data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of perfor... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31200",
"split": "train"
} |
math | [
{
"content": "Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the perpendiculars $C$ a... | {
"ground_truth": "180\\sqrt{2}",
"style": "rule"
} | {
"index": "31201",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "464",
"style": "rule"
} | {
"index": "31202",
"split": "train"
} |
math | [
{
"content": "Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31203",
"split": "train"
} |
math | [
{
"content": "If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \\neq x_2$, the inequality $\\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an \"ideal function.\" Among ... | {
"ground_truth": "(3)",
"style": "rule"
} | {
"index": "31204",
"split": "train"
} |
math | [
{
"content": "Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \\le 100$ for all $m$ such that $\\frac{1}{2} < m < a$, find the maximum possible value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{50}{99}",
"style": "rule"
} | {
"index": "31205",
"split": "train"
} |
math | [
{
"content": "Given a regular tetrahedron with four vertices \\(A, B, C, D\\) and each edge of length 1 meter, a bug starts at point \\(A\\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability and crawls to the oth... | {
"ground_truth": "\\frac{182}{729}",
"style": "rule"
} | {
"index": "31206",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $v_2$ when $x = 2$ for $f(x) = 3x^4 + x^3 + 2x^2 + x + 4$ using Horner's method. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31207",
"split": "train"
} |
math | [
{
"content": "Point \\( P \\) is located on the side \\( AB \\) of the square \\( ABCD \\) such that \\( AP: PB = 2:3 \\). Point \\( Q \\) lies on the side \\( BC \\) of the square and divides it in the ratio \\( BQ: QC = 3 \\). Lines \\( DP \\) and \\( AQ \\) intersect at point \\( E \\). Find the ratio of len... | {
"ground_truth": "4:9",
"style": "rule"
} | {
"index": "31208",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of trapezoid $EFCD$ if the height from $E$ to line $CD$ is 10 units. Plea... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "31209",
"split": "train"
} |
math | [
{
"content": "For a $5 \\times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "31210",
"split": "train"
} |
math | [
{
"content": "One million bucks (i.e. one million male deer) are in different cells of a $1000 \\times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the b... | {
"ground_truth": "1200000",
"style": "rule"
} | {
"index": "31211",
"split": "train"
} |
math | [
{
"content": "Given that the sequence $\\{a_n\\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\\sqrt{e})$$ (where $e$ is the base of the natural logarithm),\n1. Find the general formula for $\\{a_n\\}$.\n2. Let $b_n = \\ln a_n$, an... | {
"ground_truth": "n = 4",
"style": "rule"
} | {
"index": "31212",
"split": "train"
} |
math | [
{
"content": "Use the six digits 0, 1, 2, 3, 4, 5 to form four-digit even numbers without repeating any digit. Calculate the total number of such numbers that can be formed. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "156",
"style": "rule"
} | {
"index": "31213",
"split": "train"
} |
math | [
{
"content": "Given that $a > 0$, $b > 0$, and $a + b = 1$, find the minimum value of $\\frac{2}{a} + \\frac{3}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 + 2\\sqrt{6}",
"style": "rule"
} | {
"index": "31214",
"split": "train"
} |
math | [
{
"content": "Read the following material and then answer the following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\\frac{5}{\\sqrt{3}}$, $\\frac{2}{\\sqrt{3}+1}$, in fact, we can further simplify them:<br/>$($1) $\\frac{5}{\\sqrt{3}}=\\frac{5×\\sqrt{3}}{\\sqrt{3... | {
"ground_truth": "\\frac{3\\sqrt{11}-1}{2}",
"style": "rule"
} | {
"index": "31215",
"split": "train"
} |
math | [
{
"content": "Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1, and the larger rhombus has area 9. Compute the side length of the larger rhombus. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{15}",
"style": "rule"
} | {
"index": "31216",
"split": "train"
} |
math | [
{
"content": "$\\frac{10^7}{5\\times 10^4}=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "31217",
"split": "train"
} |
math | [
{
"content": "There are 6 blue, 7 red, and 9 white light bulbs. In how many ways can you arrange them (using all the light bulbs) in a garland so that no two white light bulbs are consecutive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3435432",
"style": "rule"
} | {
"index": "31218",
"split": "train"
} |
math | [
{
"content": "Let $a = \\pi/2008$. Find the smallest positive integer $n$ such that \\[2[\\cos(a)\\sin(a) + \\cos(4a)\\sin(2a) + \\cos(9a)\\sin(3a) + \\cdots + \\cos(n^2a)\\sin(na)]\\] is an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "251",
"style": "rule"
} | {
"index": "31219",
"split": "train"
} |
math | [
{
"content": "Let $\\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\\mathrm {P}=r(\\cos{\\theta^{\\circ}}+i\\sin{\\theta^{\\circ}})$, where $0<r$ and $0\\leq \\theta <360$. Find $\\theta$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "276",
"style": "rule"
} | {
"index": "31220",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\sqrt{3}\\sin ωx−2{(\\sin \\dfrac{ωx}{2})}^{2}(ω > 0)$ with the smallest positive period of $3π$.\n\n(1) Find the maximum and minimum values of the function $f(x)$ in the interval $[-{\\dfrac{3π }{4}},π]$.\n\n(2) Given $a$, $b$, $c$ are the sides opposite to angles $A... | {
"ground_truth": "\\dfrac {3+ \\sqrt {3}}{3}",
"style": "rule"
} | {
"index": "31221",
"split": "train"
} |
math | [
{
"content": "The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.\n\nHow many fruits in total grew on the miracle tree? Pl... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31222",
"split": "train"
} |
math | [
{
"content": "Calculate the volumes of the bodies bounded by the surfaces.\n\n$$\nz = 2x^2 + 18y^2, \\quad z = 6\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\pi",
"style": "rule"
} | {
"index": "31223",
"split": "train"
} |
math | [
{
"content": "We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31224",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. P... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "31225",
"split": "train"
} |
math | [
{
"content": "How many of the positive divisors of 3240 are multiples of 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "31226",
"split": "train"
} |
math | [
{
"content": "For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal t... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31227",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be an odd function defined on $\\mathbb{R}$. When $x > 0$, $f(x)=x^{2}+2x-1$.\n(1) Find $f(-2)$;\n(2) Find the expression of $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "31228",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "31229",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\displaystyle\\frac{235^2-221^2}{14}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "456",
"style": "rule"
} | {
"index": "31230",
"split": "train"
} |
math | [
{
"content": "The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "31231",
"split": "train"
} |
math | [
{
"content": "$5y$ varies inversely as the square of $x$. When $y=16$, $x=1$. When $x=8$, $y$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "31232",
"split": "train"
} |
math | [
{
"content": "In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \\( a \\) and \\( b \\). What is the maximum number of marked points that can lie on a line that does not coincide with \\( a \\) or \\( b \\)? Please output the final answer within \\boxed{}."... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "31233",
"split": "train"
} |
math | [
{
"content": "Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "31234",
"split": "train"
} |
math | [
{
"content": "Let set \\( A = \\{a \\mid a = 3k + 2, k \\leqslant 2000, k \\in \\mathbf{N}_{+}\\} \\) and set \\( B = \\{b \\mid b = 4k - 1, k \\leqslant 2000, k \\in \\mathbf{N}_{+}\\} \\). How many elements are in \\( A \\cap B \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "31235",
"split": "train"
} |
math | [
{
"content": "Let \\(D\\) be a point on side \\(BC\\) of \\(\\triangle ABC\\). Points \\(E\\) and \\(F\\) are the centroids of \\(\\triangle ABD\\) and \\(\\triangle ACD\\), respectively. The line segment \\(EF\\) intersects \\(AD\\) at point \\(G\\). Find the value of \\(\\frac{DG}{GA}\\). Please output the fi... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "31236",
"split": "train"
} |
math | [
{
"content": "The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "31237",
"split": "train"
} |
math | [
{
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\\cos B=b\\cos C$, and the dot product $\\vec{AB}\\cdot \\vec{BC} = -3$,\n1. Find the area of $\\triangle ABC$;\n2. Find the minimum value of side $AC$. Please output the f... | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "31238",
"split": "train"
} |
math | [
{
"content": "Eric is taking a biology class. His problem sets are worth 100 points in total, his three midterms are worth 100 points each, and his final is worth 300 points. If he gets a perfect score on his problem sets and scores $60 \\%, 70 \\%$, and $80 \\%$ on his midterms respectively, what is the minimu... | {
"ground_truth": "60 \\%",
"style": "rule"
} | {
"index": "31239",
"split": "train"
} |
math | [
{
"content": "How many polynomials of degree exactly 5 with real coefficients send the set \\{1,2,3,4,5,6\\} to a permutation of itself? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "714",
"style": "rule"
} | {
"index": "31240",
"split": "train"
} |
math | [
{
"content": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be f... | {
"ground_truth": "2(w+h)^2",
"style": "rule"
} | {
"index": "31241",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \\frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{31}{16}",
"style": "rule"
} | {
"index": "31242",
"split": "train"
} |
math | [
{
"content": "From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written? Please o... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31243",
"split": "train"
} |
math | [
{
"content": "Compute the number of nonempty subsets $S \\subseteq\\{-10,-9,-8, \\ldots, 8,9,10\\}$ that satisfy $|S|+\\min (S)$. $\\max (S)=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "335",
"style": "rule"
} | {
"index": "31244",
"split": "train"
} |
math | [
{
"content": "The orthocenter of triangle $ABC$ divides altitude $\\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calculate $\\tan A \\tan B.$\n\n[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E, F, H;\n\nA = (0,0);\nB = (5,0);\nC = (4,4);\nD = (A + reflect(B,C)*(A))/2;\nE = (B + reflect(C,A)*(... | {
"ground_truth": "\\frac{7}{2}",
"style": "rule"
} | {
"index": "31245",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, define $s(n)$ to equal the sum of the digits of $n$. The number of integers $n$ with $100 \\leq n \\leq 999$ and $7 \\leq s(n) \\leq 11$ is $S$. What is the integer formed by the rightmost two digits of $S$? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31246",
"split": "train"
} |
math | [
{
"content": "In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced by any digit from $0, 1, 2, 3, 4, 5, 6, 7, 8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. In how many ways can this be done? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1458",
"style": "rule"
} | {
"index": "31247",
"split": "train"
} |
math | [
{
"content": "A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples.\nWhat is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31248",
"split": "train"
} |
math | [
{
"content": "Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(3, 3)",
"style": "rule"
} | {
"index": "31249",
"split": "train"
} |
math | [
{
"content": "How many distinct equilateral triangles can be constructed by connecting three different vertices of a regular dodecahedron? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31250",
"split": "train"
} |
math | [
{
"content": "Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31251",
"split": "train"
} |
math | [
{
"content": "Find the smallest $n$ such that $n!$ ends with 10 zeroes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "31252",
"split": "train"
} |
math | [
{
"content": "Eight identical cubes with of size $1 \\times 1 \\times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the face opposite numbe... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "31253",
"split": "train"
} |
math | [
{
"content": "Calculate the remainder when $1 + 11 + 11^2 + \\cdots + 11^{1024}$ is divided by $500$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "31254",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and vectors $\\overrightarrow{m}=(1-\\cos (A+B),\\cos \\frac {A-B}{2})$ and $\\overrightarrow{n}=( \\frac {5}{8},\\cos \\frac {A-B}{2})$ with $\\overrightarrow{m}\\cdot \\overrightarro... | {
"ground_truth": "-\\frac {3}{8}",
"style": "rule"
} | {
"index": "31255",
"split": "train"
} |
math | [
{
"content": "Given a $4 \\times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainte... | {
"ground_truth": "\\frac{1}{65536}",
"style": "rule"
} | {
"index": "31256",
"split": "train"
} |
math | [
{
"content": "Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can... | {
"ground_truth": "1152",
"style": "rule"
} | {
"index": "31257",
"split": "train"
} |
math | [
{
"content": "Given the plane rectangular coordinate system $(xOy)$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C_{1}$ is $\\rho = 4\\cos \\theta$, and the line $l$: $\\begin{cases} x = 1 - \\frac{2\\... | {
"ground_truth": "\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "31258",
"split": "train"
} |
math | [
{
"content": "A lattice point in the plane is a point of the form $(n, m)$, where $n$ and $m$ are integers. Consider a set $S$ of lattice points. We construct the transform of $S$, denoted by $S^{\\prime}$, by the following rule: the pair $(n, m)$ is in $S^{\\prime}$ if and only if any of $(n, m-1),(n, m+1),(n-... | {
"ground_truth": "421",
"style": "rule"
} | {
"index": "31259",
"split": "train"
} |
math | [
{
"content": "If \\( 3^x = 5 \\), what is the value of \\( 3^{x+2} \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "31260",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new ... | {
"ground_truth": "245",
"style": "rule"
} | {
"index": "31261",
"split": "train"
} |
math | [
{
"content": "There are 7 line segments with integer lengths in centimeters: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, satisfying $a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7$, and any three of these line segments cannot form a triangle. If $a_1 = 1$ cm and $a_7 = 21$ cm, then $a_6 = \\ $. Please output the ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31262",
"split": "train"
} |
math | [
{
"content": "Reading material: In class, the teacher explained the following two problems on the blackboard:\n\nProblem 1: Calculate: $77.7\\times 11-77.7\\times 5+77.7\\times 4$.\nSolution: $77.7\\times 11-77.7\\times 5+77.7\\times 4=77.7\\times \\left(11-5+4\\right)=777$.\nThis problem uses the distributive ... | {
"ground_truth": "-\\frac{1}{3}",
"style": "rule"
} | {
"index": "31263",
"split": "train"
} |
math | [
{
"content": "John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "31264",
"split": "train"
} |
math | [
{
"content": "Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$? Please output th... | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "31265",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $n\\in\\mathbb{N}^*$.\n\n (1) Find $a_n$;\n\n (2) If the sequence $\\{b_n\\}$ satisfies $b_n=\\dfrac{1}{(1+\\log_{3}a_n)(3+\\log_{3}a_n)}$, and the sum of the fi... | {
"ground_truth": "\\dfrac{3}{4}",
"style": "rule"
} | {
"index": "31266",
"split": "train"
} |
math | [
{
"content": "If $f(x)=\\log \\left(\\frac{1+x}{1-x}\\right)$ for $-1<x<1$, then $f\\left(\\frac{3x+x^3}{1+3x^2}\\right)$ in terms of $f(x)$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3f(x)",
"style": "rule"
} | {
"index": "31267",
"split": "train"
} |
math | [
{
"content": "$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31268",
"split": "train"
} |
math | [
{
"content": "A square with side length 1 is rotated about one vertex by an angle of $\\alpha,$ where $0^\\circ < \\alpha < 90^\\circ$ and $\\cos \\alpha = \\frac{4}{5}.$ Find the area of the shaded region that is common to both squares.\n\n[asy]\nunitsize(3 cm);\n\npair A, B, C, D, Bp, Cp, Dp, P;\n\nA = (0,0)... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31269",
"split": "train"
} |
math | [
{
"content": "Say that an integer $A$ is yummy if there exist several consecutive integers, including $A$, that add up to 2014. What is the smallest yummy integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2013",
"style": "rule"
} | {
"index": "31270",
"split": "train"
} |
math | [
{
"content": "An entrepreneur invested \\$12,000 in a three-month savings certificate that paid a simple annual interest rate of $8\\%$. After three months, she invested the total value of her investment in another three-month certificate. After three more months, the investment was worth \\$12,980. If the annu... | {
"ground_truth": "24\\%",
"style": "rule"
} | {
"index": "31271",
"split": "train"
} |
math | [
{
"content": "Determine the value of \n\\[3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9006",
"style": "rule"
} | {
"index": "31272",
"split": "train"
} |
math | [
{
"content": "Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increas... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31273",
"split": "train"
} |
math | [
{
"content": "A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31274",
"split": "train"
} |
math | [
{
"content": "Given an obtuse triangle \\(ABC\\) with obtuse angle \\(C\\). Points \\(P\\) and \\(Q\\) are marked on its sides \\(AB\\) and \\(BC\\) respectively, such that \\(\\angle ACP = CPQ = 90^\\circ\\). Find the length of segment \\(PQ\\) if it is known that \\(AC = 25\\), \\(CP = 20\\), and \\(\\angle A... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31275",
"split": "train"
} |
math | [
{
"content": "In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\\overrightarrow{m}\\left(a, \\sqrt{3b}\\right)$ is parallel to $\\overrightarrow{n}=\\left(\\cos A,\\sin B\\right)$.\n\n(1) Find $A$;\n\n(2) If $a= \\sqrt{7},b=2$, find the area of $∆ABC$.... | {
"ground_truth": "\\dfrac{3 \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "31276",
"split": "train"
} |
math | [
{
"content": "Let the function \\( f(x) = 3 \\sin x + 2 \\cos x + 1 \\). If real numbers \\( a, b, c \\) satisfy \\( a f(x) + b f(x-c) = 1 \\) for all real numbers \\( x \\), find the value of \\( \\frac{b \\cos c}{a} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "31277",
"split": "train"
} |
math | [
{
"content": "Given the vectors $\\overrightarrow{a} = (\\cos 25^\\circ, \\sin 25^\\circ)$, $\\overrightarrow{b} = (\\sin 20^\\circ, \\cos 20^\\circ)$, and $\\overrightarrow{u} = \\overrightarrow{a} + t\\overrightarrow{b}$, where $t\\in\\mathbb{R}$, find the minimum value of $|\\overrightarrow{u}|$. Please outp... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31278",
"split": "train"
} |
math | [
{
"content": "For a nonnegative integer \\(n\\), let \\(r_{11}(7n)\\) stand for the remainder left when \\(n\\) is divided by \\(11.\\) For example, \\(r_{11}(7 \\cdot 3) = 10.\\)\n\nWhat is the \\(15^{\\text{th}}\\) entry in an ordered list of all nonnegative integers \\(n\\) that satisfy $$r_{11}(7n) \\leq 5~... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "31279",
"split": "train"
} |
math | [
{
"content": "One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "31280",
"split": "train"
} |
math | [
{
"content": "Given $$(5x- \\frac {1}{ \\sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "31281",
"split": "train"
} |
math | [
{
"content": "In Geometrytown, all streets are $30$ feet wide and the blocks they enclose are rectangles with side lengths of $300$ feet and $500$ feet. Anne runs around the block on the $300$-foot side of the street, while Bob runs on the opposite side of the street. How many more feet than Anne does Bob run f... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "31282",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\cos A= \\frac{4}{5}$.\n\n(1) Find the value of $\\sin ^{2} \\frac{B+C}{2}+\\cos 2A$;\n\n(2) If $b=2$, the area of $\\triangle ABC$ is $S=3$, find $a$. Please output the final answer withi... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "31283",
"split": "train"
} |
math | [
{
"content": "A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longest interval of time, ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31284",
"split": "train"
} |
math | [
{
"content": "What is the units digit in the product of all natural numbers from 1 to 99, inclusive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "31285",
"split": "train"
} |
math | [
{
"content": "(1) Simplify: $\\dfrac{\\sin(\\pi -\\alpha)\\cos(\\pi +\\alpha)\\sin(\\dfrac{\\pi}{2}+\\alpha)}{\\sin(-\\alpha)\\sin(\\dfrac{3\\pi}{2}+\\alpha)}$.\n\n(2) Given $\\alpha \\in (\\dfrac{\\pi}{2}, \\pi)$, and $\\sin(\\pi -\\alpha) + \\cos \\alpha = \\dfrac{7}{13}$, find $\\tan \\alpha$. Please output ... | {
"ground_truth": "-\\dfrac{12}{5}",
"style": "rule"
} | {
"index": "31286",
"split": "train"
} |
math | [
{
"content": "Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends 1024 dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going ... | {
"ground_truth": "1023",
"style": "rule"
} | {
"index": "31287",
"split": "train"
} |
math | [
{
"content": "What is $(3^{12} \\times 9^{-3})^2$? Write your answer as an integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "531441",
"style": "rule"
} | {
"index": "31288",
"split": "train"
} |
math | [
{
"content": "All triangles have the same value, and all circles have the same value. What is the sum of three circles? \\begin{align*}\n\\Delta + \\bigcirc + \\Delta + \\bigcirc + \\Delta&= 21\\\\\n\\bigcirc + \\Delta+\\bigcirc+\\Delta+\\bigcirc &= 19\\\\\n\\bigcirc + \\bigcirc + \\bigcirc &= \\ ?\n\\end{align... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "31289",
"split": "train"
} |
math | [
{
"content": "A piece of platinum, which has a density of $2.15 \\cdot 10^{4} \\mathrm{kg} / \\mathrm{m}^{3}$, is connected to a piece of cork wood (density $2.4 \\cdot 10^{2} \\mathrm{kg} / \\mathrm{m}^{3}$). The density of the combined system is $4.8 \\cdot 10^{2} \\mathrm{kg} / \\mathrm{m}^{3}$. What is the ... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "31290",
"split": "train"
} |
math | [
{
"content": "There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "31291",
"split": "train"
} |
math | [
{
"content": "Given four points \\( K, L, M, N \\) that are not coplanar. A sphere touches the planes \\( K L M \\) and \\( K L N \\) at points \\( M \\) and \\( N \\) respectively. Find the surface area of the sphere, knowing that \\( M L = 1 \\), \\( K M = 2 \\), \\( \\angle M N L = 60^\\circ \\), and \\( \\a... | {
"ground_truth": "\\frac{64\\pi}{11}",
"style": "rule"
} | {
"index": "31292",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer \\( n \\) such that\n\\[\n\\sqrt{5 n}-\\sqrt{5 n-4}<0.01\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8001",
"style": "rule"
} | {
"index": "31293",
"split": "train"
} |
math | [
{
"content": "Let the function $f(x)=\\ln x- \\frac{1}{2}ax^{2}-bx$.\n\n$(1)$ When $a=b= \\frac{1}{2}$, find the maximum value of the function $f(x)$;\n\n$(2)$ Let $F(x)=f(x)+ \\frac{1}{2}ax^{2}+bx+ \\frac{a}{x}$, $(0 < x\\leqslant 3)$, the slope of the tangent line at any point $P(x_{0},y_{0})$ on its graph is... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31294",
"split": "train"
} |
math | [
{
"content": "A random sample of $10$ households was taken from a residential area, and the monthly income data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\\sum_{i=1}^{10} x_i = 80$, $\\sum_{i=1}^{10} y_i = 20$, $\\sum_{i=1}^{10} ... | {
"ground_truth": "1.7",
"style": "rule"
} | {
"index": "31295",
"split": "train"
} |
math | [
{
"content": "Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \\[ \\dfrac {xy+yz+zx}{x^2+y^2+z^2}. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac12",
"style": "rule"
} | {
"index": "31296",
"split": "train"
} |
math | [
{
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31297",
"split": "train"
} |
math | [
{
"content": "Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "31298",
"split": "train"
} |
math | [
{
"content": "Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "31299",
"split": "train"
} |
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