data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Misha is the 50th best as well as the 50th worst student in her grade. How many students are in Misha's grade? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "5100",
"split": "train"
} |
math | [
{
"content": "Find the total number of solutions to the equation $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$ where $a, b, c$ are positive integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1755",
"style": "rule"
} | {
"index": "5101",
"split": "train"
} |
math | [
{
"content": "At an observation station $C$, the distances to two lighthouses $A$ and $B$ are $300$ meters and $500$ meters, respectively. Lighthouse $A$ is observed at $30^{\\circ}$ north by east from station $C$, and lighthouse $B$ is due west of station $C$. Calculate the distance between the two lighthouses... | {
"ground_truth": "700",
"style": "rule"
} | {
"index": "5102",
"split": "train"
} |
math | [
{
"content": "Triangle $PQR$ has sides $\\overline{PQ}$, $\\overline{QR}$, and $\\overline{RP}$ of length 47, 14, and 50, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle PQR$ and let $S$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline{RP}$ that is not on... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5103",
"split": "train"
} |
math | [
{
"content": "Determine the value of\n\\[2002 + \\frac{1}{2} \\left( 2001 + \\frac{1}{2} \\left( 2000 + \\dots + \\frac{1}{2} \\left( 3 + \\frac{1}{2} \\cdot 2 \\right) \\right) \\dotsb \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4002",
"style": "rule"
} | {
"index": "5104",
"split": "train"
} |
math | [
{
"content": "Three Triangles: Within triangle \\(ABC\\), a random point \\(M\\) is chosen. What is the probability that the area of one of the triangles \\(ABM, BCM,\\) and \\(CAM\\) will be greater than the sum of the areas of the other two? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "0.75",
"style": "rule"
} | {
"index": "5105",
"split": "train"
} |
math | [
{
"content": "The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "5106",
"split": "train"
} |
math | [
{
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ which satisfy the following equality for all $x,y\\in\\mathbb{R}$ \\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\\][i] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x^2 + 1",
"style": "rule"
} | {
"index": "5107",
"split": "train"
} |
math | [
{
"content": "Let \\( A \\) be a 4-digit integer. When both the first digit (left-most) and the third digit are increased by \\( n \\), and the second digit and the fourth digit are decreased by \\( n \\), the new number is \\( n \\) times \\( A \\). Find the value of \\( A \\). Please output the final answer w... | {
"ground_truth": "1818",
"style": "rule"
} | {
"index": "5108",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers with $x+y=31$, what is the largest possible value of $x y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "5109",
"split": "train"
} |
math | [
{
"content": "Compute the expressions \n\\[ C = 3 \\times 4 + 5 \\times 6 + 7 \\times 8 + \\cdots + 43 \\times 44 + 45 \\]\nand \n\\[ D = 3 + 4 \\times 5 + 6 \\times 7 + \\cdots + 42 \\times 43 + 44 \\times 45 \\]\nand find the positive difference between integers $C$ and $D$. Please output the final answer wit... | {
"ground_truth": "882",
"style": "rule"
} | {
"index": "5110",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), \\( M \\) is the midpoint of side \\( AC \\), \\( D \\) is a point on side \\( BC \\) such that \\( AD \\) is the angle bisector of \\( \\angle BAC \\), and \\( P \\) is the point of intersection of \\( AD \\) and \\( BM \\). Given that \\( AB = 10 \\, \\text{cm... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5111",
"split": "train"
} |
math | [
{
"content": "In this 5x5 square array of 25 dots, five dots are to be chosen at random. What is the probability that the five dots will be collinear? Express your answer as a common fraction.\n\n[asy]\nsize(59);\nfor(int i = 0; i < 5; ++i)\nfor(int j = 0; j < 5; ++j)\ndot((i, j), linewidth(7));\n[/asy] Please ... | {
"ground_truth": "\\frac{12}{53130}",
"style": "rule"
} | {
"index": "5112",
"split": "train"
} |
math | [
{
"content": "Given the function $y=x^{2}+bx+3$ (where $b$ is a real number), the range of $y$ is $\\left[0,+\\infty \\right)$. Find the value of the real number $c$ if the solution set of the inequality $x^{2}+bx+3 \\lt c$ is $m-8 \\lt x \\lt m$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5113",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(\\cos x, \\sin x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos x, 2\\cos x-\\sqrt{3}\\sin x)$, let $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$. \n$(1)$ Find the interval where $f(x)$ is monotonically decreasing. \n$(2)$ If the maximum value of the functio... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "5114",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "227.052227052227",
"style": "rule"
} | {
"index": "5115",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\\overline{AC}$ and $\\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\\overline{AB}$ and $\\overline{BC}$. No point of circle $Q$ lies outside of $\\triangle ABC$. The radiu... | {
"ground_truth": "254",
"style": "rule"
} | {
"index": "5116",
"split": "train"
} |
math | [
{
"content": "Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.\n\nCalculate the probability that the product is odd. What is the probability that the product of the numb... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5117",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}=(2\\cos \\alpha,\\sin ^{2}\\alpha)$, $\\overrightarrow{b}=(2\\sin \\alpha,t)$, where $\\alpha\\in(0, \\frac {\\pi}{2})$, and $t$ is a real number. \n$(1)$ If $\\overrightarrow{a}- \\overrightarrow{b}=( \\frac {2}{5},0)$, find the value of $t$; \n$(2)$ If $t=1$,... | {
"ground_truth": "\\frac {23}{7}",
"style": "rule"
} | {
"index": "5118",
"split": "train"
} |
math | [
{
"content": "What is the area of the region defined by the equation $x^2 + y^2 - 10 = 4y - 10x + 4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43\\pi",
"style": "rule"
} | {
"index": "5119",
"split": "train"
} |
math | [
{
"content": "A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(... | {
"ground_truth": "240000",
"style": "rule"
} | {
"index": "5120",
"split": "train"
} |
math | [
{
"content": "In a paper, a $4 \\times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.\n\nObserve that the diagonal $AB$ intersects the grid at 9 points.\n\nIf the grid were of size $12 \\times 17$, at how many points would the diagonal $AB$ intersect the grid? Please output the final answe... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "5121",
"split": "train"
} |
math | [
{
"content": "The coefficients of the polynomial\n\\[x^4 + bx^3 + cx^2 + dx + e = 0\\]are all integers. Let $n$ be the exact number of integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x + 3)^2 (x^2 + 4x + 11) = 0$ has two integer roots counting multiplicity, because the ro... | {
"ground_truth": "0, 1, 2, 4",
"style": "rule"
} | {
"index": "5122",
"split": "train"
} |
math | [
{
"content": "Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test? $\\textbf{(A)} 92 \\qqua... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "5123",
"split": "train"
} |
math | [
{
"content": "What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16.67\\%",
"style": "rule"
} | {
"index": "5124",
"split": "train"
} |
math | [
{
"content": "Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate ... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "5125",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=3$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt 2+\\sqrt 5",
"style": "rule"
} | {
"index": "5126",
"split": "train"
} |
math | [
{
"content": "Define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$ . Define $\\{x \\} = x - \\lfloor x \\rfloor$ . For example, $\\{ 3 \\} = 3-3 = 0$ , $\\{ \\pi \\} = \\pi - 3$ , and $\\{ - \\pi \\} = 4-\\pi$ . If $\\{n\\} + \\{ 3n\\} = 1.4$ , then find the sum of all possible ... | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "5127",
"split": "train"
} |
math | [
{
"content": "In a convex quadrilateral \\(ABCD\\), the midpoint of side \\(AD\\) is marked as point \\(M\\). Segments \\(BM\\) and \\(AC\\) intersect at point \\(O\\). It is known that \\(\\angle ABM = 55^\\circ\\), \\(\\angle AMB = 70^\\circ\\), \\(\\angle BOC = 80^\\circ\\), and \\(\\angle ADC = 60^\\circ\\)... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "5128",
"split": "train"
} |
math | [
{
"content": "Given that 10 is the arithmetic mean of the set $\\{6, 13, 18, 4, x\\}$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "5129",
"split": "train"
} |
math | [
{
"content": "How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "5130",
"split": "train"
} |
math | [
{
"content": "How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not.\n(a) 10\n(b) 13\n(c) 18\n(d) 22\n(e) 25 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5131",
"split": "train"
} |
math | [
{
"content": "In the provided polygon, each side is perpendicular to its adjacent sides, and all 36 of the sides are congruent. The perimeter of the polygon is 72. Inside, the polygon is divided into rectangles instead of squares. Find the area of the polygon. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "5132",
"split": "train"
} |
math | [
{
"content": "Solve\n\\[\\arcsin x + \\arcsin 2x = \\frac{\\pi}{3}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{21}}{14}",
"style": "rule"
} | {
"index": "5133",
"split": "train"
} |
math | [
{
"content": "A club is creating a special series of membership cards that includes a code consisting of five characters selected from the letters B, E, T, A and the digits in 2023. No character should appear in a code more often than it appears in \"BETA\" or \"2023\" (B, E, T, A appear once each; 2 is twice; ... | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "5134",
"split": "train"
} |
math | [
{
"content": "The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12.8470588235",
"style": "rule"
} | {
"index": "5135",
"split": "train"
} |
math | [
{
"content": "Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \\(a, b, c, d\\), and \\(e\\) respectively. We know that:\n\n1. \\(c = b \\cdot e\\)\n2. \\(a + b = d\\)\n3. \\(e - d = a\\)\n\nFrankie picks up the card ... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "5136",
"split": "train"
} |
math | [
{
"content": "A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \\leq x \\leq 500$. Please output the final answer within \\boxed{}.... | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "5137",
"split": "train"
} |
math | [
{
"content": "If $M(\\frac{p}{2}$,$p)$ is a point on the parabola $y^{2}=2px\\left(p \\gt 0\\right)$, and the distance from $M$ to the point $\\left(1,0\\right)$ is 1 greater than the distance to the $y$-axis, find:<br/>\n$(1)$ The equation of the parabola.<br/>\n$(2)$ Let line $l$ intersect the parabola at poi... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "5138",
"split": "train"
} |
math | [
{
"content": "Find the greatest constant $M,$ so that\n\\[\\frac{a^2 + b^2}{c^2} > M\\]whenever $a,$ $b,$ and $c$ are the sides of a triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5139",
"split": "train"
} |
math | [
{
"content": "A quadrilateral pyramid \\(SABCD\\) is given, with a base that is a trapezoid \\(ABCD\\). The ratio of the bases \\(AD\\) and \\(BC\\) of this trapezoid is 2. Construct the cross-section of the pyramid with a plane passing through point \\(D\\) and the midpoints of the edges \\(SA\\) and \\(SB\\).... | {
"ground_truth": "2:1",
"style": "rule"
} | {
"index": "5140",
"split": "train"
} |
math | [
{
"content": "Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5141",
"split": "train"
} |
math | [
{
"content": "Points $A, B, C$ in the plane satisfy $\\overline{A B}=2002, \\overline{A C}=9999$. The circles with diameters $A B$ and $A C$ intersect at $A$ and $D$. If $\\overline{A D}=37$, what is the shortest distance from point $A$ to line $B C$? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "5142",
"split": "train"
} |
math | [
{
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "5143",
"split": "train"
} |
math | [
{
"content": "A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\\_1$ are $\\begin{cases} x=\\sqrt{2}\\sin(\\alpha + \\frac{\\pi}{4}) \\\\ y=\\sin(2\\alpha) + 1 \\end{cases}$, where $\\alpha$ is the parameter. Establi... | {
"ground_truth": "\\frac{\\sqrt{7}}{2} - 1",
"style": "rule"
} | {
"index": "5144",
"split": "train"
} |
math | [
{
"content": "Find all real numbers $x$ such that \\[3 \\le \\frac{x}{2x-5} < 8.\\](Give your answer in interval notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(\\tfrac83, 3]",
"style": "rule"
} | {
"index": "5145",
"split": "train"
} |
math | [
{
"content": "If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5146",
"split": "train"
} |
math | [
{
"content": "Given the points M(2,0) and N(a,b) in the Cartesian coordinate system, with the Manhattan distance between M and N defined as d(M,N) = |x₁ - x₂| + |y₁ - y₂|, and d(M,N) = 2, find the sum of the minimum and maximum values of a² + b² - 4a. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5147",
"split": "train"
} |
math | [
{
"content": "Given that a shop advertises everything as \"half price in today's sale,\" and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both... | {
"ground_truth": "20\\%",
"style": "rule"
} | {
"index": "5148",
"split": "train"
} |
math | [
{
"content": "On a complex plane map of a fictional continent, city A is located at the origin $0$, city B is at $3900i$, and city C is at $1170 + 1560i$. Calculate the distance from city C to city A on this plane. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1950",
"style": "rule"
} | {
"index": "5149",
"split": "train"
} |
math | [
{
"content": "A line passing through one focus F_1 of the ellipse 4x^{2}+2y^{2}=1 intersects the ellipse at points A and B. Then, points A, B, and the other focus F_2 of the ellipse form ∆ABF_2. Calculate the perimeter of ∆ABF_2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt {2}",
"style": "rule"
} | {
"index": "5150",
"split": "train"
} |
math | [
{
"content": "If $g(x) = 2x^2 - 3$ and $h(x) = 4x^3 +1$, what is the value of $g(h(-1))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5151",
"split": "train"
} |
math | [
{
"content": "On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{16}",
"style": "rule"
} | {
"index": "5152",
"split": "train"
} |
math | [
{
"content": "Football tickets cost $\\$13.50$ each. What is the maximum number of tickets Jane can buy with $\\$100.00$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "5153",
"split": "train"
} |
math | [
{
"content": "Given triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ lying on the line $x + 2y = 8$, find the area of triangle $ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "5154",
"split": "train"
} |
math | [
{
"content": "Find the length of the parametric curve described by\n\\[(x,y) = (2 \\sin t, 2 \\cos t)\\]from $t = 0$ to $t = \\pi.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\pi",
"style": "rule"
} | {
"index": "5155",
"split": "train"
} |
math | [
{
"content": "Solve the application problem by setting up equations:<br/>A gift manufacturing factory receives an order for a batch of teddy bears and plans to produce them in a certain number of days. If they produce $20$ teddy bears per day, they will be $100$ short of the order. If they produce $23$ teddy be... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "5156",
"split": "train"
} |
math | [
{
"content": "\\( P \\) is a moving point on the ellipse \\(\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\\). The tangent line to the ellipse at point \\( P \\) intersects the circle \\(\\odot O\\): \\(x^{2}+y^{2}=12\\) at points \\( M \\) and \\( N \\). The tangents to \\(\\odot O\\) at \\( M \\) and \\( N \\) intersect... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "5157",
"split": "train"
} |
math | [
{
"content": "In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "693",
"style": "rule"
} | {
"index": "5158",
"split": "train"
} |
math | [
{
"content": "I am dining at a Mexican restaurant with a friend who is vegan and allergic to nuts. The restaurant menu lists 8 dishes that are vegan. These vegan options constitute one-fourth of the entire menu. However, 5 of these vegan dishes contain nuts. What fraction of the menu can my friend eat? Please o... | {
"ground_truth": "\\frac{3}{32}",
"style": "rule"
} | {
"index": "5159",
"split": "train"
} |
math | [
{
"content": "In the diagram, $\\angle PQR=\\angle PRQ$. If $QR=5$ and $PR=7$, what is the perimeter of $\\triangle PQR$? [asy]\ndraw((0,0)--(2.5,7.43)--(5,0)--cycle);\nlabel(\"5\",(2.5,0),S);\nlabel(\"$Q$\",(0,0),SW);\nlabel(\"$R$\",(5,0),SE);\nlabel(\"$P$\",(2.5,7.43),N);\nlabel(\"7\",(4.2,3.7));\n[/asy] Ple... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "5160",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\angle A = 90^\\circ$ and $\\tan C = 3$. If $BC = 90$, what is the length of $AB$, and what is the perimeter of triangle ABC? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36\\sqrt{10} + 90",
"style": "rule"
} | {
"index": "5161",
"split": "train"
} |
math | [
{
"content": "In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2-\\sqrt{3}",
"style": "rule"
} | {
"index": "5162",
"split": "train"
} |
math | [
{
"content": "If I have a $4\\times 4$ chess board, in how many ways can I place four distinct pawns on the board such that each column and row of the board contains no more than one pawn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "5163",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ that satisfy $|\\overrightarrow {a}|= \\sqrt {3}$, $|\\overrightarrow {b}|=2$, and $(\\overrightarrow {a}- \\overrightarrow {b}) \\perp \\overrightarrow {a}$, find the projection of $\\overrightarrow {a}$ on $\\overrightarrow {b}$. Pl... | {
"ground_truth": "\\frac {3}{2}",
"style": "rule"
} | {
"index": "5164",
"split": "train"
} |
math | [
{
"content": "How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "343",
"style": "rule"
} | {
"index": "5165",
"split": "train"
} |
math | [
{
"content": "Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "5166",
"split": "train"
} |
math | [
{
"content": "Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \\[ A.\\ \\ \\frac{2006}{2005}+\\frac{2006}{2007} \\qquad B.\\ \\ \\frac{2006}{2007}+\\frac{2008}{2007} \\qquad C.\\ \\ \\frac{2007}{2006}+\\frac{2007}{2008} \\] Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\text{A}",
"style": "rule"
} | {
"index": "5167",
"split": "train"
} |
math | [
{
"content": "Given circle $O$, points $E$ and $F$ are on the same side of diameter $\\overline{AB}$, $\\angle AOE = 60^\\circ$, and $\\angle FOB = 90^\\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "5168",
"split": "train"
} |
math | [
{
"content": "(1) Given $0 < x < \\frac{1}{2}$, find the maximum value of $y= \\frac{1}{2}x(1-2x)$;\n\n(2) Given $x > 0$, find the maximum value of $y=2-x- \\frac{4}{x}$;\n\n(3) Given $x$, $y\\in\\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\\frac{1}{x}+ \\frac{3}{y}$. Please output the final answer... | {
"ground_truth": "1+ \\frac{ \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "5169",
"split": "train"
} |
math | [
{
"content": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 5,$ then find\n\\[\\begin{vmatrix} a - c & b - d \\\\ c & d \\end{vmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5170",
"split": "train"
} |
math | [
{
"content": "The sequence consists of 19 ones and 49 zeros arranged in a random order. A group is defined as the maximal subsequence of identical symbols. For example, in the sequence 110001001111, there are five groups: two ones, then three zeros, then one one, then two zeros, and finally four ones. Find the ... | {
"ground_truth": "2.83",
"style": "rule"
} | {
"index": "5171",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \\neq 1$. <br/> $(1)$ Discuss the monotonicity of the function $f(x)$; <br/> $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \\in [0,3]$, find the minimum value of $g(x)$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5172",
"split": "train"
} |
math | [
{
"content": "If $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse, find $g^{-1}(f(10))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{14}",
"style": "rule"
} | {
"index": "5173",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be angles such that:\n\\[\n\\sin a = \\cot b, \\quad \\sin b = \\cot c, \\quad \\sin c = \\cot a.\n\\]\nFind the largest possible value of \\(\\cos a\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{3 - \\sqrt{5}}{2}}",
"style": "rule"
} | {
"index": "5174",
"split": "train"
} |
math | [
{
"content": "Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible dis... | {
"ground_truth": "630",
"style": "rule"
} | {
"index": "5175",
"split": "train"
} |
math | [
{
"content": "For each of the possible outcomes of rolling three coins (TTT, THT, TTH, HHT, HTH, HHH), a fair die is rolled for each two heads that appear together. What is the probability that the sum of the die rolls is odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "5176",
"split": "train"
} |
math | [
{
"content": "$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{63}{4}",
"style": "rule"
} | {
"index": "5177",
"split": "train"
} |
math | [
{
"content": "A traffic light cycles repeatedly in the following order: green for 45 seconds, then yellow for 5 seconds, and red for 50 seconds. Cody picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching? Please output the final answer... | {
"ground_truth": "\\frac{3}{20}",
"style": "rule"
} | {
"index": "5178",
"split": "train"
} |
math | [
{
"content": "Equilateral triangle $ABC$ has a side length of $\\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \\sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\\angle ABD_1 = 30^\\circ$ and $... | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "5179",
"split": "train"
} |
math | [
{
"content": "Subtract $111.11$ from $333.33.$ Express the result as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "222.22",
"style": "rule"
} | {
"index": "5180",
"split": "train"
} |
math | [
{
"content": "The circle is divided by points \\(A, B, C, D\\) such that \\(\\sim AB: BC: CD: DA = 2: 3: 5: 6\\). Chords \\(AC\\) and \\(BD\\) are drawn, intersecting at point \\(M\\).\n\nFind the angle \\(AMB\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78.75",
"style": "rule"
} | {
"index": "5181",
"split": "train"
} |
math | [
{
"content": "The number of four-digit even numbers formed without repeating digits from the numbers $2$, $0$, $1$, $7$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5182",
"split": "train"
} |
math | [
{
"content": "Given $\\boldsymbol{a} = (\\cos \\alpha, \\sin \\alpha)$ and $\\boldsymbol{b} = (\\cos \\beta, \\sin \\beta)$, the relationship between $\\boldsymbol{a}$ and $\\boldsymbol{b}$ is given by $|k \\boldsymbol{a} + \\boldsymbol{b}| - \\sqrt{3}|\\boldsymbol{a} - k \\boldsymbol{b}|$, where $k > 0$. Find ... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5183",
"split": "train"
} |
math | [
{
"content": "Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5184",
"split": "train"
} |
math | [
{
"content": "On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest ... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5185",
"split": "train"
} |
math | [
{
"content": "How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248",
"style": "rule"
} | {
"index": "5186",
"split": "train"
} |
math | [
{
"content": "The sum of sides \\( AB \\) and \\( BC \\) of triangle \\( ABC \\) is 11, angle \\( B \\) is \\( 60^\\circ \\), and the radius of the inscribed circle is \\(\\frac{2}{\\sqrt{3}}\\). It is also known that side \\( AB \\) is longer than side \\( BC \\). Find the height of the triangle dropped from v... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "5187",
"split": "train"
} |
math | [
{
"content": "Use all digits from 1 to 9 to form three three-digit numbers such that their product is:\n\na) the smallest; \nb) the largest. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "941 \\times 852 \\times 763",
"style": "rule"
} | {
"index": "5188",
"split": "train"
} |
math | [
{
"content": "Determine the minimum possible value of the sum\n\\[\n\\frac{a}{3b} + \\frac{b}{6c} + \\frac{c}{9a},\n\\]\nwhere \\(a\\), \\(b\\), and \\(c\\) are positive real numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{\\sqrt[3]{162}}",
"style": "rule"
} | {
"index": "5189",
"split": "train"
} |
math | [
{
"content": "A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a p... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "5190",
"split": "train"
} |
math | [
{
"content": "The Gropkas of Papua New Guinea have ten letters in their alphabet: A, E, G, I, K, O, R, U, and V. Suppose license plates of four letters use only the letters in the Gropka alphabet. How many possible license plates are there of four letters that begin with either A or E, end with V, cannot contai... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "5191",
"split": "train"
} |
math | [
{
"content": "A high school is holding a speech contest with 10 participants. There are 3 students from Class 1, 2 students from Class 2, and 5 students from other classes. Using a draw to determine the speaking order, what is the probability that the 3 students from Class 1 are placed consecutively (in consecu... | {
"ground_truth": "$\\frac{1}{20}$",
"style": "rule"
} | {
"index": "5192",
"split": "train"
} |
math | [
{
"content": "What is the largest digit $N$ for which $2345N$ is divisible by 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5193",
"split": "train"
} |
math | [
{
"content": "Determine the exact value of the series\n\\[\n\\frac{1}{3 + 1} + \\frac{2}{3^2 + 1} + \\frac{4}{3^4 + 1} + \\frac{8}{3^8 + 1} + \\frac{16}{3^{16} + 1} + \\dotsb.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5194",
"split": "train"
} |
math | [
{
"content": "The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17.67",
"style": "rule"
} | {
"index": "5195",
"split": "train"
} |
math | [
{
"content": "Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "65.31\\%",
"style": "rule"
} | {
"index": "5196",
"split": "train"
} |
math | [
{
"content": "Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{35}{128}",
"style": "rule"
} | {
"index": "5197",
"split": "train"
} |
math | [
{
"content": "Find $\\tan \\frac{9 \\pi}{4}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5198",
"split": "train"
} |
math | [
{
"content": "A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1221",
"style": "rule"
} | {
"index": "5199",
"split": "train"
} |
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