data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2003",
"style": "rule"
} | {
"index": "35600",
"split": "train"
} |
math | [
{
"content": "While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{11}{16}",
"style": "rule"
} | {
"index": "35601",
"split": "train"
} |
math | [
{
"content": "The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks. \n\nHow ... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "35602",
"split": "train"
} |
math | [
{
"content": "For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "35603",
"split": "train"
} |
math | [
{
"content": "Compute the value of the expression:\n\\[ 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": "1022",
"style": "rule"
} | {
"index": "35604",
"split": "train"
} |
math | [
{
"content": "A four-digit palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $9$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "35605",
"split": "train"
} |
math | [
{
"content": "The value of the expression $\\frac{\\sin 10^{\\circ}}{1-\\sqrt{3}\\tan 10^{\\circ}}$ can be simplified using trigonometric identities and calculated exactly. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "35606",
"split": "train"
} |
math | [
{
"content": "The imaginary part of the complex number $z = -4i + 3$ is $-4i$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "35607",
"split": "train"
} |
math | [
{
"content": "What is the smallest square number whose first five digits are 4 and the sixth digit is 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "666667",
"style": "rule"
} | {
"index": "35608",
"split": "train"
} |
math | [
{
"content": "In a rectangle $P Q R S$ with $P Q=5$ and $Q R=3$, $P R$ is divided into three segments of equal length by points $T$ and $U$. What is the area of quadrilateral $S T Q U$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "35609",
"split": "train"
} |
math | [
{
"content": "If $200\\%$ of $x$ is equal to $50\\%$ of $y$, and $x = 16$, what is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "35610",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the angle bisectors $AA_{1}$ and $BB_{1}$ intersect at point $O$. Find the ratio $AA_{1} : OA_{1}$ given $AB=6, BC=5$, and $CA=4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 : 1",
"style": "rule"
} | {
"index": "35611",
"split": "train"
} |
math | [
{
"content": "Given six senior students (including 4 boys and 2 girls) are arranged to intern at three schools, A, B, and C, with two students at each school, and the two girls cannot be at the same school or at school C, and boy A cannot go to school A, calculate the total number of different arrangements. Ple... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35612",
"split": "train"
} |
math | [
{
"content": "Two distinct positive integers $a$ and $b$ are factors of 48. If $a\\cdot b$ is not a factor of 48, what is the smallest possible value of $a\\cdot b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "35613",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ have an angle of $30^{\\circ}$ between them, and $|\\overrightarrow{a}|=\\sqrt{3}$, $|\\overrightarrow{b}|=1$,\n\n$(1)$ Find the value of $|\\overrightarrow{a}-2\\overrightarrow{b}|$\n\n$(2)$ Let vector $\\overrightarrow{p}=\\overrighta... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35614",
"split": "train"
} |
math | [
{
"content": "Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\\{A,B\\}$ of subsets of $S$ such that $A \\cup B = S$, $A \\cap B = \\emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be th... | {
"ground_truth": "\\binom{n}{2} + 1",
"style": "rule"
} | {
"index": "35615",
"split": "train"
} |
math | [
{
"content": "Calculate: (1) $( \\frac {3}{2})^{-2}-(-4.5)^{0}-( \\frac {8}{27})^{ \\frac {2}{3}}$; \n(2) $\\frac {2}{3}\\text{lg}8+\\text{lg}25-3^{2\\log_{3}5}+16^{ \\frac {3}{4}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "35616",
"split": "train"
} |
math | [
{
"content": "In a regional frisbee league, teams have 7 members and each of the 5 teams takes turns hosting matches. At each match, each team selects three members of that team to be on the match committee, except the host team, which selects four members. How many possible 13-member match committees are there... | {
"ground_truth": "262,609,375",
"style": "rule"
} | {
"index": "35617",
"split": "train"
} |
math | [
{
"content": "Given the set \\( M = \\{1, 2, \\cdots, 2017\\} \\) which consists of the first 2017 positive integers, if one element is removed from \\( M \\) such that the sum of the remaining elements is a perfect square, what is the removed element? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "1677",
"style": "rule"
} | {
"index": "35618",
"split": "train"
} |
math | [
{
"content": "A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle. Please ou... | {
"ground_truth": "4:1",
"style": "rule"
} | {
"index": "35619",
"split": "train"
} |
math | [
{
"content": "When the two-digit integer $MM$, with equal digits, is multiplied by the one-digit integer $M$, the result is the three-digit integer $NPM$. What is the greatest possible value of $NPM$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "396",
"style": "rule"
} | {
"index": "35620",
"split": "train"
} |
math | [
{
"content": "The base of a pyramid is an equilateral triangle with a side length of 1. Out of the three vertex angles at the apex of the pyramid, two are right angles.\n\nFind the maximum volume of the pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "35621",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circu... | {
"ground_truth": "147/10",
"style": "rule"
} | {
"index": "35622",
"split": "train"
} |
math | [
{
"content": "How many distinct pairs of integers \\(x, y\\) are there between 1 and 1000 such that \\(x^{2} + y^{2}\\) is divisible by 49? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10153",
"style": "rule"
} | {
"index": "35623",
"split": "train"
} |
math | [
{
"content": "What is $1010101_2 + 111000_2$? Write your answer in base $10$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "141",
"style": "rule"
} | {
"index": "35624",
"split": "train"
} |
math | [
{
"content": "Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\\times 3 \\times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\\frac{p^a}{q^br^c},$ where $p,... | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "35625",
"split": "train"
} |
math | [
{
"content": "So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35626",
"split": "train"
} |
math | [
{
"content": "Two \\(1 \\times 1\\) squares are removed from a \\(5 \\times 5\\) grid as shown. Determine the total number of squares of various sizes on the grid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "35627",
"split": "train"
} |
math | [
{
"content": "Given that $θ$ is a real number, if the complex number $z=\\sin 2θ-1+i( \\sqrt {2}\\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35628",
"split": "train"
} |
math | [
{
"content": "The average of the data $x_1, x_2, \\ldots, x_8$ is 6, and the standard deviation is 2. Then, the average and the variance of the data $2x_1-6, 2x_2-6, \\ldots, 2x_8-6$ are Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "35629",
"split": "train"
} |
math | [
{
"content": "There are 8 Olympic volunteers, among them volunteers $A_{1}$, $A_{2}$, $A_{3}$ are proficient in Japanese, $B_{1}$, $B_{2}$, $B_{3}$ are proficient in Russian, and $C_{1}$, $C_{2}$ are proficient in Korean. One volunteer proficient in Japanese, Russian, and Korean is to be selected from them to f... | {
"ground_truth": "\\dfrac {5}{6}",
"style": "rule"
} | {
"index": "35630",
"split": "train"
} |
math | [
{
"content": "Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly one of the triplets is in the starting lineup? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1386",
"style": "rule"
} | {
"index": "35631",
"split": "train"
} |
math | [
{
"content": "Let P_{1}, P_{2}, \\ldots, P_{6} be points in the complex plane, which are also roots of the equation x^{6}+6 x^{3}-216=0. Given that P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} is a convex hexagon, determine the area of this hexagon. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "9 \\sqrt{3}",
"style": "rule"
} | {
"index": "35632",
"split": "train"
} |
math | [
{
"content": "In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated ... | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "35633",
"split": "train"
} |
math | [
{
"content": "Through two vertices of an equilateral triangle \\(ABC\\) with an area of \\(21 \\sqrt{3} \\ \\text{cm}^2\\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{7}",
"style": "rule"
} | {
"index": "35634",
"split": "train"
} |
math | [
{
"content": "The function \\( f \\) satisfies the equation \\((x-1) f(x) + f\\left(\\frac{1}{x}\\right) = \\frac{1}{x-1}\\) for each value of \\( x \\) not equal to 0 and 1. Find \\( f\\left(\\frac{2016}{2017}\\right) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "35635",
"split": "train"
} |
math | [
{
"content": "Given a unit square region $R$ and an integer $n \\geq 4$, determine how many points are $80$-ray partitional but not $50$-ray partitional. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7062",
"style": "rule"
} | {
"index": "35636",
"split": "train"
} |
math | [
{
"content": "In the tetrahedron \\( P-ABC \\), \\( \\triangle ABC \\) is an equilateral triangle with a side length of \\( 2\\sqrt{3} \\), \\( PB = PC = \\sqrt{5} \\), and the dihedral angle between \\( P-BC \\) and \\( BC-A \\) is \\( 45^\\circ \\). Find the surface area of the circumscribed sphere around the... | {
"ground_truth": "25\\pi",
"style": "rule"
} | {
"index": "35637",
"split": "train"
} |
math | [
{
"content": "A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining white squares in the row above. How ma... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "35638",
"split": "train"
} |
math | [
{
"content": "The circles whose equations are $x^2 + y^2 - 4x + 2y - 11 = 0$ and $x^2 + y^2 - 14x + 12y + 60 = 0$ intersect in the points $A$ and $B.$ Compute the slope of $\\overline{AB}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "35639",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=4\\sin({8x-\\frac{π}{9}})$, $x\\in \\left[0,+\\infty \\right)$, determine the initial phase of this harmonic motion. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\pi}{9}",
"style": "rule"
} | {
"index": "35640",
"split": "train"
} |
math | [
{
"content": "There is a stack of 200 cards, numbered from 1 to 200 from top to bottom. Starting with the top card, the following operations are performed in sequence: remove the top card, then place the next card at the bottom of the stack; remove the new top card, then place the next card at the bottom of the... | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "35641",
"split": "train"
} |
math | [
{
"content": "Tetrahedron $A B C D$ with volume 1 is inscribed in circumsphere $\\omega$ such that $A B=A C=A D=2$ and $B C \\cdot C D \\cdot D B=16$. Find the radius of $\\omega$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "35642",
"split": "train"
} |
math | [
{
"content": "\"Modulo $m$ graph paper\" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \\equiv 3y + 1 \\pmod{1... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "35643",
"split": "train"
} |
math | [
{
"content": "A stalker throws a small nut from the Earth's surface at an angle of $\\alpha=30^{\\circ}$ to the horizontal with an initial speed $v_{0}=10 \\, \\mathrm{m}/\\mathrm{s}$. The normal acceleration due to gravity is $g=10 \\, \\mathrm{m}/\\mathrm{s}^{2}$. At the highest point of its trajectory, the n... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "35644",
"split": "train"
} |
math | [
{
"content": "Parallelogram $ABCD$ has area $1,\\!000,\\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is ... | {
"ground_truth": "784",
"style": "rule"
} | {
"index": "35645",
"split": "train"
} |
math | [
{
"content": "Given the curve $x^{2}-y-2\\ln \\sqrt{x}=0$ and the line $4x+4y+1=0$, find the shortest distance from any point $P$ on the curve to the line. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\sqrt{2}(1+\\ln2)}{2}",
"style": "rule"
} | {
"index": "35646",
"split": "train"
} |
math | [
{
"content": "A public bus departs on schedule at 6:30, 7:00, and 7:30. Student Xiao Ming arrives at the station between 6:50 and 7:30 to catch the bus, and his time of arrival is random. The probability that his waiting time is no more than 10 minutes is ______. Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "35647",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\\left\\{\\begin{array}{l}x=\\cos\\beta\\\\ y=1+\\sin\\beta\\end{array}\\right.$ ($\\beta$ is the parameter). Establish a polar coordinate system with the coor... | {
"ground_truth": "2\\sqrt{5} + 2",
"style": "rule"
} | {
"index": "35648",
"split": "train"
} |
math | [
{
"content": "Given that $M(2,5)$ is the midpoint of $\\overline{AB}$ and $A(3,1)$ is one endpoint, what is the product of the coordinates of point $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "35649",
"split": "train"
} |
math | [
{
"content": "I am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \\le n \\le 25$, I take one step forward if $n$ is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to my original starting point. How many steps long is ... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "35650",
"split": "train"
} |
math | [
{
"content": "A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the s... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "35651",
"split": "train"
} |
math | [
{
"content": "Given the number 2550, calculate the sum of its prime factors. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "35652",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among\n\\[|r_1|, |r_2|, \\dots, |r_{2006}|.\\]What is the minimum number of real roots that $f(x)$ can have? Please output the final an... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "35653",
"split": "train"
} |
math | [
{
"content": "A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed \"9\" and you pressed [+1], it would display \"10.\" If you then pressed [x2], it would display \"20.\" Start... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "35654",
"split": "train"
} |
math | [
{
"content": "If the positive numbers $a$ and $b$ satisfy the equation $2+\\log_{2}a=3+\\log_{3}b=\\log_{6}(a+b)$, find the value of $\\frac{1}{a}+\\frac{1}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "35655",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4104",
"style": "rule"
} | {
"index": "35656",
"split": "train"
} |
math | [
{
"content": "Among the triangles with natural number side lengths, a perimeter not exceeding 100, and the difference between the longest and shortest sides not greater than 2, there are a total of different triangles that are not congruent to each other. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "190",
"style": "rule"
} | {
"index": "35657",
"split": "train"
} |
math | [
{
"content": "Given $x > 0$, $y > 0$, and the inequality $2\\log_{\\frac{1}{2}}[(a-1)x+ay] \\leq 1 + \\log_{\\frac{1}{2}}(xy)$ always holds, find the minimum value of $4a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}+\\sqrt{2}",
"style": "rule"
} | {
"index": "35658",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the coordinates of points $E'$ and $F'$ are $(0, \\sqrt{3})$ and $(0, -\\sqrt{3})$, respectively. A moving point $G$ satisfies: the product of the slopes of line $E'G$ and line $F'G$ is $-\\frac{3}{4}$. \n(1) Find the equation of the trajectory of the mov... | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "35659",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, if $\\sin^{2}A + \\sin^{2}B = \\sin^{2}C - \\sqrt{2}\\sin A\\sin B$, find the maximum value of $\\sin 2A\\tan^{2}B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | {
"index": "35660",
"split": "train"
} |
math | [
{
"content": "Determine the product of the digits $X$ and $Y$ such that the number $590\\,X34\\,52Y$ is divisible by $45$. What is the sum of all distinct possible products $XY$, with each product counted only once regardless of the pair ($X$, $Y$) variations that produce it? Please output the final answer with... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35661",
"split": "train"
} |
math | [
{
"content": "The vertices of an equilateral triangle lie on the hyperbola \\( xy = 3 \\). The centroid of this triangle is at the origin, \\( (0,0) \\). What is the square of the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "35662",
"split": "train"
} |
math | [
{
"content": "Let $p = 2027$ be the smallest prime greater than $2018$ , and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let $\\mathrm{GF}(p)$ be the integers modulo $p$ , and let $\\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\\mathrm{GF}(p)$ (so th... | {
"ground_truth": "4114810",
"style": "rule"
} | {
"index": "35663",
"split": "train"
} |
math | [
{
"content": "Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$ . Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$ . Let $DE$ be the diameter through $D$ . If $[XYZ]$ denotes the area of the triangle $XYZ$ , find... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "35664",
"split": "train"
} |
math | [
{
"content": "The line $y = -\\frac{2}{3}x + 6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\\triangle POQ$ is four times the area of $\\triangle TOP$, then what is the value of $r+s$? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "8.25",
"style": "rule"
} | {
"index": "35665",
"split": "train"
} |
math | [
{
"content": "Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\\%$ of the difference between $A$ and the cost ... | {
"ground_truth": "23\\%",
"style": "rule"
} | {
"index": "35666",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) and \\( y \\) be real numbers such that\n\\[\n 1 < \\frac{x - y}{x + y} < 3.\n\\]\nIf \\( \\frac{x}{y} \\) is an integer, what is its value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35667",
"split": "train"
} |
math | [
{
"content": "A number's cube is a five-digit number; the first digit is 4, and the fifth digit is 6. What is this number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35668",
"split": "train"
} |
math | [
{
"content": "How many integers $m$ are there from 1 to 1996, such that $\\frac{m^{2}+7}{m+4}$ is not a reduced fraction? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "35669",
"split": "train"
} |
math | [
{
"content": "Rain drops fall vertically at a speed of $v = 2 \\, \\text{m/s}$. The rear window of a car is inclined at an angle of $\\alpha = 60^{\\circ}$ to the horizontal. At what speed $u$ must the car travel on a horizontal, flat road so that its rear window remains dry? Please output the final answer with... | {
"ground_truth": "1.15",
"style": "rule"
} | {
"index": "35670",
"split": "train"
} |
math | [
{
"content": "Calculate the number of four-digit numbers without repeating digits that can be formed by taking any two odd numbers and two even numbers from the six digits 0, 1, 2, 3, 4, 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "35671",
"split": "train"
} |
math | [
{
"content": "How many interior diagonals does an icosahedron have? (An $\\emph{icosahedron}$ is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex. An $\\emph{interior}$ diagonal is a segment connecting two vertices which do not lie on a common face.) Please ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35672",
"split": "train"
} |
math | [
{
"content": "Nathan will roll two six-sided dice. What is the probability that he will roll a number less than three on the first die and a number greater than three on the second die? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "35673",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $\\sin A = \\frac{3}{5}$ and $\\cos B = \\frac{5}{13}.$ Find $\\cos C.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{65}",
"style": "rule"
} | {
"index": "35674",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = \\frac{x^6 - 2}{4}$. Find $f^{-1}\\left(-\\frac{1}{8}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\frac{3}{2}\\right)^{\\frac{1}{6}}",
"style": "rule"
} | {
"index": "35675",
"split": "train"
} |
math | [
{
"content": "The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35676",
"split": "train"
} |
math | [
{
"content": "If\n\\[\n\\begin{pmatrix}\n1 & 3 & b \\\\\n0 & 1 & 5 \\\\\n0 & 0 & 1\n\\end{pmatrix}^m = \\begin{pmatrix}\n1 & 33 & 6006 \\\\\n0 & 1 & 55 \\\\\n0 & 0 & 1\n\\end{pmatrix},\n\\]\nthen find $b + m.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "35677",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "35678",
"split": "train"
} |
math | [
{
"content": "Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time? Please output the ... | {
"ground_truth": "8.4",
"style": "rule"
} | {
"index": "35679",
"split": "train"
} |
math | [
{
"content": "An investor placed $\\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\\$15,735$. Determine the ... | {
"ground_truth": "11.36",
"style": "rule"
} | {
"index": "35680",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the square one of whose diagonals has endpoints $(1/10,7/10)$ and $(-1/10,-7/10)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0 \\le x \\le 2012$ and $0\\le y\\le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. Wh... | {
"ground_truth": "\\frac{4}{25}",
"style": "rule"
} | {
"index": "35681",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $x$ when the arithmetic mean of the following five expressions is 30: $$x + 10 \\hspace{.5cm} 3x - 5 \\hspace{.5cm} 2x \\hspace{.5cm} 18 \\hspace{.5cm} 2x + 6$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15.125",
"style": "rule"
} | {
"index": "35682",
"split": "train"
} |
math | [
{
"content": "Express the following sum as a simple fraction in lowest terms.\n\n$$\\frac{1}{1\\times2} + \\frac{1}{2\\times3} + \\frac{1}{3\\times4} + \\frac{1}{4\\times5} + \\frac{1}{5\\times6}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "35683",
"split": "train"
} |
math | [
{
"content": "For some positive integer $k$, when 60 is divided by $k^2$, the remainder is 6. What is the remainder when 100 is divided by $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "35684",
"split": "train"
} |
math | [
{
"content": "Find the product of the three smallest prime factors of 180. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "35685",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $XY = 4$, $XZ = 3$, and $YZ = 5$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{47}{15}",
"style": "rule"
} | {
"index": "35686",
"split": "train"
} |
math | [
{
"content": "Jaclyn, Marcelle, Susanna and Wanda are tutors in their school math lab. Their schedule is as follows: Jaclyn works every third school day, Marcelle works every fourth school day, Susanna works every sixth school day, and Wanda works every seventh school day. Today, they are all working in the mat... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "35687",
"split": "train"
} |
math | [
{
"content": "In an equilateral triangle $ABC$ with side length of 10, a similar process of division by midpoints and shading of one of these triangles occurs. If this dividing and shading process is repeated indefinitely, and the first triangle to be shaded is the triangle involving vertex $A$, the total shade... | {
"ground_truth": "\\frac{25\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "35688",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer divisible by $10$, $11$, and $12$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "660",
"style": "rule"
} | {
"index": "35689",
"split": "train"
} |
math | [
{
"content": "Factor $t^2-49$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(t-7)(t+7)",
"style": "rule"
} | {
"index": "35690",
"split": "train"
} |
math | [
{
"content": "For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ? Please output the final answer within \\b... | {
"ground_truth": "\\phi=\\frac{1+\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "35691",
"split": "train"
} |
math | [
{
"content": "Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Define $A = \\left[\\frac{7}{8}\\right] + \\left[\\frac{7^2}{8}\\right] + \\cdots + \\left[\\frac{7^{2016}}{8}\\right]$. Find the remainder when $A$ is divided by 50. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "35692",
"split": "train"
} |
math | [
{
"content": "Given two lines $l_1: y = m$ and $l_2: y = \\frac{8}{2m+1}$ ($m > 0$), line $l_1$ intersects the graph of the function $y = |\\log_2 x|$ from left to right at points $A$ and $B$, and line $l_2$ intersects the graph of the function $y = |\\log_2 x|$ from left to right at points $C$ and $D$. The len... | {
"ground_truth": "8\\sqrt{2}",
"style": "rule"
} | {
"index": "35693",
"split": "train"
} |
math | [
{
"content": "If the number of zeros in the integer equal to $(10^{100}) \times (100^{10})$ is sought, what is this number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "35694",
"split": "train"
} |
math | [
{
"content": "Arrange 2002 students numbered from 1 to 2002 in a row from left to right. Counting from left to right from 1 to 11, the student who is counted as 11 remains in place, while the others leave the row. Then, the remaining students count from left to right from 1 to 11 again, and the student who is c... | {
"ground_truth": "1331",
"style": "rule"
} | {
"index": "35695",
"split": "train"
} |
math | [
{
"content": "Consider the multiplication of the two numbers $1,002,000,000,000,000,000$ and $999,999,999,999,999,999$. Calculate the number of digits in the product of these two numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "35696",
"split": "train"
} |
math | [
{
"content": "Solve the equation: $x^{2}+5x-1=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-5-\\sqrt{29}}{2}",
"style": "rule"
} | {
"index": "35697",
"split": "train"
} |
math | [
{
"content": "The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 65^\\circ$ and $\\angle QRC = 30^\\circ$, find $\\angle QPR$, in degrees.\n\n[asy]\nunitsize(1.0 cm);\n\npair Q, P, R, C;\n\nQ = (2.43,3.46);\nP = (0,0);\nR = (4.43,0);\nC = incenter(Q,P,R);\n\n... | {
"ground_truth": "55^\\circ",
"style": "rule"
} | {
"index": "35698",
"split": "train"
} |
math | [
{
"content": "Given that the sine and cosine values of angle $α$ are both negative, and $\\cos(75^{\\circ}+α)=\\frac{1}{3}$, find the value of $\\cos(105^{\\circ}-α)+\\sin(α-105^{\\circ})$ = \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2}-1}{3}",
"style": "rule"
} | {
"index": "35699",
"split": "train"
} |
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