{
"content": "How many rectangles are there whose four vertices are points on this grid? [asy]\nsize(50);\ndot((0,0));\ndot((5,0));\ndot((10,0));\ndot((0,5));\ndot((0,10));\ndot((5,5));\ndot((5,10));\ndot((10,5));\ndot((10,10));\n[/asy] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38883",
"split": "train"
} |
math | [
{
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i = -8; i <= 8; ++i)... | {
"ground_truth": "\\text{B}",
"style": "rule"
} | {
"index": "38884",
"split": "train"
} |
math | [
{
"content": "Xiaoming saw a tractor pulling a rope slowly on the road and decided to measure the length of the rope. If Xiaoming walks in the direction the tractor is moving, it takes him a total of 140 steps to walk from one end of the rope to the other. If Xiaoming walks in the opposite direction to the trac... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "38885",
"split": "train"
} |
math | [
{
"content": "Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "38886",
"split": "train"
} |
math | [
{
"content": "A Sudoku matrix is defined as a $9 \\times 9$ array with entries from \\{1,2, \\ldots, 9\\} and with the constraint that each row, each column, and each of the nine $3 \\times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen at random (so that ev... | {
"ground_truth": "\\frac{2}{21}",
"style": "rule"
} | {
"index": "38887",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be an isosceles right triangle with $\\angle A=90^o$ . Point $D$ is the midpoint of the side $[AC]$ , and point $E \\in [AC]$ is so that $EC = 2AE$ . Calculate $\\angle AEB + \\angle ADB$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "135",
"style": "rule"
} | {
"index": "38888",
"split": "train"
} |
math | [
{
"content": "Given the curve $C:\\begin{cases}x=2\\cos a \\\\ y= \\sqrt{3}\\sin a\\end{cases} (a$ is the parameter) and the fixed point $A(0,\\sqrt{3})$, ${F}_1,{F}_2$ are the left and right foci of this curve, respectively. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a... | {
"ground_truth": "\\dfrac{12\\sqrt{3}}{13}",
"style": "rule"
} | {
"index": "38889",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{5}+ax^{3}+bx-8$, if $f(-2)=10$, find the value of $f(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-26",
"style": "rule"
} | {
"index": "38890",
"split": "train"
} |
math | [
{
"content": "Let the set $U=\\{1, 3a+5, a^2+1\\}$, $A=\\{1, a+1\\}$, and $\\mathcal{C}_U A=\\{5\\}$. Find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "38891",
"split": "train"
} |
math | [
{
"content": "The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "38892",
"split": "train"
} |
math | [
{
"content": "Zachary paid for a $\\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "38893",
"split": "train"
} |
math | [
{
"content": "In the complex plane, the distance between the points corresponding to the complex numbers $-3+i$ and $1-i$ is $\\boxed{\\text{answer}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{20}",
"style": "rule"
} | {
"index": "38894",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38895",
"split": "train"
} |
math | [
{
"content": "A certain school sends two students, A and B, to form a \"youth team\" to participate in a shooting competition. In each round of the competition, A and B each shoot once. It is known that the probability of A hitting the target in each round is $\\frac{1}{2}$, and the probability of B hitting the... | {
"ground_truth": "\\frac{7}{24}",
"style": "rule"
} | {
"index": "38896",
"split": "train"
} |
math | [
{
"content": "If $t = \\frac{1}{1 - \\sqrt[4]{2}}$, then $t$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$-(1+\\sqrt[4]{2})(1+\\sqrt{2})$",
"style": "rule"
} | {
"index": "38897",
"split": "train"
} |
math | [
{
"content": "The flea Kuzya can make a jump on the plane in any direction for a distance of exactly 17 mm. Its task is to get from point \\( A \\) to point \\( B \\) on the plane, the distance between which is 1947 cm. What is the minimum number of jumps it must make? Please output the final answer within \\bo... | {
"ground_truth": "1146",
"style": "rule"
} | {
"index": "38898",
"split": "train"
} |
math | [
{
"content": "The sampling group size is 10. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "38899",
"split": "train"
} |
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