data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many triangles with integer sides have a perimeter equal to 27? (Triangles that differ only in the order of sides, such as 7, 10, 10 and 10, 10, 7, are considered the same triangle.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38500",
"split": "train"
} |
math | [
{
"content": "We marked the midpoints of all sides and diagonals of a regular 1976-sided polygon. What is the maximum number of these points that can lie on a single circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1976",
"style": "rule"
} | {
"index": "38501",
"split": "train"
} |
math | [
{
"content": "A cube with a side of 10 is divided into 1000 smaller cubes each with an edge of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (along any of the three directions) equals 0. In one of the cubes (denoted as A), the number 1 is written. There ar... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "38502",
"split": "train"
} |
math | [
{
"content": "Given that \\( P Q R S \\) is a square, and point \\( O \\) is on the line \\( R Q \\) such that the distance from \\( O \\) to point \\( P \\) is 1, what is the maximum possible distance from \\( O \\) to point \\( S \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38503",
"split": "train"
} |
math | [
{
"content": "If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "38504",
"split": "train"
} |
math | [
{
"content": "Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ and $0$ respectively... | {
"ground_truth": "0.525",
"style": "rule"
} | {
"index": "38505",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $$\\rho=4 \\sqrt {2}\\sin\\left( \\frac {3\\pi}{4}-\\theta\\right)$$ \n(1) Convert the polar equation of circle $C$ into a Cartesian coord... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "38506",
"split": "train"
} |
math | [
{
"content": "Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes? Please output th... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "38507",
"split": "train"
} |
math | [
{
"content": "In a survey, 100 students were asked if they like lentils and were also asked if they like chickpeas. A total of 68 students like lentils. A total of 53 like chickpeas. A total of 6 like neither lentils nor chickpeas. How many of the 100 students like both lentils and chickpeas? Please output the ... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38508",
"split": "train"
} |
math | [
{
"content": "When drawing 20 numbers from 2005 numbers using systematic sampling, calculate the interval of sampling. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38509",
"split": "train"
} |
math | [
{
"content": "Let $2^x$ be the greatest power of $2$ that is a factor of $144$, and let $3^y$ be the greatest power of $3$ that is a factor of $144$. Evaluate the following expression: $$\\left(\\frac15\\right)^{y - x}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "38510",
"split": "train"
} |
math | [
{
"content": "Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "38511",
"split": "train"
} |
math | [
{
"content": "In a regular quadrilateral pyramid $P-ABCD$, if $\\angle APC = 60^{\\circ}$, find the cosine of the dihedral angle between planes $A-PB-C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$-\\frac{1}{7}$",
"style": "rule"
} | {
"index": "38512",
"split": "train"
} |
math | [
{
"content": "Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$. What is the largest real value that $x + y$ can have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38513",
"split": "train"
} |
math | [
{
"content": "Given that \\( F \\) is the right focus of the hyperbola \\( x^{2} - y^{2} = 1 \\), \\( l \\) is the right directrix of the hyperbola, and \\( A \\) and \\( B \\) are two moving points on the right branch of the hyperbola such that \\( A F \\perp B F \\). The projection of the midpoint \\( M \\) o... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "38514",
"split": "train"
} |
math | [
{
"content": "Express 2.175 billion yuan in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.175 \\times 10^9",
"style": "rule"
} | {
"index": "38515",
"split": "train"
} |
math | [
{
"content": "Given a point P $(x, y)$ moves on the circle $x^2 + (y-1)^2 = 1$, the maximum value of $\\frac{y-1}{x-2}$ is \\_\\_\\_\\_\\_\\_, and the minimum value is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "38516",
"split": "train"
} |
math | [
{
"content": "On a section of the map, three roads form a right triangle. When motorcyclists were asked about the distance between $A$ and $B$, one of them responded that after traveling from $A$ to $B$, then to $C$, and back to $A$, his odometer showed 60 km. The second motorcyclist added that he knew by chanc... | {
"ground_truth": "22.5",
"style": "rule"
} | {
"index": "38517",
"split": "train"
} |
math | [
{
"content": "Compute the sum of all positive integers $n<2048$ such that $n$ has an even number of 1's in its binary representation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1048064",
"style": "rule"
} | {
"index": "38518",
"split": "train"
} |
math | [
{
"content": "Four whole numbers, when added three at a time, give the sums $180, 197, 208$ and $222$. What is the largest of the four numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "38519",
"split": "train"
} |
math | [
{
"content": "Compute the value of $252^2 - 248^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": "38520",
"split": "train"
} |
math | [
{
"content": "Determine the integer root of the polynomial \\[2x^3 + ax^2 + bx + c = 0,\\] where $a, b$, and $c$ are rational numbers. The equation has $4-2\\sqrt{3}$ as a root and another root whose sum with $4-2\\sqrt{3}$ is 8. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "38521",
"split": "train"
} |
math | [
{
"content": "Calculate the nearest integer to $(3+\\sqrt{5})^6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20608",
"style": "rule"
} | {
"index": "38522",
"split": "train"
} |
math | [
{
"content": "Values for $A, B, C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not ... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "38523",
"split": "train"
} |
math | [
{
"content": "Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \\[a^2+b^2+16c^2 = 9k^2 + 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(3, 3, 2, 3), (3, 37, 3, 13), (37, 3, 3, 13), (3, 17, 3, 7), (17, 3, 3, 7)",
"style": "rule"
} | {
"index": "38524",
"split": "train"
} |
math | [
{
"content": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 180$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac1{72}",
"style": "rule"
} | {
"index": "38525",
"split": "train"
} |
math | [
{
"content": "The minimum possible sum of the three dimensions of a rectangular box with a volume of $1729$ in$^3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "38526",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The perimeter of $\\triangle ABC$ is $\\sqrt {2}+1$, and $\\sin A + \\sin B = \\sqrt {2} \\sin C$.\n\n(I) Find the length of side $c$.\n(II) If the area of $\\triangle ABC$ is $\\frac {1}{5} \\s... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "38527",
"split": "train"
} |
math | [
{
"content": "In a chess match between players A and B, the probabilities of A winning, B winning, and a tie are $0.5$, $0.3$, and $0.2$, respectively. Find the probability of B winning at least one match against A after two matches. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.51",
"style": "rule"
} | {
"index": "38528",
"split": "train"
} |
math | [
{
"content": "Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minutes to reach her frie... | {
"ground_truth": "2.8",
"style": "rule"
} | {
"index": "38529",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the function \\( f(x) = x^{2} + 2x + \\frac{6}{x} + \\frac{9}{x^{2}} + 4 \\) for \\( x > 0 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 + 4 \\sqrt{3}",
"style": "rule"
} | {
"index": "38530",
"split": "train"
} |
math | [
{
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "38531",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, curve $C_1$ passes through point $P(a, 1)$ with parametric equations $$\\begin{cases} x=a+ \\frac { \\sqrt {2}}{2}t \\\\ y=1+ \\frac { \\sqrt {2}}{2}t\\end{cases}$$ where $t$ is a parameter and $a \\in \\mathbb{R}$. In the polar coordinate system with th... | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "38532",
"split": "train"
} |
math | [
{
"content": "Find the smallest composite number that has no prime factors less than 15. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "289",
"style": "rule"
} | {
"index": "38533",
"split": "train"
} |
math | [
{
"content": "If \\( a(x+1)=x^{3}+3x^{2}+3x+1 \\), find \\( a \\) in terms of \\( x \\).\n\nIf \\( a-1=0 \\), then the value of \\( x \\) is \\( 0 \\) or \\( b \\). What is \\( b \\) ?\n\nIf \\( p c^{4}=32 \\), \\( p c=b^{2} \\), and \\( c \\) is positive, what is the value of \\( c \\) ?\n\n\\( P \\) is an ope... | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "38534",
"split": "train"
} |
math | [
{
"content": "Given four points O, A, B, C on a plane satisfying OA=4, OB=3, OC=2, and $\\overrightarrow{OB} \\cdot \\overrightarrow{OC} = 3$, find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "38535",
"split": "train"
} |
math | [
{
"content": "Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{83}{3}",
"style": "rule"
} | {
"index": "38536",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\vec{m}=(2a\\cos x,\\sin x)$ and $\\vec{n}=(\\cos x,b\\cos x)$, the function $f(x)=\\vec{m}\\cdot \\vec{n}-\\frac{\\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\\frac{\\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\\left(\\frac{\\pi}{12},1\\right)$.... | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "38537",
"split": "train"
} |
math | [
{
"content": "Compute $\\cos 240^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "38538",
"split": "train"
} |
math | [
{
"content": "Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "5.5",
"style": "rule"
} | {
"index": "38539",
"split": "train"
} |
math | [
{
"content": "Given a class of 50 students with exam scores following a normal distribution $N(100,10^2)$, and $P(90 ≤ ξ ≤ 100) = 0.3$, estimate the number of students who scored above 110 points. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38540",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{2^a 3^b 5^c}.\\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1624}",
"style": "rule"
} | {
"index": "38541",
"split": "train"
} |
math | [
{
"content": "Points $C(1,1)$ and $D(8,6)$ are the endpoints of a diameter of a circle on a coordinate plane. Calculate both the area and the circumference of the circle. Express your answers in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{74}\\pi",
"style": "rule"
} | {
"index": "38542",
"split": "train"
} |
math | [
{
"content": "Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10(q - 1)",
"style": "rule"
} | {
"index": "38543",
"split": "train"
} |
math | [
{
"content": "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. Please output the final answer w... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "38544",
"split": "train"
} |
math | [
{
"content": "What is the greatest two-digit multiple of 13? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "38545",
"split": "train"
} |
math | [
{
"content": "Let $F_{1}$ and $F_{2}$ be the foci of the ellipse $C_{1}$: $\\frac{x^{2}}{6}+\\frac{y^{2}}{2}=1$, and let $P$ be a point of intersection between the ellipse $C_{1}$ and the hyperbola $C_{2}$: $\\frac{x^{2}}{3}-y^{2}=1$. Find the value of $\\cos \\angle F_{1}PF_{2}$. Please output the final answer... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38546",
"split": "train"
} |
math | [
{
"content": "In the quadrilateral pyramid \\( S A B C D \\):\n\n- The lateral faces \\( S A B \\), \\( S B C \\), \\( S C D \\), and \\( S D A \\) have areas 9, 9, 27, 27 respectively;\n- The dihedral angles at the edges \\( A B \\), \\( B C \\), \\( C D \\), \\( D A \\) are equal;\n- The quadrilateral \\( A B... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "38547",
"split": "train"
} |
math | [
{
"content": "Distribute 16 identical books among 4 students so that each student gets at least one book, and each student gets a different number of books. How many distinct ways can this be done? (Answer with a number.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "38548",
"split": "train"
} |
math | [
{
"content": "In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path $p$, let $A_{p}$ denote the number of unit squares under the path $p$. Compute the sum of $A_{p}$ over all up-right pa... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "38549",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "38550",
"split": "train"
} |
math | [
{
"content": "Given a point P in the plane satisfying $|PM| - |PN| = 2\\sqrt{2}$, with $M(-2,0)$, $N(2,0)$, and $O(0,0)$,\n(1) Find the locus S of point P;\n(2) A straight line passing through the point $(2,0)$ intersects with S at points A and B. Find the minimum value of the area of triangle $\\triangle OAB$.... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "38551",
"split": "train"
} |
math | [
{
"content": "The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38552",
"split": "train"
} |
math | [
{
"content": "In an acute-angled triangle, the sides $a$ and $b$ are the roots of the equation $x^{2}-2 \\sqrt {3}x+2=0$. The angles $A$ and $B$ satisfy the equation $2\\sin (A+B)- \\sqrt {3}=0$. Find the value of the side $c$ and the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\dfrac { \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "38553",
"split": "train"
} |
math | [
{
"content": "In the village of Znoynoe, there are exactly 1000 inhabitants, which exceeds the average population of the villages in the valley by 90 people.\n\nHow many inhabitants are there in the village of Raduzhny, which is also located in the Sunny Valley? Please output the final answer within \\boxed{}."... | {
"ground_truth": "900",
"style": "rule"
} | {
"index": "38554",
"split": "train"
} |
math | [
{
"content": "Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \\sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex $A$ to face $B C D$ can be written as $\\frac{a \\sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is square-free, and $\\operatorname{gcd}(a, c)=1$. Find $100 a+10... | {
"ground_truth": "2851",
"style": "rule"
} | {
"index": "38555",
"split": "train"
} |
math | [
{
"content": "How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55440",
"style": "rule"
} | {
"index": "38556",
"split": "train"
} |
math | [
{
"content": "Suppose \\(198 \\cdot 963 \\equiv m \\pmod{50}\\), where \\(0 \\leq m < 50\\).\n\nWhat is the value of \\(m\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38557",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\\frac mn$, where $m$ and $n$ are relatively prime positive intege... | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "38558",
"split": "train"
} |
math | [
{
"content": "Given that the area of $\\triangle ABC$ is $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n(I) Find the value of $\\tan 2A$;\n(II) If $\\cos C = \\frac{3}{5}$, and $|\\overrightarrow{AC} - \\overrightarrow{AB}| = 2$, find the area of $\\triangle ABC$. Please output the final answ... | {
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | {
"index": "38559",
"split": "train"
} |
math | [
{
"content": "How many ways are there to paint each of the integers $2, 3, \\cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "38560",
"split": "train"
} |
math | [
{
"content": "Shirley has a magical machine. If she inputs a positive even integer $n$ , the machine will output $n/2$ , but if she inputs a positive odd integer $m$ , the machine will output $m+3$ . The machine keeps going by automatically using its output as a new input, stopping immediately before it ob... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "38561",
"split": "train"
} |
math | [
{
"content": "It is currently 3:15:15 PM on a 12-hour digital clock. After 196 hours, 58 minutes, and 16 seconds, what will the time be in the format $A:B:C$? What is the sum $A + B + C$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "38562",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "12\\sqrt2",
"style": "rule"
} | {
"index": "38563",
"split": "train"
} |
math | [
{
"content": "In right triangle $XYZ$, we have $\\angle X = \\angle Z$ and $XZ = 8\\sqrt{2}$. What is the area of $\\triangle XYZ$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "38564",
"split": "train"
} |
math | [
{
"content": "If $\\frac{a}{10^x-1}+\\frac{b}{10^x+2}=\\frac{2 \\cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "38565",
"split": "train"
} |
math | [
{
"content": "Solve for $x$:\n\n$$x^2 + 4x + 3 = -(x + 3)(x + 5).$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "38566",
"split": "train"
} |
math | [
{
"content": "Calculate $f(x) = 3x^5 + 5x^4 + 6x^3 - 8x^2 + 35x + 12$ using the Horner's Rule when $x = -2$. Find the value of $v_4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "38567",
"split": "train"
} |
math | [
{
"content": "How many positive four-digit integers are divisible by both 7 and 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "215",
"style": "rule"
} | {
"index": "38568",
"split": "train"
} |
math | [
{
"content": "Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blue buckets, and 1 puc... | {
"ground_truth": "\\frac{89}{243}",
"style": "rule"
} | {
"index": "38569",
"split": "train"
} |
math | [
{
"content": "A bag contains four balls of identical shape and size, numbered $1$, $2$, $3$, $4$.\n(I) A ball is randomly drawn from the bag, its number recorded as $a$, then another ball is randomly drawn from the remaining three, its number recorded as $b$. Find the probability that the quadratic equation $x^... | {
"ground_truth": "\\frac {1}{4}",
"style": "rule"
} | {
"index": "38570",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\cos A= \\frac{\\sqrt{3}}{3}$, $c=\\sqrt{3}$, and $a=3\\sqrt{2}$. Find the value of $\\sin C$ and the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "38571",
"split": "train"
} |
math | [
{
"content": "What is the product of $\\frac{1}{5}$ and $\\frac{3}{7}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{35}",
"style": "rule"
} | {
"index": "38572",
"split": "train"
} |
math | [
{
"content": "Leticia has a $9\\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green... | {
"ground_truth": "486",
"style": "rule"
} | {
"index": "38573",
"split": "train"
} |
math | [
{
"content": "How many positive 3-digit numbers are divisible by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "38574",
"split": "train"
} |
math | [
{
"content": "Compute \n\\[\n\\sin^2 3^\\circ + \\sin^2 6^\\circ + \\sin^2 9^\\circ + \\dots + \\sin^2 177^\\circ.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "38575",
"split": "train"
} |
math | [
{
"content": "Given in the polar coordinate system, the equation of curve Ω is $\\rho=6\\cos\\theta$. Taking the pole as the origin of the Cartesian coordinate system, with the polar axis as the positive half-axis of the x-axis, and using the same unit of length in both coordinate systems, establish a Cartesian... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38576",
"split": "train"
} |
math | [
{
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(3, 2, 1), (3, 1, 2), (2, 3, 1), (2, 1, 3), (1, 3, 2), (1, 2, 3)",
"style": "rule"
} | {
"index": "38577",
"split": "train"
} |
math | [
{
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "38578",
"split": "train"
} |
math | [
{
"content": "Suppose \\(\\triangle A B C\\) has lengths \\(A B=5, B C=8\\), and \\(C A=7\\), and let \\(\\omega\\) be the circumcircle of \\(\\triangle A B C\\). Let \\(X\\) be the second intersection of the external angle bisector of \\(\\angle B\\) with \\(\\omega\\), and let \\(Y\\) be the foot of the perpe... | {
"ground_truth": "\\frac{13}{2}",
"style": "rule"
} | {
"index": "38579",
"split": "train"
} |
math | [
{
"content": "Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many differe... | {
"ground_truth": "592",
"style": "rule"
} | {
"index": "38580",
"split": "train"
} |
math | [
{
"content": "Given that quadrilateral \\(ABCD\\) is an isosceles trapezoid with \\(AB \\parallel CD\\), \\(AB = 6\\), and \\(CD = 16\\). Triangle \\(ACE\\) is a right-angled triangle with \\(\\angle AEC = 90^\\circ\\), and \\(CE = BC = AD\\). Find the length of \\(AE\\). Please output the final answer within \... | {
"ground_truth": "4\\sqrt{6}",
"style": "rule"
} | {
"index": "38581",
"split": "train"
} |
math | [
{
"content": "In a right triangle, medians are drawn from point $A$ to segment $\\overline{BC}$, which is the hypotenuse, and from point $B$ to segment $\\overline{AC}$. The lengths of these medians are 5 and $3\\sqrt{5}$ units, respectively. Calculate the length of segment $\\overline{AB}$. Please output the f... | {
"ground_truth": "2\\sqrt{14}",
"style": "rule"
} | {
"index": "38582",
"split": "train"
} |
math | [
{
"content": "John is thinking of a number. He gives the following 3 clues. ``My number has 125 as a factor. My number is a multiple of 30. My number is between 800 and 2000.'' What is John's number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1500",
"style": "rule"
} | {
"index": "38583",
"split": "train"
} |
math | [
{
"content": "We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{39}",
"style": "rule"
} | {
"index": "38584",
"split": "train"
} |
math | [
{
"content": "Find the radius of the circle with equation $9x^2-18x+9y^2+36y+44=0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38585",
"split": "train"
} |
math | [
{
"content": "The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the oth... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "38586",
"split": "train"
} |
math | [
{
"content": "The *cross* of a convex $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \\times 4$ rectangle is $\\sqrt{ \\dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \\dfrac{5}{3} \\sqrt{6}$ .\n\nSuppose $S$ is a dodecagon ( $12$ -gon... | {
"ground_truth": "\\frac{2\\sqrt{66}}{11}",
"style": "rule"
} | {
"index": "38587",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{17}",
"style": "rule"
} | {
"index": "38588",
"split": "train"
} |
math | [
{
"content": "Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38589",
"split": "train"
} |
math | [
{
"content": "In a certain ellipse, the center is at $(-3,1),$ one focus is at $(-3,0),$ and one endpoint of a semi-major axis is at $(-3,3).$ Find the semi-minor axis of the ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38590",
"split": "train"
} |
math | [
{
"content": "Roll three dice once each, and let event A be \"the three numbers are all different,\" and event B be \"at least one 1 is rolled.\" Then the conditional probabilities P(A|B) and P(B|A) are respectively ( ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38591",
"split": "train"
} |
math | [
{
"content": "What number is placed in the shaded circle if each of the numbers $1,5,6,7,13,14,17,22,26$ is placed in a different circle, the numbers 13 and 17 are placed as shown, and Jen calculates the average of the numbers in the first three circles, the average of the numbers in the middle three circles, a... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "38592",
"split": "train"
} |
math | [
{
"content": "The function \\( f(x) \\) has a domain of \\( \\mathbf{R} \\). For any \\( x \\in \\mathbf{R} \\) and \\( y \\neq 0 \\), \\( f(x+y)=f\\left(x y-\\frac{x}{y}\\right) \\), and \\( f(x) \\) is a periodic function. Find one of its positive periods. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38593",
"split": "train"
} |
math | [
{
"content": "The inverse of $f(x) = \\frac{3x - 2}{x + 4}$ may be written in the form $f^{-1}(x)=\\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are real numbers. Find $a/c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "38594",
"split": "train"
} |
math | [
{
"content": "A beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ according to the rule: ... | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "38595",
"split": "train"
} |
math | [
{
"content": "Let $a,b,c,d$ be real numbers such that $b-d \\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\boxed{16}",
"style": "rule"
} | {
"index": "38596",
"split": "train"
} |
math | [
{
"content": "Let $S$ be a subset of $\\{1,2,\\dots,2017\\}$ such that for any two distinct elements in $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "865",
"style": "rule"
} | {
"index": "38597",
"split": "train"
} |
math | [
{
"content": "How many regions of the plane are bounded by the graph of $$x^{6}-x^{5}+3 x^{4} y^{2}+10 x^{3} y^{2}+3 x^{2} y^{4}-5 x y^{4}+y^{6}=0 ?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "38598",
"split": "train"
} |
math | [
{
"content": "How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "38599",
"split": "train"
} |
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