data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let \\(x\\) and \\(y\\) be real numbers such that \\(2(x^3 + y^3) = x + y\\). Find the maximum value of \\(x - y\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "1600",
"split": "train"
} |
math | [
{
"content": "Determine the value of the following expressions:\n$(1)(2 \\frac{7}{9})^{0.5}+0.1^{-2}+(2 \\frac{10}{27})\\,^{- \\frac{2}{3}}-3π^{0}+ \\frac{37}{48}$;\n$(2)(-3 \\frac{3}{8})\\,^{- \\frac{2}{3}}+(0.002)\\,^{- \\frac{1}{2}}-10( \\sqrt{5}-2)^{-1}+( \\sqrt{2}- \\sqrt{3})^{0}$. Please output the final ... | {
"ground_truth": "- \\frac{167}{9}",
"style": "rule"
} | {
"index": "1601",
"split": "train"
} |
math | [
{
"content": "What is the measure of an angle, in degrees, if its supplement is six times its complement? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72^{\\circ}",
"style": "rule"
} | {
"index": "1602",
"split": "train"
} |
math | [
{
"content": "$2\\left(1-\\frac{1}{2}\\right) + 3\\left(1-\\frac{1}{3}\\right) + 4\\left(1-\\frac{1}{4}\\right) + \\cdots + 10\\left(1-\\frac{1}{10}\\right)=$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "1603",
"split": "train"
} |
math | [
{
"content": "A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.2",
"style": "rule"
} | {
"index": "1604",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) + \\left( \\frac{y}{x} + \\frac{z}{y} + \\frac{x}{z} \\right) = 8.\\]Find the minimum value of\n\\[\\left( \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} \\right) \\left( \\frac{y}{x} +... | {
"ground_truth": "22 \\sqrt{11} - 57",
"style": "rule"
} | {
"index": "1605",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer \\(n\\) such that \\(\\frac{n}{n+51}\\) is equal to a terminating decimal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "74",
"style": "rule"
} | {
"index": "1606",
"split": "train"
} |
math | [
{
"content": "The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.\n\n$$\nAB = 18, EO = 7, AE = 2 BE, R = ?\n$$ Please output the final answer within \\... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "1607",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40400",
"style": "rule"
} | {
"index": "1608",
"split": "train"
} |
math | [
{
"content": "In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by F... | {
"ground_truth": "0.665",
"style": "rule"
} | {
"index": "1609",
"split": "train"
} |
math | [
{
"content": "For any set \\( S \\), let \\( |S| \\) represent the number of elements in set \\( S \\) and let \\( n(S) \\) represent the number of subsets of set \\( S \\). If \\( A \\), \\( B \\), and \\( C \\) are three finite sets such that:\n\n(1) \\( |A|=|B|=2016 \\);\n(2) \\( n(A) + n(B) + n(C) = n(A \\c... | {
"ground_truth": "2015",
"style": "rule"
} | {
"index": "1610",
"split": "train"
} |
math | [
{
"content": "Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlight and Boris from th... | {
"ground_truth": "163",
"style": "rule"
} | {
"index": "1611",
"split": "train"
} |
math | [
{
"content": "Let $\\pi$ be a permutation of the numbers from 2 through 2012. Find the largest possible value of $\\log _{2} \\pi(2) \\cdot \\log _{3} \\pi(3) \\cdots \\log _{2012} \\pi(2012)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "1612",
"split": "train"
} |
math | [
{
"content": "For a natural number \\( N \\), if at least five of the natural numbers from 1 to 9 can divide \\( N \\) evenly, then \\( N \\) is called a \"Five-Divisible Number.\" Find the smallest \"Five-Divisible Number\" that is greater than 2000. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "1613",
"split": "train"
} |
math | [
{
"content": "In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ . Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "1614",
"split": "train"
} |
math | [
{
"content": "What is the eighth term in the arithmetic sequence $\\frac 23, 1, \\frac 43, \\dots$? Express your answer in simplest form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1615",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a\\_n\\}$ satisfying $a\\_1=2$, $a\\_2=6$, and $a_{n+2} - 2a_{n+1} + a\\_n = 2$, find the value of $\\left\\lfloor \\frac{2017}{a\\_1} + \\frac{2017}{a\\_2} + \\ldots + \\frac{2017}{a_{2017}} \\right\\rfloor$, where $\\lfloor x \\rfloor$ represents the greatest integer not g... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "1616",
"split": "train"
} |
math | [
{
"content": "The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and ent... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "1617",
"split": "train"
} |
math | [
{
"content": "Side $\\overline{AB}$ of $\\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "1618",
"split": "train"
} |
math | [
{
"content": "Let $x, y, n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^{n}-y^{n}=2^{100}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "1619",
"split": "train"
} |
math | [
{
"content": "Point O is the circumcenter of acute triangle ABC, with AB=6, AC=2. Determine the value of $\\overrightarrow {AO}\\cdot ( \\overrightarrow {AB}+ \\overrightarrow {AC})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1620",
"split": "train"
} |
math | [
{
"content": "Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "1621",
"split": "train"
} |
math | [
{
"content": "When a granary records the arrival of 30 tons of grain as \"+30\", determine the meaning of \"-30\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-30",
"style": "rule"
} | {
"index": "1622",
"split": "train"
} |
math | [
{
"content": "Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n (a) the number $n$ is divisible by $5,$\n (b) the first and last digits of $n$ are equal, and\n (c) the sum of the digits of $n$ is divisible by $5.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "1623",
"split": "train"
} |
math | [
{
"content": "Let \\( r_{1}, r_{2}, \\cdots, r_{20} \\) be the roots of the polynomial \\( x^{20}-7x^{3}+1 \\). If \\(\\frac{1}{r_{1}^{2}+1}+\\frac{1}{r_{2}^{2}+1}+\\cdots+\\frac{1}{r_{20}^{2}+1} \\) can be expressed in the form \\( \\frac{m}{n} \\) (with \\( m \\) and \\( n \\) coprime), find the value of \\( ... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "1624",
"split": "train"
} |
math | [
{
"content": "Let \\( D \\) be the midpoint of the hypotenuse \\( BC \\) of the right triangle \\( ABC \\). On the leg \\( AC \\), a point \\( M \\) is chosen such that \\(\\angle AMB = \\angle CMD\\). Find the ratio \\(\\frac{AM}{MC}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "1:2",
"style": "rule"
} | {
"index": "1625",
"split": "train"
} |
math | [
{
"content": "Suppose \\( a \\) is an integer. A sequence \\( x_1, x_2, x_3, x_4, \\ldots \\) is constructed with:\n\n- \\( x_1 = a \\),\n- \\( x_{2k} = 2x_{2k-1} \\) for every integer \\( k \\geq 1 \\),\n- \\( x_{2k+1} = x_{2k} - 1 \\) for every integer \\( k \\geq 1 \\).\n\nFor example, if \\( a = 2 \\), then... | {
"ground_truth": "1409",
"style": "rule"
} | {
"index": "1626",
"split": "train"
} |
math | [
{
"content": "How many positive three-digit integers less than 700 have at least two digits that are the same and none of the digits can be zero? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "1627",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "1628",
"split": "train"
} |
math | [
{
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that t... | {
"ground_truth": "405",
"style": "rule"
} | {
"index": "1629",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $\\sqrt{\\frac{\\sqrt{81} + \\sqrt{81}}{2}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1630",
"split": "train"
} |
math | [
{
"content": "Let $\\theta$ be an acute angle, and let\n\\[\\sin \\frac{\\theta}{2} = \\sqrt{\\frac{x - 1}{2x}}.\\]Express $\\tan \\theta$ in terms of $x.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{x^2 - 1}",
"style": "rule"
} | {
"index": "1631",
"split": "train"
} |
math | [
{
"content": "The fare in Moscow with the \"Troika\" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles? Please output the final answer within \\boxed{}."... | {
"ground_truth": "157",
"style": "rule"
} | {
"index": "1632",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\\cos B - b\\cos A = c$, and $C = \\frac{π}{5}$, calculate the value of $\\angle B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{10}",
"style": "rule"
} | {
"index": "1633",
"split": "train"
} |
math | [
{
"content": "The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of... | {
"ground_truth": "790",
"style": "rule"
} | {
"index": "1634",
"split": "train"
} |
math | [
{
"content": "Find the sum of $111_4+323_4+132_4$. Express your answer in base $4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1232_4",
"style": "rule"
} | {
"index": "1635",
"split": "train"
} |
math | [
{
"content": "For real numbers \\( x \\) and \\( y \\) such that \\( x + y = 1 \\), determine the maximum value of the expression \\( A(x, y) = x^4 y + x y^4 + x^3 y + x y^3 + x^2 y + x y^2 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "1636",
"split": "train"
} |
math | [
{
"content": "Find $\\sec \\frac{5 \\pi}{3}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "1637",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = 2\\sin x\\cos x - 2\\sin^2 x$ \n(1) Find the smallest positive period of the function $f(x)$. \n(2) Let $\\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$, respectively, and satisfy $f(A) = 0$, $c = 1$, $b = \\sqrt{2}$. Find the area of $\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "1638",
"split": "train"
} |
math | [
{
"content": "Find the projection of the vector $\\begin{pmatrix} 4 \\\\ 5 \\end{pmatrix}$ onto the vector $\\begin{pmatrix} 2 \\\\ 0 \\end{pmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}",
"style": "rule"
} | {
"index": "1639",
"split": "train"
} |
math | [
{
"content": "Jarris the triangle is playing in the \\((x, y)\\) plane. Let his maximum \\(y\\) coordinate be \\(k\\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \\(x\\)-axis, find the minimum possible value of \\(k\\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "24/5",
"style": "rule"
} | {
"index": "1640",
"split": "train"
} |
math | [
{
"content": "The graph of the polynomial\n\n$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$\n\nhas five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?\n\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\textbf{(D)}\\ d \\qquad... | {
"ground_truth": "\\text{(D)}",
"style": "rule"
} | {
"index": "1641",
"split": "train"
} |
math | [
{
"content": "A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?\n- **A)** $\\sqrt{84}$\n- **B)** $\\sqrt{85}$\n- **C)** $\\sqrt{8... | {
"ground_truth": "\\sqrt{91}",
"style": "rule"
} | {
"index": "1642",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$, leg $\\overline{BC}$ is perpendicular to bases $\\overline{AB}$ and $\\overline{CD}$, and diagonals $\\overline{AC}$ and $\\overline{BD}$ are perpendicular. Given that $AB=\\sqrt{11}$ and $AD=\\sqrt{1001}$, find $BC^2$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "1643",
"split": "train"
} |
math | [
{
"content": "Distinct points $A, B, C, D$ are given such that triangles $A B C$ and $A B D$ are equilateral and both are of side length 10 . Point $E$ lies inside triangle $A B C$ such that $E A=8$ and $E B=3$, and point $F$ lies inside triangle $A B D$ such that $F D=8$ and $F B=3$. What is the area of quadri... | {
"ground_truth": "\\frac{91 \\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "1644",
"split": "train"
} |
math | [
{
"content": "Find $k$ where $2^k$ is the largest power of $2$ that divides the product \\[2008\\cdot 2009\\cdot 2010\\cdots 4014.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2007",
"style": "rule"
} | {
"index": "1645",
"split": "train"
} |
math | [
{
"content": "What number can be added to both the numerator and denominator of $\\frac{3}{5}$ so that the resulting fraction will be equivalent to $\\frac{5}{6}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "1646",
"split": "train"
} |
math | [
{
"content": "Without using any tables, find the exact value of the product:\n\\[ P = \\cos \\frac{\\pi}{15} \\cos \\frac{2\\pi}{15} \\cos \\frac{3\\pi}{15} \\cos \\frac{4\\pi}{15} \\cos \\frac{5\\pi}{15} \\cos \\frac{6\\pi}{15} \\cos \\frac{7\\pi}{15}. \\] Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1/128",
"style": "rule"
} | {
"index": "1647",
"split": "train"
} |
math | [
{
"content": "Suppose two distinct numbers are chosen from between 6 and 20, inclusive. What is the probability that their product is even, or exactly one of the numbers is a prime? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{94}{105}",
"style": "rule"
} | {
"index": "1648",
"split": "train"
} |
math | [
{
"content": "Calculate the argument of the sum:\n\\[ e^{5\\pi i/36} + e^{11\\pi i/36} + e^{17\\pi i/36} + e^{23\\pi i/36} + e^{29\\pi i/36} \\]\nin the form $r e^{i \\theta}$, where $0 \\le \\theta < 2\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17\\pi}{36}",
"style": "rule"
} | {
"index": "1649",
"split": "train"
} |
math | [
{
"content": "For any number $y$, define the operations $\\&y = 2(7-y)$ and $\\&y = 2(y-7)$. What is the value of $\\&(-13\\&)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "1650",
"split": "train"
} |
math | [
{
"content": "The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "1651",
"split": "train"
} |
math | [
{
"content": "Given points A, B, and C on the surface of a sphere O, the height of the tetrahedron O-ABC is 2√2 and ∠ABC=60°, with AB=2 and BC=4. Find the surface area of the sphere O. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48\\pi",
"style": "rule"
} | {
"index": "1652",
"split": "train"
} |
math | [
{
"content": "Find the number of $x$-intercepts on the graph of $y = \\sin \\frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2865",
"style": "rule"
} | {
"index": "1653",
"split": "train"
} |
math | [
{
"content": "Consider the largest solution to the equation \\[\\log_{10x^2} 10 + \\log_{100x^3} 10 = -2.\\]Find the value of $\\frac{1}{x^{12}},$ writing your answer in decimal representation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000000",
"style": "rule"
} | {
"index": "1654",
"split": "train"
} |
math | [
{
"content": "Factor the following expression: $45x+30$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15(3x+2)",
"style": "rule"
} | {
"index": "1655",
"split": "train"
} |
math | [
{
"content": "The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "1656",
"split": "train"
} |
math | [
{
"content": "Place the terms of the sequence $\\{2n-1\\}$ ($n\\in\\mathbb{N}^+$) into brackets according to the following pattern: the first bracket contains the first term, the second bracket contains the second and third terms, the third bracket contains the fourth, fifth, and sixth terms, the fourth bracket... | {
"ground_truth": "1251",
"style": "rule"
} | {
"index": "1657",
"split": "train"
} |
math | [
{
"content": "Let $x = \\sqrt{\\frac{\\sqrt{53}}{2} + \\frac{3}{2}}.$ There exist unique positive integers $a,$ $b,$ $c$ such that\n\\[x^{100} = 2x^{98} + 14x^{96} + 11x^{94} - x^{50} + ax^{46} + bx^{44} + cx^{40}.\\]Find $a + b + c.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "157",
"style": "rule"
} | {
"index": "1658",
"split": "train"
} |
math | [
{
"content": "Draw a square of side length 1. Connect its sides' midpoints to form a second square. Connect the midpoints of the sides of the second square to form a third square. Connect the midpoints of the sides of the third square to form a fourth square. And so forth. What is the sum of the areas of all th... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "1659",
"split": "train"
} |
math | [
{
"content": "Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20201",
"style": "rule"
} | {
"index": "1660",
"split": "train"
} |
math | [
{
"content": "Find the coefficient of \\(x^5\\) in the expansion of \\(\\left(1+2x+3x^2+4x^3\\right)^5\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1772",
"style": "rule"
} | {
"index": "1661",
"split": "train"
} |
math | [
{
"content": "Given a triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \\frac{a^{2}+b^{2}-c^{2}}{4}$ and $\\sin A= \\frac{3}{5}$.\n\n1. Find $\\sin B$.\n2. If side $c=5$, find the area of $\\triangle ABC$, denoted ... | {
"ground_truth": "\\frac{21}{2}",
"style": "rule"
} | {
"index": "1662",
"split": "train"
} |
math | [
{
"content": "In $\\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\\sqrt{2}$, and $C=\\frac{\\pi}{4}$, find the area of $\\Delta ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3} +1",
"style": "rule"
} | {
"index": "1663",
"split": "train"
} |
math | [
{
"content": "In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100,000",
"style": "rule"
} | {
"index": "1664",
"split": "train"
} |
math | [
{
"content": "In a regular tetrahedron \\( ABCD \\) with side length \\( \\sqrt{2} \\), it is known that \\( \\overrightarrow{AP} = \\frac{1}{2} \\overrightarrow{AB} \\), \\( \\overrightarrow{AQ} = \\frac{1}{3} \\overrightarrow{AC} \\), and \\( \\overrightarrow{AR} = \\frac{1}{4} \\overrightarrow{AD} \\). If po... | {
"ground_truth": "\\frac{1}{36}",
"style": "rule"
} | {
"index": "1665",
"split": "train"
} |
math | [
{
"content": "Yangyang leaves home for school. If she walks 60 meters per minute, she arrives at school at 6:53. If she walks 75 meters per minute, she arrives at school at 6:45. What time does Yangyang leave home? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6:13",
"style": "rule"
} | {
"index": "1666",
"split": "train"
} |
math | [
{
"content": "In 2003, the average monthly rainfall in Mathborough was $41.5\\text{ mm.}$ In 2004, the average monthly rainfall in Mathborough was $2\\text{ mm}$ more than in 2003. What was the total amount of rain that fell in Mathborough in 2004? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "522",
"style": "rule"
} | {
"index": "1667",
"split": "train"
} |
math | [
{
"content": "For $\\pi \\le \\theta < 2\\pi$, let\n\\begin{align*} P &= \\frac12\\cos\\theta - \\frac14\\sin 2\\theta - \\frac18\\cos 3\\theta + \\frac{1}{16}\\sin 4\\theta + \\frac{1}{32} \\cos 5\\theta - \\frac{1}{64} \\sin 6\\theta - \\frac{1}{128} \\cos 7\\theta + \\cdots \\end{align*}\n and\n\\begin{align... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1668",
"split": "train"
} |
math | [
{
"content": "Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exa... | {
"ground_truth": "32.5",
"style": "rule"
} | {
"index": "1669",
"split": "train"
} |
math | [
{
"content": "Select 4 students from 5 female and 4 male students to participate in a speech competition. \n(1) If 2 male and 2 female students are to be selected, how many different selections are there? \n(2) If at least 1 male and 1 female student must be selected, and male student A and female student B c... | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "1670",
"split": "train"
} |
math | [
{
"content": "What is the sum of all of the possibilities for Sam's number if Sam thinks of a 5-digit number, Sam's friend Sally tries to guess his number, Sam writes the number of matching digits beside each of Sally's guesses, and a digit is considered \"matching\" when it is the correct digit in the correct ... | {
"ground_truth": "526758",
"style": "rule"
} | {
"index": "1671",
"split": "train"
} |
math | [
{
"content": "Given the set \\( S = \\{1, 2, \\cdots, 2005\\} \\), and a subset \\( A \\subseteq S \\) such that the sum of any two numbers in \\( A \\) is not divisible by 117, determine the maximum value of \\( |A| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1003",
"style": "rule"
} | {
"index": "1672",
"split": "train"
} |
math | [
{
"content": "Given six thin wooden sticks, the two longer ones are $\\sqrt{3}a$ and $\\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges. Please output the final answer within \\boxe... | {
"ground_truth": "\\frac{\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "1673",
"split": "train"
} |
math | [
{
"content": "For every $n$ the sum of $n$ terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6r - 1",
"style": "rule"
} | {
"index": "1674",
"split": "train"
} |
math | [
{
"content": "How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "1675",
"split": "train"
} |
math | [
{
"content": "Let $a_{n+1} = \\frac{4}{7}a_n + \\frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\\lim_{n \\to \\infty} a_n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.7",
"style": "rule"
} | {
"index": "1676",
"split": "train"
} |
math | [
{
"content": "What integer $n$ satisfies $0 \\leq n < 201$ and $$200n \\equiv 144 \\pmod {101}~?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "1677",
"split": "train"
} |
math | [
{
"content": "In the game of preference, each of the three players is dealt 10 cards, and two cards are placed in the kitty. How many different arrangements are possible in this game? (Consider possible distributions without accounting for which 10 cards go to each specific player.) Please output the final answ... | {
"ground_truth": "\\frac{32!}{(10!)^3 \\cdot 2! \\cdot 3!}",
"style": "rule"
} | {
"index": "1678",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid \\(ABCD\\), \\(AB\\) is parallel to \\(CD\\), \\(AB = 6\\), \\(CD = 14\\), \\(\\angle AEC\\) is a right angle, and \\(CE = CB\\). What is \\(AE^2\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "1679",
"split": "train"
} |
math | [
{
"content": "Eight congruent copies of the parabola \\( y = x^2 \\) are arranged symmetrically around a circle such that each vertex is tangent to the circle, and each parabola is tangent to its two neighbors. Find the radius of the circle. Assume that one of the tangents to the parabolas corresponds to the li... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "1680",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=\\frac{\\cos 2x-(a+2)\\cos x+a+1}{\\sin x}.$ For constant $a$ such that $\\lim_{x\\rightarrow 0} \\frac{f(x)}{x}=\\frac 12$ , evaluate $\\int_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}} \\frac{1}{f(x)}dx.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "1681",
"split": "train"
} |
math | [
{
"content": "Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 399, what is the sum of $a$ and $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "1682",
"split": "train"
} |
math | [
{
"content": "$A B C$ is a right triangle with $\\angle A=30^{\\circ}$ and circumcircle $O$. Circles $\\omega_{1}, \\omega_{2}$, and $\\omega_{3}$ lie outside $A B C$ and are tangent to $O$ at $T_{1}, T_{2}$, and $T_{3}$ respectively and to $A B, B C$, and $C A$ at $S_{1}, S_{2}$, and $S_{3}$, respectively. Lin... | {
"ground_truth": "\\frac{\\sqrt{3}+1}{2}",
"style": "rule"
} | {
"index": "1683",
"split": "train"
} |
math | [
{
"content": "For $y=\\frac{x+2}{5x-7}$, at what $x$-value is there a vertical asymptote? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{5}",
"style": "rule"
} | {
"index": "1684",
"split": "train"
} |
math | [
{
"content": "Each cell of a $3 \\times 3$ grid is labeled with a digit in the set $\\{1,2,3,4,5\\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2664",
"style": "rule"
} | {
"index": "1685",
"split": "train"
} |
math | [
{
"content": "Fill the numbers 1 to 6 into the six boxes in the image. The smallest result you can get is ______ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "342",
"style": "rule"
} | {
"index": "1686",
"split": "train"
} |
math | [
{
"content": "In a labor and technical competition among five students: A, B, C, D, and E, the rankings from first to fifth place were determined. When A and B asked about their results, the respondent told A, \"Unfortunately, both you and B did not win the championship\"; and told B, \"You certainly are not th... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1687",
"split": "train"
} |
math | [
{
"content": "When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{72}",
"style": "rule"
} | {
"index": "1688",
"split": "train"
} |
math | [
{
"content": "$A$ and $B$ are on a circle of radius $20$ centered at $C$ , and $\\angle ACB = 60^\\circ$ . $D$ is chosen so that $D$ is also on the circle, $\\angle ACD = 160^\\circ$ , and $\\angle DCB = 100^\\circ$ . Let $E$ be the intersection of lines $AC$ and $BD$ . What is $DE$ ? Please ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1689",
"split": "train"
} |
math | [
{
"content": "In the binomial expansion of $(x-1)^n$ ($n\\in\\mathbb{N}_{+}$), if only the 5th binomial coefficient is the largest, find the constant term in the binomial expansion of $(2\\sqrt{x}-\\frac{1}{\\sqrt{x}})^n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1120",
"style": "rule"
} | {
"index": "1690",
"split": "train"
} |
math | [
{
"content": "Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\\frac{(x-a)(x-b)}{x}=\\frac{{x}^{2}-(a+b)x+ab}{x}=x+\\frac{ab}{x}-\\left(a+b\\right)$, the solu... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1691",
"split": "train"
} |
math | [
{
"content": "In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\frac{2}{3}",
"style": "rule"
} | {
"index": "1692",
"split": "train"
} |
math | [
{
"content": "A circle is inscribed in quadrilateral $EFGH$, tangent to $\\overline{EF}$ at $R$ and to $\\overline{GH}$ at $S$. Given that $ER=24$, $RF=31$, $GS=40$, and $SH=29$, find the square of the radius of the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "945",
"style": "rule"
} | {
"index": "1693",
"split": "train"
} |
math | [
{
"content": "Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "1694",
"split": "train"
} |
math | [
{
"content": "If $x-y=15$ and $xy=4$, what is the value of $x^2+y^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "1695",
"split": "train"
} |
math | [
{
"content": "In a kennel with 60 dogs, 9 dogs like watermelon, 48 dogs like salmon, and 5 like both salmon and watermelon. How many dogs in the kennel will not eat either? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "1696",
"split": "train"
} |
math | [
{
"content": "Jenny places a total of 18 red Easter eggs in several green baskets and a total of 24 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 4 eggs in each basket. How many eggs did Jenny put in each basket? Please output the final answer wit... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "1697",
"split": "train"
} |
math | [
{
"content": "Darren has borrowed $100$ clams from Ethan at a $10\\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments in that time period... | {
"ground_truth": "20\\text{ days}",
"style": "rule"
} | {
"index": "1698",
"split": "train"
} |
math | [
{
"content": "Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\\overline{AB}$, and $OA=2\\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\\overline{EF}$ is a common tangent. What is the area of the shaded regi... | {
"ground_truth": "8\\sqrt{2}-4-\\pi",
"style": "rule"
} | {
"index": "1699",
"split": "train"
} |
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