data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In $\\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\\overline{PQ},$ $\\overline{QR},$ and $\\overline{PR},$ respectively, such that $\\overline{MN}$ and $\\overline{NO}$ are parallel to $\\overline{PR}$ and $\\overline{PQ},$ respectively. What is the perimete... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "3100",
"split": "train"
} |
math | [
{
"content": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "100!",
"style": "rule"
} | {
"index": "3101",
"split": "train"
} |
math | [
{
"content": "What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3102",
"split": "train"
} |
math | [
{
"content": "Compute the largest integer $k$ such that $2004^k$ divides $2004!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3103",
"split": "train"
} |
math | [
{
"content": "Determine all functions $f : \\mathbb{R}^2 \\to\\mathbb {R}$ for which \\[f(A)+f(B)+f(C)+f(D)=0,\\]whenever $A,B,C,D$ are the vertices of a square with side-length one. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$f(x)=0$",
"style": "rule"
} | {
"index": "3104",
"split": "train"
} |
math | [
{
"content": "A geometric progression with 10 terms starts with the first term as 2 and has a common ratio of 3. Calculate the sum of the new geometric progression formed by taking the reciprocal of each term in the original progression.\nA) $\\frac{29523}{59049}$\nB) $\\frac{29524}{59049}$\nC) $\\frac{29525}{5... | {
"ground_truth": "\\frac{29524}{59049}",
"style": "rule"
} | {
"index": "3105",
"split": "train"
} |
math | [
{
"content": "The isosceles triangle and the square shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the side length of the square, $s$?\n\n[asy]\ndraw((0,0)--(0,10)--(10,10)--(10,0)--cycle);\nfill((0,0)--(17,5)--(0,10)--cycle,white);\ndraw((0,0)--(17,5)--(0,10... | {
"ground_truth": "2s",
"style": "rule"
} | {
"index": "3106",
"split": "train"
} |
math | [
{
"content": "Compute the sum of the geometric series $-3 + 6 - 12 + 24 - \\cdots - 768$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "514",
"style": "rule"
} | {
"index": "3107",
"split": "train"
} |
math | [
{
"content": "What is the greatest prime factor of $15! + 18!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4897",
"style": "rule"
} | {
"index": "3108",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point in coordinate space, where all the coordinates of $P$ are positive. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\\alpha,$ $\\beta,$ and $\\gamma,$ respectively. If $\\cos \\alpha = \\frac{1}{3}$ and $\\cos \\... | {
"ground_truth": "\\frac{\\sqrt{191}}{15}",
"style": "rule"
} | {
"index": "3109",
"split": "train"
} |
math | [
{
"content": "A survey of participants was conducted at the Olympiad. $50\\%$ of the participants liked the first round, $60\\%$ of the participants liked the second round, $70\\%$ of the participants liked the opening of the Olympiad. It is known that each participant liked either one option or all three... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "3110",
"split": "train"
} |
math | [
{
"content": "James writes down three integers. Alex picks some two of those integers, takes the average of them, and adds the result to the third integer. If the possible final results Alex could get are 42, 13, and 37, what are the three integers James originally chose? Please output the final answer within \... | {
"ground_truth": "-20, 28, 38",
"style": "rule"
} | {
"index": "3111",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is an even function defined on $\\mathbb{R}$, and $g(x)$ is an odd function defined on $\\mathbb{R}$ that passes through the point $(-1, 3)$ and $g(x) = f(x-1)$, find the value of $f(2007) + f(2008)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "3112",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the side \\(BC\\) is 19 cm. The perpendicular \\(DF\\), drawn to side \\(AB\\) through its midpoint \\(D\\), intersects side \\(BC\\) at point \\(F\\). Find the perimeter of triangle \\(AFC\\) if side \\(AC\\) is 10 cm. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "3113",
"split": "train"
} |
math | [
{
"content": "A school arranges for five people, \\( A \\), \\( B \\), \\( C \\), \\( D \\), and \\( E \\), to enter into three classes, with each class having at least one person, and \\( A \\) and \\( B \\) cannot be in the same class. Calculate the total number of different arrangements. Please output the fi... | {
"ground_truth": "114",
"style": "rule"
} | {
"index": "3114",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be an acute triangle with orthocenter $H$. Let $D, E$ be the feet of the $A, B$-altitudes respectively. Given that $A H=20$ and $H D=15$ and $B E=56$, find the length of $B H$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "3115",
"split": "train"
} |
math | [
{
"content": "In the \"Nine Chapters on the Mathematical Art,\" a tetrahedron with all four faces being right-angled triangles is referred to as a \"turtle's knee.\" Given that in the turtle's knee $M-ABC$, $MA \\perp$ plane $ABC$, and $MA=AB=BC=2$, the sum of the surface areas of the circumscribed sphere and t... | {
"ground_truth": "24\\pi-8\\sqrt{2}\\pi",
"style": "rule"
} | {
"index": "3116",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin(\\omega x+\\varphi)$ is monotonically increasing on the interval ($\\frac{π}{6}$,$\\frac{{2π}}{3}$), and the lines $x=\\frac{π}{6}$ and $x=\\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, evaluate $f(-\\frac{{5π}}{{12}})$. Please out... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "3117",
"split": "train"
} |
math | [
{
"content": "We randomly throw a 10-filér coin onto a $40 \\times 40$ cm chessboard. What is the probability that the coin:\n\na) will be entirely inside one square?\n\nb) will partially cover one and only one square edge?\n\nc) will partially cover two square edges but not a vertex?\n\nd) will cover a vertex?... | {
"ground_truth": "0.1134",
"style": "rule"
} | {
"index": "3118",
"split": "train"
} |
math | [
{
"content": "Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "3119",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $a=2$, $b=3$, $\\cos C=\\frac{1}{3}$. The radius of the circumcircle is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9 \\sqrt{2}}{8}",
"style": "rule"
} | {
"index": "3120",
"split": "train"
} |
math | [
{
"content": "Given $\\sin(3\\pi + \\alpha) = -\\frac{1}{2}$, find the value of $\\cos\\left(\\frac{7\\pi}{2} - \\alpha\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "3121",
"split": "train"
} |
math | [
{
"content": "Given that positive real numbers $x$ and $y$ satisfy $4x+3y=4$, find the minimum value of $\\frac{1}{2x+1}+\\frac{1}{3y+2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}+\\frac{\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "3122",
"split": "train"
} |
math | [
{
"content": "In a park, 10,000 trees are planted in a square grid pattern (100 rows of 100 trees). What is the maximum number of trees that can be cut down such that if one stands on any stump, no other stumps are visible? (Trees can be considered thin enough for this condition.) Please output the final answer... | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "3123",
"split": "train"
} |
math | [
{
"content": "The infinite sequence $S=\\{s_1,s_2,s_3,\\ldots\\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "3124",
"split": "train"
} |
math | [
{
"content": "There are 30 people in a room, 60\\% of whom are men. If no men enter or leave the room, how many women must enter the room so that 40\\% of the total number of people in the room are men? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "3125",
"split": "train"
} |
math | [
{
"content": "Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\\frac{2}{3}$ of its games and team $B$ has won $\\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "3126",
"split": "train"
} |
math | [
{
"content": "Given a square grid of size $2023 \\times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the ... | {
"ground_truth": "338",
"style": "rule"
} | {
"index": "3127",
"split": "train"
} |
math | [
{
"content": "Given a convex quadrilateral \\( ABCD \\) with an interior point \\( P \\) such that \\( P \\) divides \\( ABCD \\) into four triangles \\( ABP, BCP, CDP, \\) and \\( DAP \\). Let \\( G_1, G_2, G_3, \\) and \\( G_4 \\) denote the centroids of these triangles, respectively. Determine the ratio of t... | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "3128",
"split": "train"
} |
math | [
{
"content": "Find all positive integers \\( n \\) such that \\( n \\) is equal to 100 times the number of positive divisors of \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": "3129",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface are... | {
"ground_truth": "40 \\sqrt{1925}",
"style": "rule"
} | {
"index": "3130",
"split": "train"
} |
math | [
{
"content": "If the area of $\\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\\sin A$ is equal to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "3131",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3132",
"split": "train"
} |
math | [
{
"content": "The three vertices of an inscribed triangle in a circle divide the circumference into three arcs of lengths $3$, $4$, and $5$. The area of this triangle is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{9}{\\pi^{2}}(\\sqrt{3}+3)$",
"style": "rule"
} | {
"index": "3133",
"split": "train"
} |
math | [
{
"content": "I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all $n$ ... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "3134",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lfloor (12.1)^2 \\rfloor - \\lfloor 12.1 \\rfloor \\cdot \\lfloor 12.1 \\rfloor .$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "3135",
"split": "train"
} |
math | [
{
"content": "If $\\overline{AD} \\| \\overline{FG}$, how many degrees are in angle $EFG$?\n\n[asy]\nimport olympiad;\n\npair A = (-15,20);\npair B = (-12,35);\npair C = (35,50);\npair D = (35,20);\npair E = (14,20);\npair F = (0,0);\npair G = (40,0);\n\ndraw(F--G);\ndraw(F--C);\ndraw(A--D);\ndraw(B--E);\n\nlab... | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "3136",
"split": "train"
} |
math | [
{
"content": "Given the following conditions:①$\\left(2b-c\\right)\\cos A=a\\cos C$, ②$a\\sin\\ \\ B=\\sqrt{3}b\\cos A$, ③$a\\cos C+\\sqrt{3}c\\sin A=b+c$, choose one of these three conditions and complete the solution below.<br/>Question: In triangle $\\triangle ABC$, with sides $a$, $b$, and $c$ opposite to a... | {
"ground_truth": "\\frac{7\\sqrt{481}}{481}",
"style": "rule"
} | {
"index": "3137",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}=(1+\\cos ^{2} \\frac {nπ}{2})a_{n}+\\sin ^{2} \\frac {nπ}{2}$, find the sum of the first 12 terms of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "3138",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac{\\cos \\alpha}{1 + \\sin \\alpha} = \\sqrt{3}$, find the value of $\\frac{\\cos \\alpha}{\\sin \\alpha - 1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "3139",
"split": "train"
} |
math | [
{
"content": "In the expansion of $\\left(\\frac{1}{ \\sqrt[3]{x}} +2x \\sqrt {x}\\right)^{7}$, the coefficient of $x^{5}$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "560",
"style": "rule"
} | {
"index": "3140",
"split": "train"
} |
math | [
{
"content": "Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to asse... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "3141",
"split": "train"
} |
math | [
{
"content": "The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24.75",
"style": "rule"
} | {
"index": "3142",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system xOy, point P(x0, y0) is on the unit circle O. Suppose the angle ∠xOP = α, and if α ∈ (π/3, 5π/6), and sin(α + π/6) = 3/5, determine the value of x0. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3-4\\sqrt{3}}{10}",
"style": "rule"
} | {
"index": "3143",
"split": "train"
} |
math | [
{
"content": "In a $3 \\times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3144",
"split": "train"
} |
math | [
{
"content": "Provided $x$ is a multiple of $27720$, determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+7)(3x+8)$ and $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "3145",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $T_n$. It is known that $a_1=2$, $3S_n=(n+2)a_n$, and $a_nb_n=\\frac{1}{2}$. If $\\lambda > T_n$ holds true for any $n\\in\\mathbb{N}^*$, find the minimum value... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3146",
"split": "train"
} |
math | [
{
"content": "Ava's machine takes four-digit positive integers as input. When the four-digit integer $ABCD$ is input, the machine outputs the integer $A \times B + C \times D$. What is the output when the input is 2023? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3147",
"split": "train"
} |
math | [
{
"content": "Two ants crawled along their own closed routes on a $7 \\times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ant crawled a... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3148",
"split": "train"
} |
math | [
{
"content": "Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute ... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "3149",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer \\( n \\) that satisfies the following condition: If the set \\( A = \\{1, 2, 3, \\ldots, n\\} \\) is arbitrarily divided into 63 pairwise disjoint subsets \\( A_1, A_2, A_3, \\ldots, A_{63} \\) (which are non-empty and their union is the set \\( A \\)), then the... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "3150",
"split": "train"
} |
math | [
{
"content": "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "3151",
"split": "train"
} |
math | [
{
"content": "If two circles $(x-m)^2+y^2=4$ and $(x+1)^2+(y-2m)^2=9$ are tangent internally, then the real number $m=$ ______ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | {
"index": "3152",
"split": "train"
} |
math | [
{
"content": "Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "3153",
"split": "train"
} |
math | [
{
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2014} \\) be a permutation of the positive integers \\( 1, 2, \\cdots, 2014 \\). Define\n\\[ S_{k} = a_{1} + a_{2} + \\cdots + a_{k} \\quad (k=1, 2, \\cdots, 2014). \\]\n\nWhat is the maximum number of odd numbers among \\( S_{1}, S_{2}, \\cdots, S_{2014} \\)? Ple... | {
"ground_truth": "1511",
"style": "rule"
} | {
"index": "3154",
"split": "train"
} |
math | [
{
"content": "Let $A, B, C$, and $D$ be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines \\overline{A B}, \\overline{A C}, \\overline{A D}, \\overline{B C}, \\overline{B D}$, and \\overline{C D}$ all have positive slope? Please output the fi... | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"index": "3155",
"split": "train"
} |
math | [
{
"content": "Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "280",
"style": "rule"
} | {
"index": "3156",
"split": "train"
} |
math | [
{
"content": "In a survey of $150$ employees at a tech company, it is found that:\n- $90$ employees are working on project A.\n- $50$ employees are working on project B.\n- $30$ employees are working on both project A and B.\n\nDetermine what percent of the employees surveyed are not working on either project. ... | {
"ground_truth": "26.67\\%",
"style": "rule"
} | {
"index": "3157",
"split": "train"
} |
math | [
{
"content": "Find the sum of all real solutions to the equation \\[\\sqrt{x} + \\sqrt{\\frac{4}{x}} + \\sqrt{x + \\frac{4}{x}} = 6.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{64}{9}",
"style": "rule"
} | {
"index": "3158",
"split": "train"
} |
math | [
{
"content": "Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "430",
"style": "rule"
} | {
"index": "3159",
"split": "train"
} |
math | [
{
"content": "Evaluate the integral $\\int_{2}^{3}{\\frac{x-2}{\\left( x-1 \\right)\\left( x-4 \\right)}dx}=$\n\nA) $-\\frac{1}{3}\\ln 2$\nB) $\\frac{1}{3}\\ln 2$\nC) $-\\ln 2$\nD) $\\ln 2$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{3}\\ln 2",
"style": "rule"
} | {
"index": "3160",
"split": "train"
} |
math | [
{
"content": "The ratio of girls to boys in Ms. Snow's math class is 3:2. If there is a total of 45 students, how many girls are in Ms. Snow's math class? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "3161",
"split": "train"
} |
math | [
{
"content": "Given the polar equation of curve C is $\\rho - 6\\cos\\theta + 2\\sin\\theta + \\frac{1}{\\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through poi... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "3162",
"split": "train"
} |
math | [
{
"content": "Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "3163",
"split": "train"
} |
math | [
{
"content": "If $a$, $b\\in \\{-1,1,2\\}$, then the probability that the function $f\\left(x\\right)=ax^{2}+2x+b$ has a zero point is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "3164",
"split": "train"
} |
math | [
{
"content": "Let \\( M = \\{1, 2, \\cdots, 1995\\} \\). Suppose \\( A \\) is a subset of \\( M \\) that satisfies the condition: if \\( x \\in A \\), then \\( 15x \\notin A \\). What is the maximum number of elements in \\( A \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1870",
"style": "rule"
} | {
"index": "3165",
"split": "train"
} |
math | [
{
"content": "Using systematic sampling, extract a sample of size 12 from a population of 123 individuals. The sampling interval is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3166",
"split": "train"
} |
math | [
{
"content": "Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height, dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$.\n\n$A_{1}(2, -1, -2)$\n\n$A_{2}(1, 2, 1)$\n\n$A_{3}(5, 0, -6)$\n\n$A_{4}(-10, 9, -7)$ Please output the final answer within \\... | {
"ground_truth": "4\\sqrt{14}",
"style": "rule"
} | {
"index": "3167",
"split": "train"
} |
math | [
{
"content": "If $3015x + 3020y = 3025$ and $3018x + 3024y = 3030$, what is the value of $x - y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11.1167",
"style": "rule"
} | {
"index": "3168",
"split": "train"
} |
math | [
{
"content": "The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3169",
"split": "train"
} |
math | [
{
"content": "For what values of $z$ is $z^2-40z+340\\le 4$? Express your answer in interval notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "[12,28]",
"style": "rule"
} | {
"index": "3170",
"split": "train"
} |
math | [
{
"content": "How many positive integers \\( n \\) exist such that both \\(\\frac{n+1}{3}\\) and \\(3n+1\\) are three-digit integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3171",
"split": "train"
} |
math | [
{
"content": "In 2010, the sum of the year's digits is $3$. What is the first year after 2010 in which the sum of the digits is 3 again? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2100",
"style": "rule"
} | {
"index": "3172",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=|2x+1|-|x|-2$.\n(1) Solve the inequality $f(x)\\geqslant 0$;\n(2) If there exists a real number $x$ such that $f(x)-a\\leqslant |x|$, find the minimum value of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "3173",
"split": "train"
} |
math | [
{
"content": "The quadratic equation $ax^2+8x+c=0$ has exactly one solution. If $a+c=10$, and $a<c$ find the ordered pair $(a,c)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,8)",
"style": "rule"
} | {
"index": "3174",
"split": "train"
} |
math | [
{
"content": "How many integers satisfy $-4 \\le 2x+1 \\le 6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "3175",
"split": "train"
} |
math | [
{
"content": "On the side \\( AB \\) of the parallelogram \\( ABCD \\), a point \\( F \\) is chosen, and on the extension of the side \\( BC \\) beyond the vertex \\( B \\), a point \\( H \\) is taken such that \\( \\frac{AB}{BF} = \\frac{BC}{BH} = 5 \\). The point \\( G \\) is chosen such that \\( BFGH \\) for... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "3176",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x)=\\cos x + m \\left(x+\\frac{\\pi}{2}\\right) \\sin x \\) where \\( m \\leqslant 1 \\):\n\n(1) Discuss the number of zeros of \\( f(x) \\) in the interval \\( (-\\pi, 0) \\).\n\n(2) If there exists \\( t > 0 \\) such that \\( |f(x)| < -2x - \\pi \\) holds for \\( x \\in ... | {
"ground_truth": "m = -1",
"style": "rule"
} | {
"index": "3177",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line $\\sqrt{3}x+y-1=0$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "3178",
"split": "train"
} |
math | [
{
"content": "How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\\{4, 7, 10, 13, \\ldots, 46\\}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "3179",
"split": "train"
} |
math | [
{
"content": "In the diagram, $QRS$ is a straight line. What is the measure of $\\angle RPS,$ in degrees? [asy]\npair Q=(0,0);\npair R=(1.3,0);\npair SS=(2.3,0);\npair P=(.8,1);\n\ndraw(P--Q--R--SS--P--R);\n\nlabel(\"$Q$\",Q,S);\nlabel(\"$R$\",R,S);\nlabel(\"$S$\",SS,S);\nlabel(\"$P$\",P,N);\n\nlabel(\"$48^\\ci... | {
"ground_truth": "27^\\circ",
"style": "rule"
} | {
"index": "3180",
"split": "train"
} |
math | [
{
"content": "Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "3181",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is the circumcenter of $\\triangle ABC$, and $\\overrightarrow{PA} + \\overrightarrow{PB} + \\lambda \\overrightarrow{PC} = 0$, $\\angle C = 120^{\\circ}$, determine the value of the real number $\\lambda$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "3182",
"split": "train"
} |
math | [
{
"content": "Denote $\\phi=\\frac{1+\\sqrt{5}}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a \"base-$\\phi$ \" value $p(S)$. For example, $p(1101)=\\phi^{3}+\\phi^{2}+1$. For any positive integer $n$, let $f(n)$ be the number of such strings $S$ that satisf... | {
"ground_truth": "\\frac{25+3 \\sqrt{69}}{2}",
"style": "rule"
} | {
"index": "3183",
"split": "train"
} |
math | [
{
"content": "The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "3184",
"split": "train"
} |
math | [
{
"content": "Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as... | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "3185",
"split": "train"
} |
math | [
{
"content": "Let \\( x_{1}, x_{2}, x_{3}, x_{4} \\) be non-negative real numbers satisfying the equation:\n\\[ \nx_{1} + x_{2} + x_{3} + x_{4} = 1 \n\\]\nFind the maximum value of \\( \\sum_{1 \\leq i < j \\leq 4}(x_{i} + x_{j}) \\sqrt{x_{i} x_{j}} \\), and determine the values of \\( x_{1}, x_{2}, x_{3}, x_{4... | {
"ground_truth": "3/4",
"style": "rule"
} | {
"index": "3186",
"split": "train"
} |
math | [
{
"content": "Find the minimum angle formed by any triple among five points on the plane such that the minimum angle is greater than or equal to $36^{\\circ}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ 36^{\\circ} \\]",
"style": "rule"
} | {
"index": "3187",
"split": "train"
} |
math | [
{
"content": "A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops once the dot returns to... | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "3188",
"split": "train"
} |
math | [
{
"content": "Chris received a mark of $50 \\%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \\%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test? Please output the... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3189",
"split": "train"
} |
math | [
{
"content": "The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. After a 15-minute st... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "3190",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "3191",
"split": "train"
} |
math | [
{
"content": "If the product $\\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\dfrac{7}{6}\\cdot \\ldots\\cdot \\dfrac{c}{d} = 12$, determine the sum of $c$ and $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "3192",
"split": "train"
} |
math | [
{
"content": "Calculate $7 \\cdot 9\\frac{2}{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65\\frac{4}{5}",
"style": "rule"
} | {
"index": "3193",
"split": "train"
} |
math | [
{
"content": "Dana goes first and Carl's coin lands heads with probability $\\frac{2}{7}$, and Dana's coin lands heads with probability $\\frac{3}{8}$. Find the probability that Carl wins the game. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{31}",
"style": "rule"
} | {
"index": "3194",
"split": "train"
} |
math | [
{
"content": "Express the value of the following as a common fraction:\n$\\left(1-\\frac12\\right)\\cdot\\left(1-\\frac13\\right)\\cdot\\left(1-\\frac14\\right) \\dotsm \\left(1-\\frac1{n+1}\\right) \\dotsm \\left(1-\\frac1{100}\\right)$ Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{1}{100}",
"style": "rule"
} | {
"index": "3195",
"split": "train"
} |
math | [
{
"content": "A pair of fair 8-sided dice are rolled, yielding numbers $a$ and $b$. Determine the probability that both digits $a$ and $b$ as well as the two-digit number $ab$ formed by them are all divisible by 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "3196",
"split": "train"
} |
math | [
{
"content": "Two players play alternately on a $ 5 \\times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \\times 3$ squares is calculated and the first ... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3197",
"split": "train"
} |
math | [
{
"content": "What is the greatest 4-digit base 7 positive integer that is divisible by 7? (Express your answer in base 7.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6660_7",
"style": "rule"
} | {
"index": "3198",
"split": "train"
} |
math | [
{
"content": "There are two types of products, A and B, with profits of $p$ ten-thousand yuan and $q$ ten-thousand yuan, respectively. Their relationship with the invested capital $x$ ten-thousand yuan is: $p= \\frac{1}{5}x$, $q= \\frac{3}{5} \\sqrt{x}$. Now, with an investment of 3 ten-thousand yuan in managin... | {
"ground_truth": "\\frac{21}{20}",
"style": "rule"
} | {
"index": "3199",
"split": "train"
} |
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