data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\\sin A - b\\sin B = (\\sqrt{3}a - c)\\sin C$, with $a:b = 2:3$.\n1. Find the value of $\\sin C$.\n2. If $b = 6$, find the area of $\\triangle ABC$. Please output the final answer... | {
"ground_truth": "2\\sqrt{3} + 4\\sqrt{2}",
"style": "rule"
} | {
"index": "31600",
"split": "train"
} |
math | [
{
"content": "Let \\( p, q, r, \\) and \\( s \\) be positive real numbers such that\n\\[\n\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}\np^2+q^2&=&r^2+s^2&=&2512, \\\\\npr&=&qs&=&1225.\n\\end{array}\n\\]\nIf \\( T = p+q+r+s \\), compute the value of \\( \\lfloor T \\rfloo... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "31601",
"split": "train"
} |
math | [
{
"content": "Compute $\\sqrt{(31)(30)(29)(28)+1}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "869",
"style": "rule"
} | {
"index": "31602",
"split": "train"
} |
math | [
{
"content": "For an integer $m\\geq 1$, we consider partitions of a $2^m\\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in su... | {
"ground_truth": "2^{m+2}(m+1)",
"style": "rule"
} | {
"index": "31603",
"split": "train"
} |
math | [
{
"content": "Given $a^{x}=2$ and $a^{y}=3$, find the values of $a^{x+y}$ and $a^{2x-3y}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{27}",
"style": "rule"
} | {
"index": "31604",
"split": "train"
} |
math | [
{
"content": "Compute the prime factorization of \\(1007021035035021007001\\). (You should write your answer in the form \\(p_{1}^{e_{1}} p_{2}^{e_{2}} \\ldots p_{k}^{e_{k}}\\), where \\(p_{1}, \\ldots, p_{k}\\) are distinct prime numbers and \\(e_{1}, \\ldots, e_{k}\\) are positive integers.) Please output the... | {
"ground_truth": "7^7 * 11^7 * 13^7",
"style": "rule"
} | {
"index": "31605",
"split": "train"
} |
math | [
{
"content": "Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the same time. What is t... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "31606",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{1}{x + y} + \\frac{1}{x + z} + \\frac{1}{y + z}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "31607",
"split": "train"
} |
math | [
{
"content": "In triangle \\( A B C \\), angle \\( B \\) equals \\( 45^\\circ \\) and angle \\( C \\) equals \\( 30^\\circ \\). Circles are constructed on the medians \\( B M \\) and \\( C N \\) as diameters, intersecting at points \\( P \\) and \\( Q \\). The chord \\( P Q \\) intersects side \\( B C \\) at po... | {
"ground_truth": "\\frac{1}{\\sqrt{3}}",
"style": "rule"
} | {
"index": "31608",
"split": "train"
} |
math | [
{
"content": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\dots < \\theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\\theta_... | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "31609",
"split": "train"
} |
math | [
{
"content": "Simplify $\\frac{\\sqrt{507}}{\\sqrt{48}}-\\frac{\\sqrt{175}}{\\sqrt{112}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31610",
"split": "train"
} |
math | [
{
"content": "Given that \\( \\cos A + \\cos B + \\cos C = \\sin A + \\sin B + \\sin C = 0 \\), find the value of \\( \\cos^4 A + \\cos^4 B + \\cos^4 C \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{8}",
"style": "rule"
} | {
"index": "31611",
"split": "train"
} |
math | [
{
"content": "In a polar coordinate system, the equation of curve C<sub>1</sub> is given by $\\rho^2 - 2\\rho(\\cos\\theta - 2\\sin\\theta) + 4 = 0$. With the pole as the origin and the polar axis in the direction of the positive x-axis, a Cartesian coordinate system is established using the same unit length. T... | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "31612",
"split": "train"
} |
math | [
{
"content": "If \\( N \\) is the smallest positive integer whose digits have a product of 2700, what is the sum of the digits of \\( N \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "31613",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) be the number of six-digit numbers divisible by 13 but not divisible by 17, and \\( b \\) be the number of six-digit numbers divisible by 17 but not divisible by 13.\n\nFind \\( a - b \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16290",
"style": "rule"
} | {
"index": "31614",
"split": "train"
} |
math | [
{
"content": "In a box, there are 2 one-cent coins, 4 five-cent coins, and 6 ten-cent coins. Six coins are drawn in sequence without replacement, and each coin has an equal probability of being selected. What is the probability that the total value of the drawn coins is at least 50 cents? Please output the fina... | {
"ground_truth": "$\\frac{127}{924}$",
"style": "rule"
} | {
"index": "31615",
"split": "train"
} |
math | [
{
"content": "Altitudes $B E$ and $C F$ of acute triangle $A B C$ intersect at $H$. Suppose that the altitudes of triangle $E H F$ concur on line $B C$. If $A B=3$ and $A C=4$, then $B C^{2}=\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. Please output the final answe... | {
"ground_truth": "33725",
"style": "rule"
} | {
"index": "31616",
"split": "train"
} |
math | [
{
"content": "Express $\\frac{37}{80}$ as a terminating decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.4625",
"style": "rule"
} | {
"index": "31617",
"split": "train"
} |
math | [
{
"content": "There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points) \n(Ⅰ) If the 3 girls must stand together, how many different arrangements are there? \n(Ⅱ) If no two girls are next to each other, how many different arrangements are t... | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "31618",
"split": "train"
} |
math | [
{
"content": "Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain the answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "201320132013",
"style": "rule"
} | {
"index": "31619",
"split": "train"
} |
math | [
{
"content": "Given that $\\dfrac {π}{2} < β < α < \\dfrac {3}{4}π$, $\\cos (α+β)=- \\dfrac {3}{5}$, and $\\sin (α-β)= \\dfrac {5}{13}$, find $\\cos 2β$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\dfrac {56}{65}",
"style": "rule"
} | {
"index": "31620",
"split": "train"
} |
math | [
{
"content": "4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) P... | {
"ground_truth": "\\dfrac{27}{128}",
"style": "rule"
} | {
"index": "31621",
"split": "train"
} |
math | [
{
"content": "For $-25 \\le x \\le 25,$ find the maximum value of $\\sqrt{25 + x} + \\sqrt{25 - x}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31622",
"split": "train"
} |
math | [
{
"content": "Compute $\\dbinom{7}{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "31623",
"split": "train"
} |
math | [
{
"content": "Find the greatest common factor of 5! and 6!. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "31624",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s,$ $t,$ $u$ be positive real numbers such that $p + q + r + s + t + u = 8.$ Find the minimum value of\n\\[\\frac{1}{p} + \\frac{9}{q} + \\frac{16}{r} + \\frac{25}{s} + \\frac{36}{t} + \\frac{49}{u}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "84.5",
"style": "rule"
} | {
"index": "31625",
"split": "train"
} |
math | [
{
"content": "In the isosceles trapezoid $ABCD$, $AD \\parallel BC$, $\\angle B = 45^\\circ$. Point $P$ is on the side $BC$. The area of $\\triangle PAD$ is $\\frac{1}{2}$, and $\\angle APD = 90^\\circ$. Find the minimum value of $AD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31626",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as $\\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integer... | {
"ground_truth": "251",
"style": "rule"
} | {
"index": "31627",
"split": "train"
} |
math | [
{
"content": "A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "540",
"style": "rule"
} | {
"index": "31628",
"split": "train"
} |
math | [
{
"content": "Two non-collinear vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are given, and $\\overrightarrow{a}+2\\overrightarrow{b}$ is perpendicular to $2\\overrightarrow{a}-\\overrightarrow{b}$, $\\overrightarrow{a}-\\overrightarrow{b}$ is perpendicular to $\\overrightarrow{a}$. The cosine of the... | {
"ground_truth": "\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "31629",
"split": "train"
} |
math | [
{
"content": "Two rays with common endpoint $O$ form a $30^\\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\\overline{OB}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31630",
"split": "train"
} |
math | [
{
"content": "The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?\n\nEnter your answer in the form $\\frac{a \\sqrt{b} - c}{d},$ and simplified as us... | {
"ground_truth": "\\frac{3 \\sqrt{2} - 3}{2}",
"style": "rule"
} | {
"index": "31631",
"split": "train"
} |
math | [
{
"content": "Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, \"There are no more than 16 of us in the hut,\" and then added, \"All of us are from the Uh ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31632",
"split": "train"
} |
math | [
{
"content": "If $f(x)$ is a function defined only for $0 \\le x \\le 1$, and $f(x) = ax+b$ for constants $a$ and $b$ where $a < 0$, then what is the range of $f$ in terms of $a$ and $b$? Express your answer in interval notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "[a +b, b]",
"style": "rule"
} | {
"index": "31633",
"split": "train"
} |
math | [
{
"content": "Let $c$ and $d$ be constants. Suppose that the equation \\[\\frac{(x+c)(x+d)(x-7)}{(x+4)^2} = 0\\] has exactly $3$ distinct roots, while the equation \\[\\frac{(x+2c)(x+5)(x+8)}{(x+d)(x-7)} = 0\\] has exactly $1$ distinct root. Compute $100c + d.$ Please output the final answer within \\boxed{}.",... | {
"ground_truth": "408",
"style": "rule"
} | {
"index": "31634",
"split": "train"
} |
math | [
{
"content": "$48n$ is the smallest positive integer that satisfies the following conditions:\n1. $n$ is a multiple of 75;\n2. $n$ has exactly 75 positive divisors (including 1 and itself).\n\nFind $\\frac{n}{75}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "31635",
"split": "train"
} |
math | [
{
"content": "Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price. If the ticket for a... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31636",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\\rho\\cos^2\\theta=2a\\sin\\theta$ ($a>0$). The parameter equ... | {
"ground_truth": "\\frac {3+ \\sqrt {10}}{2}",
"style": "rule"
} | {
"index": "31637",
"split": "train"
} |
math | [
{
"content": "In a taxi, one passenger can sit in the front and three in the back. In how many ways can the four passengers be seated if one of them wants to sit by the window? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31638",
"split": "train"
} |
math | [
{
"content": "The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to\n[asy]\nunitsize(3mm); defaultpen(linewidth(0.8pt));\npath p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);\npath p2=(0,1)--(1,1)--(1,0);\npath p3=(2,0)--(2,1... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "31639",
"split": "train"
} |
math | [
{
"content": "How many prime numbers between 30 and 65 have a prime remainder when divided by 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "31640",
"split": "train"
} |
math | [
{
"content": "Let $p$ be the largest prime with 2010 digits. What is the smallest positive integer $k$ such that $p^2 - k$ is divisible by 12? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "k = 1",
"style": "rule"
} | {
"index": "31641",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=e^{x}(ax+b)-x^{2}-4x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$ is $y=4x+4$.\n\n(1) Find the values of $a$ and $b$;\n\n(2) Determine the intervals of monotonicity for $f(x)$ and find the maximum value of $f(x)$. Please output the fina... | {
"ground_truth": "4(1-e^{-2})",
"style": "rule"
} | {
"index": "31642",
"split": "train"
} |
math | [
{
"content": "1. Solve the inequality $\\frac{2x+1}{3-x} \\geq 1$.\n2. Given that $x > 0$ and $y > 0$, and $x + y = 1$, find the minimum value of $\\frac{4}{x} + \\frac{9}{y}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "31643",
"split": "train"
} |
math | [
{
"content": "Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2,... | {
"ground_truth": "3895",
"style": "rule"
} | {
"index": "31644",
"split": "train"
} |
math | [
{
"content": "Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$. Now suppose that th... | {
"ground_truth": "-0.25",
"style": "rule"
} | {
"index": "31645",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ 2 \\end{pmatrix} \\right\\| = 10.\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 - 2 \\sqrt{5}",
"style": "rule"
} | {
"index": "31646",
"split": "train"
} |
math | [
{
"content": "The minimum distance from a point on the parabola $y=x^2$ to the line $2x-y-10=0$ is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "31647",
"split": "train"
} |
math | [
{
"content": "The value of \\(2 \\frac{1}{10} + 3 \\frac{11}{100} + 4 \\frac{111}{1000}\\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9.321",
"style": "rule"
} | {
"index": "31648",
"split": "train"
} |
math | [
{
"content": "From vertex $A$ of an equilateral triangle $ABC$ , a ray $Ax$ intersects $BC$ at point $D$ . Let $E$ be a point on $Ax$ such that $BA =BE$ . Calculate $\\angle AEC$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31649",
"split": "train"
} |
math | [
{
"content": "If $x$ is a real number and $x^2-7x+6<0$, what are the possible values for $x$? Use interval notation to express your answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,6)",
"style": "rule"
} | {
"index": "31650",
"split": "train"
} |
math | [
{
"content": "One more than the reciprocal of a particular number is $\\frac{7}{3}$. What is the original number expressed as a common fraction? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "31651",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers is 6. The difference of their squares is 12. What is the positive difference of the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "31652",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sum_{j = 0}^\\infty \\sum_{k = 0}^\\infty 2^{-3k - j - (k + j)^2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "31653",
"split": "train"
} |
math | [
{
"content": "Let \\( M \\) be a set of \\( n \\) points in the plane such that:\n1. There are 7 points in \\( M \\) that form the vertices of a convex heptagon.\n2. For any 5 points in \\( M \\), if these 5 points form a convex pentagon, then the interior of this convex pentagon contains at least one point fro... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31654",
"split": "train"
} |
math | [
{
"content": "Let the number of elements in the set \\( S \\) be denoted by \\( |S| \\), and the number of subsets of the set \\( S \\) be denoted by \\( n(S) \\). Given three non-empty finite sets \\( A, B, C \\) that satisfy the following conditions:\n$$\n\\begin{array}{l}\n|A| = |B| = 2019, \\\\\nn(A) + n(B)... | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": "31655",
"split": "train"
} |
math | [
{
"content": "Given that $\\{a_n\\}$ is a geometric sequence, and $S_n$ is the sum of its first $n$ terms. If $a_2 \\cdot a_3 = 2a_1$, and the arithmetic mean of $a_4$ and $2a_7$ is $\\frac{5}{4}$, then $S_6 = \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{63}{2}",
"style": "rule"
} | {
"index": "31656",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = 2^x + 3^x$ . For how many integers $1 \\leq n \\leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \\dots, f(n-1)$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31657",
"split": "train"
} |
math | [
{
"content": "A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\\geq1$, we have\n$$a_{n+1}=\\frac{n-1}{n+1}a_n-\\frac{n-2}{n^2+n}a_{n-1}.$$\nCalculate the value of $\\frac{a_1}{a_2}-\\frac{a_2}{a_3}+\\frac{a_3}{a_4}-\\frac{a_4}{a_5}+\\ldots+\\frac{a_{2013}}{a_{2014}}-\\frac{a_{... | {
"ground_truth": "3021",
"style": "rule"
} | {
"index": "31658",
"split": "train"
} |
math | [
{
"content": "Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "31659",
"split": "train"
} |
math | [
{
"content": "Given a right circular cone with a base radius of \\(1 \\, \\text{cm}\\) and a slant height of \\(3 \\, \\text{cm}\\), point \\(P\\) is on the circumference of the base. Determine the shortest distance from the vertex \\(V\\) of the cone to the shortest path from \\(P\\) back to \\(P\\). Please ou... | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "31660",
"split": "train"
} |
math | [
{
"content": "There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "197",
"style": "rule"
} | {
"index": "31661",
"split": "train"
} |
math | [
{
"content": "If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?\n[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle... | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "31662",
"split": "train"
} |
math | [
{
"content": "Suppose the product $\\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\ldots\\cdot \\dfrac{c}{d} = 16$, find the sum of $c$ and $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "31663",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any oth... | {
"ground_truth": "\\frac{\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "31664",
"split": "train"
} |
math | [
{
"content": "Given points $A(2,-5,1)$, $B(2,-2,4)$, $C(1,-4,1)$, the angle between vectors $\\overrightarrow{AB}$ and $\\overrightarrow{AC}$ is equal to ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "31665",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "31666",
"split": "train"
} |
math | [
{
"content": "Triangle $XYZ$ has vertices $X(1, 9)$, $Y(3, 1)$, and $Z(9, 1)$. A line through $Y$ cuts the area of $\\triangle XYZ$ in half. Find the sum of the slope and the $y$-intercept of this line. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "31667",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of 10-tuples $(a_0, a_1, \\dots, a_9),$ where each entry is 0 or 1, so $S$ contains $2^{10}$ 10-tuples. For each 10-tuple $s = (a_0, a_1, \\dots, a_9)$ in $S,$ let $p_s(x)$ be the polynomial of degree at most 9 such that\n\\[p_s(n) = a_n\\]for $0 \\le n \\le 9.$ For example, $... | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "31668",
"split": "train"
} |
math | [
{
"content": "The reciprocal of $-2024$ is ______; $7.8963\\approx$ ______ (rounded to the nearest hundredth). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.90",
"style": "rule"
} | {
"index": "31669",
"split": "train"
} |
math | [
{
"content": "Say that an integer $B$ is yummy if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest yummy integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2022",
"style": "rule"
} | {
"index": "31670",
"split": "train"
} |
math | [
{
"content": "Find the greatest common divisor of 4,004 and 10,010. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "31671",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, it is known that $\\overrightarrow {AB}\\cdot \\overrightarrow {AC}=9$ and $\\overrightarrow {AB}\\cdot \\overrightarrow {BC}=-16$. Find:\n 1. The value of $AB$;\n 2. The value of $\\frac {sin(A-B)}{sinC}$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\frac{7}{25}",
"style": "rule"
} | {
"index": "31672",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = 2\\sin\\omega x\\cos\\omega x - 2\\sqrt{3}\\sin^2\\omega x + \\sqrt{3}(\\_\\omega (\\_ > 0)), the lines $x = \\_x\\_{1}$ and $x = \\_x\\_{2}$ are any two symmetry axes of the graph of the function $y = f(x)$, and the minimum value of $|x\\_1 - x\\_2|$ is $\\frac{\\pi}{2}... | {
"ground_truth": "-\\frac{7}{9}",
"style": "rule"
} | {
"index": "31673",
"split": "train"
} |
math | [
{
"content": "A circle is circumscribed around $ABCD$ as follows: [asy]\npair pA, pB, pC, pD, pO;\npO = (0, 0);\npA = pO + dir(-40);\npB = pO + dir(40);\npC = pO + dir(130);\npD = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pA--pD--pC--pA);\nlabel(\"$A$\", pA, SE);\nlabel(\"$B$\", pB, NE);\nlabel(\"$C$\", pC, N... | {
"ground_truth": "110^\\circ",
"style": "rule"
} | {
"index": "31674",
"split": "train"
} |
math | [
{
"content": "Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{912}{119}",
"style": "rule"
} | {
"index": "31675",
"split": "train"
} |
math | [
{
"content": "Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 + 6\\sqrt{2}",
"style": "rule"
} | {
"index": "31676",
"split": "train"
} |
math | [
{
"content": "In how many ways can 6 distinct beads be placed on a bracelet? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31677",
"split": "train"
} |
math | [
{
"content": "Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $4$ per gallon. How many miles can Margie drive on $20$ worth of gas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "31678",
"split": "train"
} |
math | [
{
"content": "The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{123}{40}",
"style": "rule"
} | {
"index": "31679",
"split": "train"
} |
math | [
{
"content": "Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "31680",
"split": "train"
} |
math | [
{
"content": "Given point O is the circumcenter of triangle ABC, and |BA|=2, |BC|=6, calculate the value of the dot product of the vectors BO and AC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31681",
"split": "train"
} |
math | [
{
"content": "Let vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ be unit vectors, and $|\\overrightarrow{a}+\\overrightarrow{b}|=1$. Determine the angle $\\theta$ between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "31682",
"split": "train"
} |
math | [
{
"content": "The volume of a given sphere is \\(72\\pi\\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \\(\\pi\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\pi 2^{2/3}",
"style": "rule"
} | {
"index": "31683",
"split": "train"
} |
math | [
{
"content": "Let \\( \\left\\lfloor A \\right\\rfloor \\) denote the greatest integer less than or equal to \\( A \\). Given \\( A = 50 + 19 \\sqrt{7} \\), find the value of \\( A^2 - A \\left\\lfloor A \\right\\rfloor \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "31684",
"split": "train"
} |
math | [
{
"content": "Find all functions $f: R \\to R$ such that, for any $x, y \\in R$:\n$f\\left( f\\left( x \\right)-y \\right)\\cdot f\\left( x+f\\left( y \\right) \\right)={{x}^{2}}-{{y}^{2}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x",
"style": "rule"
} | {
"index": "31685",
"split": "train"
} |
math | [
{
"content": "The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\nare drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side $2/\\sqrt{3}.\\,$ How many such triangles are formed? Please output t... | {
"ground_truth": "660",
"style": "rule"
} | {
"index": "31686",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31687",
"split": "train"
} |
math | [
{
"content": "The surface area ratio of an equilateral cone to a sphere is 3:1. How many times larger is the volume of the equilateral cone compared to the volume of the sphere? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "31688",
"split": "train"
} |
math | [
{
"content": "How many solutions does the equation \\[\\frac{(x-1)(x-2)(x-3)\\dotsm(x-100)}{(x-1^2)(x-2^2)(x-3^2)\\dotsm(x-100^2)} = 0\\]have for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "31689",
"split": "train"
} |
math | [
{
"content": "The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "31690",
"split": "train"
} |
math | [
{
"content": "The cells of a $5 \\times 7$ table are filled with numbers so that in each $2 \\times 3$ rectangle (either vertical or horizontal) the sum of the numbers is zero. By paying 100 rubles, you can choose any cell and find out what number is written in it. What is the smallest amount of rubles needed t... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "31691",
"split": "train"
} |
math | [
{
"content": "In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6/5",
"style": "rule"
} | {
"index": "31692",
"split": "train"
} |
math | [
{
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$ , $4$ , and $5$ . What is the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{9}{\\pi^2}\\left(\\sqrt{3}+3\\right)$",
"style": "rule"
} | {
"index": "31693",
"split": "train"
} |
math | [
{
"content": "In rectangle ABCD, AB=30 and BC=15. Let F be a point on AB such that ∠BCF=30°. Find CF. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31694",
"split": "train"
} |
math | [
{
"content": "23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "253",
"style": "rule"
} | {
"index": "31695",
"split": "train"
} |
math | [
{
"content": "The length of a rectangular playground exceeds twice its width by 25 feet, and the perimeter of the playground is 650 feet. What is the area of the playground in square feet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22,500",
"style": "rule"
} | {
"index": "31696",
"split": "train"
} |
math | [
{
"content": "Eight balls are randomly and independently painted either black or white with equal probability. What is the probability that each ball is painted a color such that it is different from the color of at least half of the other 7 balls?\nA) $\\frac{21}{128}$\nB) $\\frac{28}{128}$\nC) $\\frac{35}{128... | {
"ground_truth": "\\frac{35}{128}",
"style": "rule"
} | {
"index": "31697",
"split": "train"
} |
math | [
{
"content": "Ninety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the tower. How many different tower heights can be achieved... | {
"ground_truth": "465",
"style": "rule"
} | {
"index": "31698",
"split": "train"
} |
math | [
{
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Please output the final answer within \\boxed{... | {
"ground_truth": "12\\%",
"style": "rule"
} | {
"index": "31699",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.