data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16.\\overline{6}\\%",
"style": "rule"
} | {
"index": "39400",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)\n\nAlexandru Mihalcu Please output the final... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "39401",
"split": "train"
} |
math | [
{
"content": "Given the power function $f(x)=kx^{\\alpha}$, its graph passes through the point $(\\frac{1}{2}, \\frac{\\sqrt{2}}{2})$. Find the value of $k+\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "39402",
"split": "train"
} |
math | [
{
"content": "$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals? Please output the final answer within \\boxed{}."... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39403",
"split": "train"
} |
math | [
{
"content": "How many distinct four-digit numbers are divisible by 3 and have 47 as their last two digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39404",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}=3$, and for $n \\geq 1$, let $a_{n+1}=(n+1) a_{n}-n$. Find the smallest $m \\geq 2005$ such that $a_{m+1}-1 \\mid a_{m}^{2}-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "39405",
"split": "train"
} |
math | [
{
"content": "Given that \\(\\tan \\frac{\\alpha+\\beta}{2}=\\frac{\\sqrt{6}}{2}\\) and \\(\\cot \\alpha \\cdot \\cot \\beta=\\frac{7}{13}\\), find the value of \\(\\cos (\\alpha-\\beta)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "39406",
"split": "train"
} |
math | [
{
"content": "In a modified SHORT BINGO, a $5\\times5$ card uses a WILD square in the middle and places 24 other numbers in the remaining squares. Numbers are placed as follows: 5 distinct numbers from $1-15$ in the first column, 5 distinct numbers from $16-30$ in the second column, 4 distinct numbers from $31-... | {
"ground_truth": "360360",
"style": "rule"
} | {
"index": "39407",
"split": "train"
} |
math | [
{
"content": "Chords $A B$ and $C D$ of a circle are perpendicular and intersect at a point $P$. If $A P=6, B P=12$, and $C D=22$, find the area of the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130 \\pi",
"style": "rule"
} | {
"index": "39408",
"split": "train"
} |
math | [
{
"content": "A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39409",
"split": "train"
} |
math | [
{
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all different from each other. What is the minimum number of distinct numbers in \\( a_{1}, a_{2}, \\cd... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "39410",
"split": "train"
} |
math | [
{
"content": "The highest power of 2 that is a factor of \\(15.13^{4} - 11^{4}\\) needs to be determined. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39411",
"split": "train"
} |
math | [
{
"content": "Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is to be made us... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39412",
"split": "train"
} |
math | [
{
"content": "A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\\sqrt[3]{2}$ cm, find the radius of the original bubble. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "39413",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39414",
"split": "train"
} |
math | [
{
"content": "In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \\leq z \\leq 1$. Let $S_{1}, S_{2}, \\ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$). The expected volume of the region $S_{1} \\cap S_{2} \\cap \\cdots \\cap S_{2022}$... | {
"ground_truth": "271619",
"style": "rule"
} | {
"index": "39415",
"split": "train"
} |
math | [
{
"content": "Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the proba... | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "39416",
"split": "train"
} |
math | [
{
"content": "Suppose $77\\cdot 88\\cdot 99 \\equiv n\\pmod{20}$, where $0\\le n<20$.\n\nWhat is the value of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39417",
"split": "train"
} |
math | [
{
"content": "For a real number \\( x \\), find the maximum value of\n\\[\n\\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{18}",
"style": "rule"
} | {
"index": "39418",
"split": "train"
} |
math | [
{
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-6, 6]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "156",
"style": "rule"
} | {
"index": "39419",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of\n\\[\n\\frac{3x + 4y + 6}{\\sqrt{x^2 + 4y^2 + 4}}\n\\]\nover all real numbers \\(x\\) and \\(y\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{61}",
"style": "rule"
} | {
"index": "39420",
"split": "train"
} |
math | [
{
"content": "Given positive numbers $x$ and $y$ satisfying $x+y=2$, calculate the minimum value of $\\frac{4}{x+2}+\\frac{3x-7}{3y+4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{16}",
"style": "rule"
} | {
"index": "39421",
"split": "train"
} |
math | [
{
"content": "The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "c+a",
"style": "rule"
} | {
"index": "39422",
"split": "train"
} |
math | [
{
"content": "On the side \\( CD \\) of the trapezoid \\( ABCD \\) with \\( AD \\parallel BC \\), a point \\( M \\) is marked. From vertex \\( A \\), a perpendicular \\( AH \\) is dropped to the segment \\( BM \\). It is known that \\( AD = HD \\). Find the length of segment \\( AD \\), given that \\( BC = 16 \... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "39423",
"split": "train"
} |
math | [
{
"content": "The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street? Please output the final answer wit... | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "39424",
"split": "train"
} |
math | [
{
"content": "In square $PQRS$ with side length 2, each of $P, Q, R$, and $S$ is the centre of a circle with radius 1. What is the area of the shaded region? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4-\\pi",
"style": "rule"
} | {
"index": "39425",
"split": "train"
} |
math | [
{
"content": "Inside a square $R_1$ with area 81, an equilateral triangle $T_1$ is inscribed such that each vertex of $T_1$ touches one side of $R_1$. Each midpoint of $T_1’s$ sides is connected to form a smaller triangle $T_2$. The process is repeated with $T_2$ to form $T_3$. Find the area of triangle $T_3$. ... | {
"ground_truth": "\\frac{81\\sqrt{3}}{256}",
"style": "rule"
} | {
"index": "39426",
"split": "train"
} |
math | [
{
"content": "Given that the sequence $\\left\\{a_n\\right\\}$ is an arithmetic sequence, if $\\dfrac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, calculate the maximum value of $n$ for which $S_n > 0$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "39427",
"split": "train"
} |
math | [
{
"content": "Mona has 12 match sticks of length 1, and she has to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make? Please output the final a... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "39428",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known that the line $l: \\rho... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "39429",
"split": "train"
} |
math | [
{
"content": "When $\\frac{1}{10101}$ is expressed as a decimal, what is the sum of the first 100 digits after the decimal point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "450",
"style": "rule"
} | {
"index": "39430",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the second triangle. The... | {
"ground_truth": "237.5\\%",
"style": "rule"
} | {
"index": "39431",
"split": "train"
} |
math | [
{
"content": "How many natural numbers greater than 10 but less than 100 are relatively prime to 21? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "39432",
"split": "train"
} |
math | [
{
"content": "A point $P(\\frac{\\pi}{12}, m)$ on the graph of the function $y = \\sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \\cos (2x - \\frac{\\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $n \\cdot m$. Please output the final answer within ... | {
"ground_truth": "\\frac{5\\pi}{48}",
"style": "rule"
} | {
"index": "39433",
"split": "train"
} |
math | [
{
"content": "Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy? Please output the final answer within \\boxe... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39434",
"split": "train"
} |
math | [
{
"content": "The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations\n$ab+bc=44$\n$ac+bc=23$\nis Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39435",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\\frac{\\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)$ ($m > \\frac{3}{4}... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "39436",
"split": "train"
} |
math | [
{
"content": "In triangle \\( A B C \\) with side \\( A C = 8 \\), a bisector \\( B L \\) is drawn. It is known that the areas of triangles \\( A B L \\) and \\( B L C \\) are in the ratio \\( 3: 1 \\). Find the bisector \\( B L \\), for which the height dropped from vertex \\( B \\) to the base \\( A C \\) wil... | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "39437",
"split": "train"
} |
math | [
{
"content": "For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$. Find the number of positive integers less th... | {
"ground_truth": "958",
"style": "rule"
} | {
"index": "39438",
"split": "train"
} |
math | [
{
"content": "Let ellipse $C$:$\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\left(a \\gt b \\gt 0\\right)$ have foci $F_{1}(-c,0)$ and $F_{2}(c,0)$. Point $P$ is the intersection point of $C$ and the circle $x^{2}+y^{2}=c^{2}$. The bisector of $\\angle PF_{1}F_{2}$ intersects $PF_{2}$ at $Q$. If $|PQ|=\\... | {
"ground_truth": "\\sqrt{3}-1",
"style": "rule"
} | {
"index": "39439",
"split": "train"
} |
math | [
{
"content": "Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39440",
"split": "train"
} |
math | [
{
"content": "A math competition problem has probabilities of being solved independently by person \\( A \\), \\( B \\), and \\( C \\) as \\( \\frac{1}{a} \\), \\( \\frac{1}{b} \\), and \\( \\frac{1}{c} \\) respectively, where \\( a \\), \\( b \\), and \\( c \\) are positive integers less than 10. When \\( A \\... | {
"ground_truth": "4/15",
"style": "rule"
} | {
"index": "39441",
"split": "train"
} |
math | [
{
"content": "Given that the vertices of triangle $\\triangle ABC$ are $A(3,2)$, the equation of the median on side $AB$ is $x-3y+8=0$, and the equation of the altitude on side $AC$ is $2x-y-9=0$. \n$(1)$ Find the coordinates of points $B$ and $C$. \n$(2)$ Find the area of $\\triangle ABC$. Please output the ... | {
"ground_truth": "\\frac{15}{2}",
"style": "rule"
} | {
"index": "39442",
"split": "train"
} |
math | [
{
"content": "Three concentric circles with radii 5 meters, 10 meters, and 15 meters, form the paths along which an ant travels moving from one point to another symmetrically. The ant starts at a point on the smallest circle, moves radially outward to the third circle, follows a path on each circle, and include... | {
"ground_truth": "\\frac{65\\pi}{3} + 20",
"style": "rule"
} | {
"index": "39443",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 349 inclusive does the decimal representation of $\\frac{n}{350}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "39444",
"split": "train"
} |
math | [
{
"content": "Bonnie constructs the frame of a cuboid using wire pieces. She uses eight pieces each of 8 inches for the length, and four pieces each of 10 inches for the width and height. Roark, on the other hand, uses 2-inch-long wire pieces to construct frames of smaller cuboids, all of the same size with dim... | {
"ground_truth": "\\frac{9}{250}",
"style": "rule"
} | {
"index": "39445",
"split": "train"
} |
math | [
{
"content": "Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\\{ 2, 3, 5, 7 \\}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1932",
"style": "rule"
} | {
"index": "39446",
"split": "train"
} |
math | [
{
"content": "In a right-angled triangle \\(ABC\\) (with right angle at \\(A\\)), the bisectors of the acute angles intersect at point \\(P\\). The distance from \\(P\\) to the hypotenuse is \\(\\sqrt{80000}\\). What is the distance from \\(P\\) to \\(A\\)? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "39447",
"split": "train"
} |
math | [
{
"content": "John has 15 marbles of different colors, including one red, one green, one blue, and one yellow marble. In how many ways can he choose 5 marbles, if at least one of the chosen marbles is red, green, or blue, but not yellow? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "1540",
"style": "rule"
} | {
"index": "39448",
"split": "train"
} |
math | [
{
"content": "Construct the cross-section of a triangular pyramid \\( A B C D \\) with a plane passing through the midpoints \\( M \\) and \\( N \\) of edges \\( A C \\) and \\( B D \\) and the point \\( K \\) on edge \\( C D \\), for which \\( C K: K D = 1: 2 \\). In what ratio does this plane divide edge \\( ... | {
"ground_truth": "1:2",
"style": "rule"
} | {
"index": "39449",
"split": "train"
} |
math | [
{
"content": "The \"One Helmet, One Belt\" safety protection campaign is a safety protection campaign launched by the Ministry of Public Security nationwide. It is also an important standard for creating a civilized city and being a civilized citizen. \"One helmet\" refers to a safety helmet. Drivers and passen... | {
"ground_truth": "6200",
"style": "rule"
} | {
"index": "39450",
"split": "train"
} |
math | [
{
"content": "Find the number of natural numbers \\( k \\) not exceeding 353500 such that \\( k^{2} + k \\) is divisible by 505. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2800",
"style": "rule"
} | {
"index": "39451",
"split": "train"
} |
math | [
{
"content": "What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105^\\circ",
"style": "rule"
} | {
"index": "39452",
"split": "train"
} |
math | [
{
"content": "The first and thirteenth terms of an arithmetic sequence are 5 and 29, respectively. What is the fiftieth term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "39453",
"split": "train"
} |
math | [
{
"content": "Suppose \\(x\\) and \\(y\\) are positive real numbers such that \\(x+\\frac{1}{y}=y+\\frac{2}{x}=3\\). Compute the maximum possible value of \\(xy\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3+\\sqrt{7}",
"style": "rule"
} | {
"index": "39454",
"split": "train"
} |
math | [
{
"content": "In the expansion of $({x-1}){({\\frac{1}{{{x^{2022}}}}+\\sqrt{x}+1})^8}$, the coefficient of $x^{2}$ is _____. (Provide your answer as a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-42",
"style": "rule"
} | {
"index": "39455",
"split": "train"
} |
math | [
{
"content": "Given a hyperbola \\( H: x^{2}-y^{2}=1 \\) with a point \\( M \\) in the first quadrant, and a line \\( l \\) tangent to the hyperbola \\( H \\) at point \\( M \\), intersecting the asymptotes of \\( H \\) at points \\( P \\) and \\( Q \\) (where \\( P \\) is in the first quadrant). If point \\( R... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "39456",
"split": "train"
} |
math | [
{
"content": "If a positive integer \\( n \\) makes the equation \\( x^{3} + y^{3} = z^{n} \\) have a positive integer solution \\( (x, y, z) \\), then \\( n \\) is called a \"good number.\" How many good numbers are there that do not exceed 2,019? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "1346",
"style": "rule"
} | {
"index": "39457",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "222",
"style": "rule"
} | {
"index": "39458",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of\n\\[8a + 3b + 5c.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7\\sqrt{2}.",
"style": "rule"
} | {
"index": "39459",
"split": "train"
} |
math | [
{
"content": "Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "39460",
"split": "train"
} |
math | [
{
"content": "Given real numbers \\( x, y, z, w \\) such that \\( x + y + z + w = 1 \\), find the maximum value of \\( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3/2",
"style": "rule"
} | {
"index": "39461",
"split": "train"
} |
math | [
{
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{12}$, what is the smallest possible value for $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "39462",
"split": "train"
} |
math | [
{
"content": "Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:\n1. The Tigers won more than the Eagles.\n2. The Patriots won more than the Cubs, but fewer than the Mounties.\n3. The Cubs won more than 15 games.\n4. The F... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "39463",
"split": "train"
} |
math | [
{
"content": "The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3\\sqrt{2}$",
"style": "rule"
} | {
"index": "39464",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of \n\\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "424",
"style": "rule"
} | {
"index": "39465",
"split": "train"
} |
math | [
{
"content": "A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\\circ$. Find the measure of the angle $\\angle ACE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40^\\circ",
"style": "rule"
} | {
"index": "39466",
"split": "train"
} |
math | [
{
"content": "How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "382",
"style": "rule"
} | {
"index": "39467",
"split": "train"
} |
math | [
{
"content": "For the four-digit number \\(\\overline{abcd}\\) where \\(1 \\leqslant a \\leqslant 9\\) and \\(0 \\leqslant b, c, d \\leqslant 9\\), if \\(a > b, b < c, c > d\\), then \\(\\overline{abcd}\\) is called a \\(P\\)-type number. If \\(a < b, b > c, c < d\\), then \\(\\overline{abcd}\\) is called a \\(... | {
"ground_truth": "285",
"style": "rule"
} | {
"index": "39468",
"split": "train"
} |
math | [
{
"content": "1. Given that $α$ and $β$ are acute angles, and $\\cos α= \\frac{4}{5}$, $\\cos (α+β)=- \\frac{16}{65}$, find the value of $\\cos β$.\n2. Given that $0 < β < \\frac{π}{4} < α < \\frac{3}{4}π$, $\\cos ( \\frac{π}{4}-α)= \\frac{3}{5}$, $\\sin ( \\frac{3π}{4}+β)= \\frac{5}{13}$, find the value of $\\... | {
"ground_truth": "\\frac{56}{65}",
"style": "rule"
} | {
"index": "39469",
"split": "train"
} |
math | [
{
"content": "Find the curve defined by the equation\n\\[r = \\frac{1}{\\sin \\theta - \\cos \\theta}.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(A)}",
"style": "rule"
} | {
"index": "39470",
"split": "train"
} |
math | [
{
"content": "Identical regular pentagons are arranged in a ring. Each of the regular pentagons has a perimeter of 65. The regular polygon formed as the inner boundary of the ring has a perimeter of \\( P \\). What is the value of \\( P \\)? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "39471",
"split": "train"
} |
math | [
{
"content": "Let \\(a \\star b=\\sin a \\cos b\\) for all real numbers \\(a\\) and \\(b\\). If \\(x\\) and \\(y\\) are real numbers such that \\(x \\star y-y \\star x=1\\), what is the maximum value of \\(x \\star y+y \\star x\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39472",
"split": "train"
} |
math | [
{
"content": "Xiao Wang's scores in three rounds of jump rope were 23, 34, and 29, respectively. Xiao Wang's final score is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "39473",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $b\\sin 2A = \\sqrt{3}a\\sin B$. \n$(Ⅰ)$ Find $\\angle A$; \n$(Ⅱ)$ If the area of $\\triangle ABC$ is $3\\sqrt{3}$, choose one of the three conditions, condition ①, condition ②, or condition ③, as the given condition to ensure the existence and uniqueness of $\\tria... | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "39474",
"split": "train"
} |
math | [
{
"content": "Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39475",
"split": "train"
} |
math | [
{
"content": "Simplify $(2x^2 + 7x - 3) - (x^2 + 5x - 12)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x^2+2x+9",
"style": "rule"
} | {
"index": "39476",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean between $\\frac {x + a}{x}$ and $\\frac {x - a}{x}$, when $x \\not = 0$, is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39477",
"split": "train"
} |
math | [
{
"content": "A group of children, numbering between 50 and 70, attended a spring math camp. To celebrate Pi Day (March 14), they decided to give each other squares if they were just acquaintances and circles if they were friends. Andrey noted that each boy received 3 circles and 8 squares, and each girl receiv... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "39478",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respective... | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "39479",
"split": "train"
} |
math | [
{
"content": "What is the largest positive multiple of $12$ that is less than $350?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "348",
"style": "rule"
} | {
"index": "39480",
"split": "train"
} |
math | [
{
"content": "Given that $y=\\left(m-2\\right)x+(m^{2}-4)$ is a direct proportion function, find the possible values of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "39481",
"split": "train"
} |
math | [
{
"content": "Determine the length of the interval of solutions of the inequality $a \\le 3x + 6 \\le b$ where the length of the interval is $15$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "39482",
"split": "train"
} |
math | [
{
"content": "In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clo... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39483",
"split": "train"
} |
math | [
{
"content": "For which integer $a$ does $x^2 - x + a$ divide $x^{13} + x + 90$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39484",
"split": "train"
} |
math | [
{
"content": "A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "187",
"style": "rule"
} | {
"index": "39485",
"split": "train"
} |
math | [
{
"content": "Given \\(0 < a < \\sqrt{3} \\cos \\theta\\), \\(\\theta \\in \\left[-\\frac{\\pi}{4}, \\frac{\\pi}{3}\\right]\\), find the minimum value of \\(f(a, \\theta) = \\cos^3 \\theta + \\frac{4}{3a \\cos^2 \\theta - a^3}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17 \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "39486",
"split": "train"
} |
math | [
{
"content": "Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B als... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "39487",
"split": "train"
} |
math | [
{
"content": "Compute $\\cos \\left( \\arctan \\frac{3}{4} \\right)$ and $\\tan \\left( \\arctan \\frac{3}{4} \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "39488",
"split": "train"
} |
math | [
{
"content": "Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another... | {
"ground_truth": "5/3",
"style": "rule"
} | {
"index": "39489",
"split": "train"
} |
math | [
{
"content": "Faces $ABC^{}_{}$ and $BCD^{}_{}$ of tetrahedron $ABCD^{}_{}$ meet at an angle of $30^\\circ$. The area of face $ABC^{}_{}$ is $120^{}_{}$, the area of face $BCD^{}_{}$ is $80^{}_{}$, and $BC=10^{}_{}$. Find the volume of the tetrahedron. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "320",
"style": "rule"
} | {
"index": "39490",
"split": "train"
} |
math | [
{
"content": "Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as \"partial fractions.\"<br/>Example: Express the fraction $\\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\\frac{{1-3x}}{{{x^2}-1}}=\\frac{M}{{x+1}}+\\frac{N}{{x-1}}$... | {
"ground_truth": "\\frac{20}{21}",
"style": "rule"
} | {
"index": "39491",
"split": "train"
} |
math | [
{
"content": "Suppose there exist constants $A$ , $B$ , $C$ , and $D$ such that \\[n^4=A\\binom n4+B\\binom n3+C\\binom n2 + D\\binom n1\\] holds true for all positive integers $n\\geq 4$ . What is $A+B+C+D$ ?\n\n*Proposed by David Altizio* Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "39492",
"split": "train"
} |
math | [
{
"content": "The repeating decimal for $\\frac{5}{13}$ is $0.cdc\\ldots$ What is the value of the sum $c+d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "39493",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB=AC$ and $D$ is a point on $\\overline{AC}$ so that $\\overline{BD}$ bisects angle $ABC$. If $BD=BC$, what is the measure, in degrees, of angle $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "39494",
"split": "train"
} |
math | [
{
"content": "What is the lowest prime number that is thirteen more than a cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "229",
"style": "rule"
} | {
"index": "39495",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a=4$, $b=6$, and $C=60^\\circ$:\n1. Calculate $\\overrightarrow{BC} \\cdot \\overrightarrow{CA}$;\n2. Find the projection of $\\overrightarrow{CA}$ onto $\\overrightarrow{B... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "39496",
"split": "train"
} |
math | [
{
"content": "A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner. Please output the final a... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "39497",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a square of side length 13. Let \\(E\\) and \\(F\\) be points on rays \\(AB\\) and \\(AD\\), respectively, so that the area of square \\(ABCD\\) equals the area of triangle \\(AEF\\). If \\(EF\\) intersects \\(BC\\) at \\(X\\) and \\(BX=6\\), determine \\(DF\\). Please output the... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "39498",
"split": "train"
} |
math | [
{
"content": "A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary. P... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "39499",
"split": "train"
} |
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