data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.2 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a mathematics class of 40 students? Please out... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33500",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ such that the $73$ fractions $\\frac{19}{n+21}, \\frac{20}{n+22},\\frac{21}{n+23},...,\\frac{91}{n+93}$ are all irreducible. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "33501",
"split": "train"
} |
math | [
{
"content": "The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "33502",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33503",
"split": "train"
} |
math | [
{
"content": "Find all integers \\( z \\) for which exactly two of the following five statements are true, and three are false:\n\n1) \\( 2z > 130 \\)\n2) \\( z < 200 \\)\n3) \\( 3z > 50 \\)\n4) \\( z > 205 \\)\n5) \\( z > 15 \\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33504",
"split": "train"
} |
math | [
{
"content": "Given a sequence \\(\\{a_n\\}\\) such that: \\(a_1 = 1\\) and \\(a_{n+1} = \\frac{a_n}{(n+1)(a_n + 1)}\\) for \\(n \\in \\mathbb{Z^+}\\), find the value of \\(\\lim_{n \\rightarrow +\\infty} n! \\cdot a_n\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{e}",
"style": "rule"
} | {
"index": "33505",
"split": "train"
} |
math | [
{
"content": "What is the probability that exactly one person gets their hat back when 6 people randomly pick hats? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{30}",
"style": "rule"
} | {
"index": "33506",
"split": "train"
} |
math | [
{
"content": "A square sheet of paper has area $6 \\text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your a... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "33507",
"split": "train"
} |
math | [
{
"content": "In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these... | {
"ground_truth": "222",
"style": "rule"
} | {
"index": "33508",
"split": "train"
} |
math | [
{
"content": "Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose th... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "33509",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "703",
"style": "rule"
} | {
"index": "33510",
"split": "train"
} |
math | [
{
"content": "The function $f$ satisfies\n\\[\nf(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3\n\\]\nfor all real numbers $x, y$. Determine the value of $f(10)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-37",
"style": "rule"
} | {
"index": "33511",
"split": "train"
} |
math | [
{
"content": "Find the ordered pair $(a,b)$ of integers such that\n\\[\\sqrt{9 - 8 \\sin 50^\\circ} = a + b \\csc 50^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(3,-1)",
"style": "rule"
} | {
"index": "33512",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{1}=1$ and $a_{n+1}=\\left\\{\\begin{array}{l}{{a}_{n}+1, n \\text{ is odd}}\\\\{{a}_{n}+2, n \\text{ is even}}\\end{array}\\right.$\n\n$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\\{b_{n}\\}$.\n\n... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "33513",
"split": "train"
} |
math | [
{
"content": "Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P_{}^{}$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertic... | {
"ground_truth": "597",
"style": "rule"
} | {
"index": "33514",
"split": "train"
} |
math | [
{
"content": "When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \\(a, b, c\\) are obtained respectively where \\(a\\) and \\(b\\) are from the standard dice and \\(c\\) is from the 8-sided die. Find the probability that \\((a-1)(b-1)(c-1) \\neq 0\\). Ple... | {
"ground_truth": "\\frac{175}{288}",
"style": "rule"
} | {
"index": "33515",
"split": "train"
} |
math | [
{
"content": "In some cells of a \\(10 \\times 10\\) board, there are fleas. Every minute, the fleas jump simultaneously to an adjacent cell (along the sides). Each flea jumps strictly in one of the four directions parallel to the sides of the board, maintaining its direction as long as possible; otherwise, it ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "33516",
"split": "train"
} |
math | [
{
"content": "Let \\( w \\) be a complex number such that \\( |w - 3 + 2i| = 4 \\). Find the minimum value of\n\\[\n|w + 1 + 2i|^2 + |w - 7 - 2i|^2.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "33517",
"split": "train"
} |
math | [
{
"content": "Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that \n- each member in $S$ is a positive integer not exceeding $2002$ ,\n- if $a,b\\in S$ (not necessarily different), then $ab\\not\\in S$ . Please output the final answer within... | {
"ground_truth": "1958",
"style": "rule"
} | {
"index": "33518",
"split": "train"
} |
math | [
{
"content": "Compute the number of triples $(f, g, h)$ of permutations on $\\{1,2,3,4,5\\}$ such that $$ \\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\\\ & g(h(f(x)))=f(h(g(x)))=h(x), \\text { and } \\\\ & h(f(g(x)))=g(f(h(x)))=f(x) \\end{aligned} $$ for all $x \\in\\{1,2,3,4,5\\}$. Please output the final ... | {
"ground_truth": "146",
"style": "rule"
} | {
"index": "33519",
"split": "train"
} |
math | [
{
"content": "In the ellipse $\\dfrac {x^{2}}{36}+ \\dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\\overrightarrow{KM} \\cdot \\overrightarrow{KN} = 0$, find the minimum value of $\\overrightarrow{KM} \\cdot \\overrightarrow{NM}$. Please output the final answe... | {
"ground_truth": "\\dfrac{23}{3}",
"style": "rule"
} | {
"index": "33520",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5 + \\sqrt{105}}{2}",
"style": "rule"
} | {
"index": "33521",
"split": "train"
} |
math | [
{
"content": "Suppose $x$ is a random real number between $1$ and $4$ , and $y$ is a random real number between $1$ and $9$ . If the expected value of \\[ \\left\\lceil \\log_2 x \\right\\rceil - \\left\\lfloor \\log_3 y \\right\\rfloor \\] can be expressed as $\\frac mn$ where $m$ and $n$ are r... | {
"ground_truth": "1112",
"style": "rule"
} | {
"index": "33522",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{F}$ be the set of all the functions $f : \\mathcal{P}(S) \\longrightarrow \\mathbb{R}$ such that for all $X, Y \\subseteq S$, we have $f(X \\cap Y) = \\min (f(X), f(Y))$, where $S$ is a finite set (and $\\mathcal{P}(S)$ is the set of its subsets). Find\n\\[\\max_{f \\in \\mathcal{F}... | {
"ground_truth": "n+1",
"style": "rule"
} | {
"index": "33523",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\(A B C\\) with \\(A B = A C\\) and \\(\\angle A = 110^{\\circ}\\). Inside the triangle, a point \\(M\\) is chosen such that \\(\\angle M B C = 30^{\\circ}\\) and \\(\\angle M C B = 25^{\\circ}\\). Find \\(\\angle A M C\\). Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "33524",
"split": "train"
} |
math | [
{
"content": "Find the mass of the body $\\Omega$ with density $\\mu = 20z$, bounded by the surfaces\n\n$$\nz = \\sqrt{1 - x^{2} - y^{2}}, \\quad z = \\sqrt{\\frac{x^{2} + y^{2}}{4}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "33525",
"split": "train"
} |
math | [
{
"content": "Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side lengths. The shortest possible length of the longest of the eight sticks is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "33526",
"split": "train"
} |
math | [
{
"content": "In an acute-angled triangle $ABC$ , the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $ . Find the value of the smallest angle of triangle $ ABC $ .\n\n(Black Maxim) Please output t... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "33527",
"split": "train"
} |
math | [
{
"content": "Given $α \\in \\left( \\frac{π}{2}, π \\right)$, and $\\sin α = \\frac{1}{3}$.\n\n$(1)$ Find the value of $\\sin 2α$; \n$(2)$ If $\\sin (α+β) = -\\frac{3}{5}$, and $β \\in (0, \\frac{π}{2})$, find the value of $\\sin β$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6\\sqrt{2}+4}{15}",
"style": "rule"
} | {
"index": "33528",
"split": "train"
} |
math | [
{
"content": "On a Cartesian plane, consider the points \\(A(-8, 2), B(-4, -2)\\) and \\(X(1, y)\\) and \\(Y(10, 3)\\). If segment \\(AB\\) is parallel to segment \\(XY\\), find the value of \\(y\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33529",
"split": "train"
} |
math | [
{
"content": "Given a quadratic function in terms of \\\\(x\\\\), \\\\(f(x)=ax^{2}-4bx+1\\\\).\n\\\\((1)\\\\) Let set \\\\(P=\\\\{1,2,3\\\\}\\\\) and \\\\(Q=\\\\{-1,1,2,3,4\\\\}\\\\), randomly pick a number from set \\\\(P\\\\) as \\\\(a\\\\) and from set \\\\(Q\\\\) as \\\\(b\\\\), calculate the probability th... | {
"ground_truth": "\\dfrac{961}{1280}",
"style": "rule"
} | {
"index": "33530",
"split": "train"
} |
math | [
{
"content": "For all positive integers $x$, let \\[f(x)=\\begin{cases}1 & \\text{if }x = 1\\\\ \\frac x{10} & \\text{if }x\\text{ is divisible by 10}\\\\ x+1 & \\text{otherwise}\\end{cases}\\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smalle... | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "33531",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) divides the side \\( BC \\) of the parallelogram \\( ABCD \\) in the ratio \\( BM:MC = 1:2 \\). The line \\( AM \\) intersects the diagonal \\( BD \\) at point \\( K \\). Find the area of the quadrilateral \\( CMKD \\) if the area of the parallelogram \\( ABCD \\) is 1. Please outp... | {
"ground_truth": "\\frac{11}{24}",
"style": "rule"
} | {
"index": "33532",
"split": "train"
} |
math | [
{
"content": "If two of the roots of \\[2x^3 + 8x^2 - 120x + k = 0\\]are equal, find the value of $k,$ given that $k$ is positive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\tfrac{6400}{27}",
"style": "rule"
} | {
"index": "33533",
"split": "train"
} |
math | [
{
"content": "Consider an arithmetic sequence where the first four terms are $x+2y$, $x-2y$, $2xy$, and $x/y$. Determine the fifth term of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-27.7",
"style": "rule"
} | {
"index": "33534",
"split": "train"
} |
math | [
{
"content": "Determine the tens digit of $17^{1993}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33535",
"split": "train"
} |
math | [
{
"content": "There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. \n$(1)$ How many unique three-digit numbers can be formed without repetition? \n$(2)$ How many unique four-digit even numbers can be formed without repetition? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "2296",
"style": "rule"
} | {
"index": "33536",
"split": "train"
} |
math | [
{
"content": "Mateo receives $20 every hour for one week, and Sydney receives $400 every day for one week. Calculate the difference between the total amounts of money that Mateo and Sydney receive over the one week period. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "560",
"style": "rule"
} | {
"index": "33537",
"split": "train"
} |
math | [
{
"content": "Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "124",
"style": "rule"
} | {
"index": "33538",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=4$ and $\\frac{\\cos B}{\\cos C} = \\frac{4}{2a - c}$.\n(1) Find the measure of angle $B$;\n(2) Find the maximum area of $\\triangle ABC$. Please output the final answer ... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "33539",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ with $a_1=1$ and $a_{n+1}= \\frac{2a_n}{a_n+2}$, find the value of $a_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{11}",
"style": "rule"
} | {
"index": "33540",
"split": "train"
} |
math | [
{
"content": "Determine how many perfect cubes exist between \\(3^6 + 1\\) and \\(3^{12} + 1\\), inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "33541",
"split": "train"
} |
math | [
{
"content": "On some cells of a 10x10 board, there is a flea. Every minute, the fleas jump simultaneously, each one to a neighboring cell (adjacent by side). Each flea jumps strictly in one of the four directions parallel to the board's sides and maintains this direction as long as possible; otherwise, it chan... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "33542",
"split": "train"
} |
math | [
{
"content": "In a bike shed, there are bicycles (two wheels), tricycles, and cars (four wheels). The number of bicycles is four times the number of cars. Several students counted the total number of wheels in the shed, but each of them obtained a different count: $235, 236, 237, 238, 239$. Among these, one cou... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "33543",
"split": "train"
} |
math | [
{
"content": "On the side \\( BC \\) of triangle \\( ABC \\), point \\( A_1 \\) is taken such that \\( BA_1 : A_1C = 2:1 \\). In what ratio does median \\( CC_1 \\) divide segment \\( AA_1 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3:1",
"style": "rule"
} | {
"index": "33544",
"split": "train"
} |
math | [
{
"content": "If $1-\\frac{4}{x}+\\frac{4}{x^2}=0$, then $\\frac{2}{x}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "33545",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\frac{2 + 6}{4^{100}} + \\frac{2 + 2 \\cdot 6}{4^{99}} + \\frac{2 + 3 \\cdot 6}{4^{98}} + \\dots + \\frac{2 + 98 \\cdot 6}{4^3} + \\frac{2 + 99 \\cdot 6}{4^2} + \\frac{2 + 100 \\cdot 6}{4}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "33546",
"split": "train"
} |
math | [
{
"content": "Given $0 \\le x_0 < 1$, let\n\\[x_n = \\begin{cases} 2x_{n-1} & \\text{ if } 2x_{n-1} < 1 \\\\ 2x_{n-1} - 1 & \\text{ if } 2x_{n-1} \\ge 1 \\end{cases}\\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "33547",
"split": "train"
} |
math | [
{
"content": "A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates o... | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "33548",
"split": "train"
} |
math | [
{
"content": "How many digits are there in the number \\(N\\) if \\(N=2^{12} \\times 5^{8}\\) ?\n\nIf \\(\\left(2^{48}-1\\right)\\) is divisible by two whole numbers between 60 and 70, find them.\n\nGiven \\(2^{\\frac{1}{2}} \\times 9^{\\frac{1}{9}}\\) and \\(3^{\\frac{1}{3}} \\times 8^{\\frac{1}{8}}\\), what i... | {
"ground_truth": "3^{\\frac{1}{3}} \\times 8^{\\frac{1}{8}}",
"style": "rule"
} | {
"index": "33549",
"split": "train"
} |
math | [
{
"content": "A box of 100 personalized pencils costs $\\$30$. How many dollars does it cost to buy 2500 pencils? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\$750",
"style": "rule"
} | {
"index": "33550",
"split": "train"
} |
math | [
{
"content": "(1) Calculate: $\\left( \\frac{1}{8} \\right)^{-\\frac{1}{3}} - 3\\log_{3}^{2}(\\log_{3}4) \\cdot (\\log_{8}27) + 2\\log_{\\frac{1}{6}} \\sqrt{3} - \\log_{6}2$\n\n(2) Calculate: $27^{\\frac{2}{3}} - 2^{\\log_{2}3} \\times \\log_{2}\\frac{1}{8} + 2\\lg \\left( \\sqrt{3+\\sqrt{5}} + \\sqrt{3-\\sqrt{... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "33551",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\\sin A = \\sqrt{7}a$.\n(1) Find the value of $\\sin B$;\n(2) If $a$, $b$, and $c$ form an arithmetic sequence with a positive common difference, find the value of $\\cos A - \\cos C$. Pl... | {
"ground_truth": "\\frac{\\sqrt{7}}{2}",
"style": "rule"
} | {
"index": "33552",
"split": "train"
} |
math | [
{
"content": "The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \\_\\_\\_\\_\\_\\_\\_\... | {
"ground_truth": "6.6",
"style": "rule"
} | {
"index": "33553",
"split": "train"
} |
math | [
{
"content": "Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\\angle ABC=\\angle DEA=90^\\circ$ and $BC+DE=1$. Compute the area of the pentagon.\n\n[i]Greece[/i] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "33554",
"split": "train"
} |
math | [
{
"content": "A bag contains 8 red balls, a number of white balls, and no other balls. If $\\frac{5}{6}$ of the balls in the bag are white, then how many white balls are in the bag? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "33555",
"split": "train"
} |
math | [
{
"content": "People have long been exploring the numerical solution of high-degree equations. Newton gave a numerical solution method for high-degree algebraic equations in his book \"Fluxions.\" This method for finding the roots of equations has been widely used in the scientific community. For example, to fi... | {
"ground_truth": "-\\frac{11}{8}",
"style": "rule"
} | {
"index": "33556",
"split": "train"
} |
math | [
{
"content": "A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy]\n/* AMC8 2003 #15 Problem */\ndraw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);\ndraw((0,1)... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "33557",
"split": "train"
} |
math | [
{
"content": "In the sequence $\\{a_{n}\\}$, $a_{1}=1$, $\\sqrt{{a}_{n+1}}-\\sqrt{{a}_{n}}=1$ ($n\\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\\{b_{n}\\}$ is $S_{n}=2^{n}-m$. For $n\\in N^{*}$, the smallest value of the real number $\\lambda$ that satisfies $\\lambda b_{n}\\geqslant a_{... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "33558",
"split": "train"
} |
math | [
{
"content": "Let\n\\[\\mathbf{M} = \\begin{pmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{pmatrix}\\]be a matrix with complex entries such that $\\mathbf{M}^2 = \\mathbf{I}.$ If $abc = 1,$ then find the possible values of $a^3 + b^3 + c^3.$ Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "2,4",
"style": "rule"
} | {
"index": "33559",
"split": "train"
} |
math | [
{
"content": "Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other.\nThe man at $R$ walks uniformly at the rate of $4\\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant\nrate of $3\\tfrac{1}{4}$ miles per hour for the first hour, at $3\\tfrac{3}{4}... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "33560",
"split": "train"
} |
math | [
{
"content": "There are \\(100\\) countries participating in an olympiad. Suppose \\(n\\) is a positive integers such that each of the \\(100\\) countries is willing to communicate in exactly \\(n\\) languages. If each set of \\(20\\) countries can communicate in exactly one common language, and no language is ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "33561",
"split": "train"
} |
math | [
{
"content": "Let $\\overrightarrow{m} = (\\sin(x - \\frac{\\pi}{3}), 1)$ and $\\overrightarrow{n} = (\\cos x, 1)$.\n\n(1) If $\\overrightarrow{m} \\parallel \\overrightarrow{n}$, find the value of $\\tan x$.\n\n(2) If $f(x) = \\overrightarrow{m} \\cdot \\overrightarrow{n}$, where $x \\in [0, \\frac{\\pi}{2}]$,... | {
"ground_truth": "1 - \\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "33562",
"split": "train"
} |
math | [
{
"content": "Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and i... | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "33563",
"split": "train"
} |
math | [
{
"content": "A circle with center $A$ and radius three inches is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),6),gray,linewidth(2));\nfilldraw(circle(3dir... | {
"ground_truth": "27\\pi",
"style": "rule"
} | {
"index": "33564",
"split": "train"
} |
math | [
{
"content": "Find the value of the expression \\(\\sum_{i=0}^{1009}(2 k+1)-\\sum_{i=1}^{1009} 2 k\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "33565",
"split": "train"
} |
math | [
{
"content": "Find the slope angle of the tangent line to the curve $f(x)=\\frac{1}{3}{x}^{3}-{x}^{2}+5$ at $x=1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "33566",
"split": "train"
} |
math | [
{
"content": "Given $f(x)= \\begin{cases} \\sin \\frac{\\pi}{3}x, & x\\leqslant 2011, \\\\ f(x-4), & x > 2011, \\end{cases}$ find $f(2012)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "33567",
"split": "train"
} |
math | [
{
"content": "The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "33568",
"split": "train"
} |
math | [
{
"content": "A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \\frac{\\text{km}}{\\text{hour}}$ on a horizontal track with a friction coefficient of $\\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$? \n\n(Note:... | {
"ground_truth": "10.3",
"style": "rule"
} | {
"index": "33569",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\angle A=55^\\circ$, $\\angle C=75^\\circ$, $D$ is on side $\\overline{AB}$ and $E$ is on side $\\overline{BC}$. If $DB=BE$, then $\\angle{BED} =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65^\\circ",
"style": "rule"
} | {
"index": "33570",
"split": "train"
} |
math | [
{
"content": "The quadratic $8x^2+12x-14$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{23}{4}",
"style": "rule"
} | {
"index": "33571",
"split": "train"
} |
math | [
{
"content": "The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "33572",
"split": "train"
} |
math | [
{
"content": "Calculate the product $\\left(\\frac{3}{6}\\right)\\left(\\frac{6}{9}\\right)\\left(\\frac{9}{12}\\right)\\cdots\\left(\\frac{2001}{2004}\\right)$. Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{668}",
"style": "rule"
} | {
"index": "33573",
"split": "train"
} |
math | [
{
"content": "The hypotenuse of an isosceles right triangle is $4\\sqrt{2}$ units. How many square units are in the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33574",
"split": "train"
} |
math | [
{
"content": "Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33575",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilater... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "33576",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\\overrightarrow{AM} = x\\overrightarrow{AB}$ and $\\overrightarrow{AN} = y\\overrightarrow{AC}$ ($x, y \\neq 0$), then the minimum value... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "33577",
"split": "train"
} |
math | [
{
"content": "Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33578",
"split": "train"
} |
math | [
{
"content": "In the number \\(2016 * * * * 02 *\\), each of the 5 asterisks needs to be replaced with any of the digits \\(0, 2, 4, 7, 8, 9\\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1728",
"style": "rule"
} | {
"index": "33579",
"split": "train"
} |
math | [
{
"content": "Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Pet... | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "33580",
"split": "train"
} |
math | [
{
"content": "It is known that\n$$\n\\sqrt{9-8 \\sin 50^{\\circ}}=a+b \\sin c^{\\circ}\n$$\nfor exactly one set of positive integers \\((a, b, c)\\), where \\(0 < c < 90\\). Find the value of \\(\\frac{b+c}{a}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33581",
"split": "train"
} |
math | [
{
"content": "Real numbers $X_1, X_2, \\dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\\min(X_1,X_2,\\dots, X_{10})^4$ can be expressed as a rational number $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?\n\n*201... | {
"ground_truth": "1002",
"style": "rule"
} | {
"index": "33582",
"split": "train"
} |
math | [
{
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "33583",
"split": "train"
} |
math | [
{
"content": "There are 20 chairs arranged in a circle. There are \\(n\\) people sitting in \\(n\\) different chairs. These \\(n\\) people stand, move \\(k\\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \\((n, k)\\) with \\(1 \\leq n \... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "33584",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive solution to\n\\[\\tan 2x + \\tan 3x = \\sec 3x\\]in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{14}",
"style": "rule"
} | {
"index": "33585",
"split": "train"
} |
math | [
{
"content": "The perimeter of a rectangle is 48. What is the largest possible area of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "33586",
"split": "train"
} |
math | [
{
"content": "Let S$_{n}$ denote the sum of the first $n$ terms of the arithmetic sequence {a$_{n}$} with a common difference d=2. The terms a$_{1}$, a$_{3}$, and a$_{4}$ form a geometric sequence. Find the value of S$_{8}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "33587",
"split": "train"
} |
math | [
{
"content": "Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "33588",
"split": "train"
} |
math | [
{
"content": "The smallest sum one could get by adding three different numbers from the set $\\{ 7,25,-1,12,-3 \\}$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33589",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point inside triangle $ABC$ such that\n\\[\\overrightarrow{PA} + 2 \\overrightarrow{PB} + 3 \\overrightarrow{PC} = \\mathbf{0}.\\]Find the ratio of the area of triangle $ABC$ to the area of triangle $APC.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "33590",
"split": "train"
} |
math | [
{
"content": "A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of br... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "33591",
"split": "train"
} |
math | [
{
"content": "A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)\n\n*Ray Li.* Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "441",
"style": "rule"
} | {
"index": "33592",
"split": "train"
} |
math | [
{
"content": "On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "33593",
"split": "train"
} |
math | [
{
"content": "Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:\n[list=1]\n[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.\n... | {
"ground_truth": "2 \\sum_{i=0}^{8} \\binom{n}{i} - 1",
"style": "rule"
} | {
"index": "33594",
"split": "train"
} |
math | [
{
"content": "Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.\n\n$A_{1}(-2, 0, -4)$\n\n$A_{2}(-1, 7, 1)$\n\n$A_{3}(4, -8, -4)$\n\n$A_{4}(1, -4, 6)$ Please output the final answer within \\boxe... | {
"ground_truth": "5\\sqrt{2}",
"style": "rule"
} | {
"index": "33595",
"split": "train"
} |
math | [
{
"content": "Find the distance between the points (0,4) and (3,0). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "33596",
"split": "train"
} |
math | [
{
"content": "Consider a $10 \\times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \\%$ of the burrito's original size and accidental... | {
"ground_truth": "71.8",
"style": "rule"
} | {
"index": "33597",
"split": "train"
} |
math | [
{
"content": "Calculate $\\sin 9^\\circ \\sin 45^\\circ \\sin 69^\\circ \\sin 81^\\circ.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{0.6293 \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "33598",
"split": "train"
} |
math | [
{
"content": "How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?\n\n[asy]\nunitsize(0.09inch);\ndraw((0,0)--(15,0)--(15,15)--(0,15)--cycle);\ndraw((5,0)--(5,15));\ndraw((10,0)--(10,15));\ndraw((0,5)--(15,5));\ndr... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "33599",
"split": "train"
} |
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