data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the derivative of the function $f(x)$ is $f'(x) = 2 + \\sin x$, and $f(0) = -1$. The sequence $\\{a_n\\}$ is an arithmetic sequence with a common difference of $\\frac{\\pi}{4}$. If $f(a_2) + f(a_3) + f(a_4) = 3\\pi$, calculate $\\frac{a_{2016}}{a_{2}}$. Please output the final answer within... | {
"ground_truth": "2015",
"style": "rule"
} | {
"index": "37900",
"split": "train"
} |
math | [
{
"content": "There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that the distance between ... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "37901",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\log_3 27\\sqrt3$. Express your answer as an improper fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac72",
"style": "rule"
} | {
"index": "37902",
"split": "train"
} |
math | [
{
"content": "If eight apples cost the same as four bananas, and two bananas cost the same as three cucumbers, how many cucumbers can Tyler buy for the price of 16 apples? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37903",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural decimal number \\(n\\) whose square starts with the digits 19 and ends with the digits 89. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1383",
"style": "rule"
} | {
"index": "37904",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{\\begin{array}{l}x=2t+1,\\\\ y=2t\\end{array}\\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis,... | {
"ground_truth": "\\frac{\\sqrt{14}}{2}",
"style": "rule"
} | {
"index": "37905",
"split": "train"
} |
math | [
{
"content": "$(1)$ When throwing two uniformly weighted dice, let $A=$\"the first time an odd number appears,\" $B=$\"the sum of the two dice is a multiple of $3$.\" Determine whether events $A$ and $B$ are independent, and explain the reason;<br/>$(2)$ Athletes A and B are participating in a shooting assessme... | {
"ground_truth": "0.2212",
"style": "rule"
} | {
"index": "37906",
"split": "train"
} |
math | [
{
"content": "A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "37907",
"split": "train"
} |
math | [
{
"content": "The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.\n\n\\b... | {
"ground_truth": "147.5",
"style": "rule"
} | {
"index": "37908",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $BC = 8$. The length of median $AD$ is 5. Let $M$ be the largest possible value of $AB^2 + AC^2$, and let $m$ be the smallest possible value. Find $M - m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "37909",
"split": "train"
} |
math | [
{
"content": "Define the Fibonacci numbers by $F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}$ for $n \\geq 2$. For how many $n, 0 \\leq n \\leq 100$, is $F_{n}$ a multiple of 13? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37910",
"split": "train"
} |
math | [
{
"content": "Suppose that $x_1+1 = x_2+2 = x_3+3 = \\cdots = x_{1000}+1000 = x_1 + x_2 + x_3 + \\cdots + x_{1000} + 1001$. Find the value of $\\left\\lfloor |S| \\right\\rfloor$, where $S = \\sum_{n=1}^{1000} x_n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "501",
"style": "rule"
} | {
"index": "37911",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, when $x > 0$, $f(x)=2^{x}+ \\ln \\frac{x}{4}$. Let $a_{n}=f(n-5)$, then the sum of the first $8$ terms of the sequence $\\{a_{n}\\}$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "37912",
"split": "train"
} |
math | [
{
"content": "Three 12 cm $\\times$12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volu... | {
"ground_truth": "864",
"style": "rule"
} | {
"index": "37913",
"split": "train"
} |
math | [
{
"content": "Compute $1+6\\cdot2-3+5\\cdot4\\div2$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37914",
"split": "train"
} |
math | [
{
"content": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probabi... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "37915",
"split": "train"
} |
math | [
{
"content": "The polynomial \\( x^{103} + Cx + D \\) is divisible by \\( x^2 + 2x + 1 \\) for some real numbers \\( C \\) and \\( D \\). Find \\( C + D \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "37916",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\cos (2x+ \\frac {\\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\\varphi|$ units, and the resulting graph i... | {
"ground_truth": "\\frac {3\\pi}{16}",
"style": "rule"
} | {
"index": "37917",
"split": "train"
} |
math | [
{
"content": "In triangle $DEF$, $DE = 6$, $EF = 8$, and $FD = 10$. Point $Q$ is randomly selected inside triangle $DEF$. What is the probability that $Q$ is closer to $D$ than it is to either $E$ or $F$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "37918",
"split": "train"
} |
math | [
{
"content": "The chord \\( AB \\) divides the circle into two arcs, with the smaller arc being \\( 130^{\\circ} \\). The larger arc is divided by chord \\( AC \\) in the ratio \\( 31:15 \\) from point \\( A \\). Find the angle \\( BAC \\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "37.5",
"style": "rule"
} | {
"index": "37919",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=|x-1|+|x+1|$.\n(I) Solve the inequality $f(x) < 3$;\n(II) If the minimum value of $f(x)$ is $m$, let $a > 0$, $b > 0$, and $a+b=m$, find the minimum value of $\\frac{1}{a}+ \\frac{2}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}+ \\sqrt{2}",
"style": "rule"
} | {
"index": "37920",
"split": "train"
} |
math | [
{
"content": "On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\\overline{2011 \\mathrm{ABCD}}$. What is $\\overline{\\mathrm{ABCD}}$? Please ou... | {
"ground_truth": "1221",
"style": "rule"
} | {
"index": "37921",
"split": "train"
} |
math | [
{
"content": "The Big Sixteen Basketball League consists of two divisions, each with eight teams. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "296",
"style": "rule"
} | {
"index": "37922",
"split": "train"
} |
math | [
{
"content": "The $\\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:\n\n\n\n- $D(1) = 0$ ;\n- $D(p)=1$ for all primes $p$ ;\n- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .\n\nFind the sum of all positive integers $n$ below $1000$ ... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "37923",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$, calculate the distance from point $F$ to the midpoint of segment $DE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "37924",
"split": "train"
} |
math | [
{
"content": "In a table tennis team of 5 players, which includes 2 veteran players and 3 new players, we need to select 3 players to be ranked as No. 1, No. 2, and No. 3 for a team competition. The selection must ensure that among the 3 chosen players, there is at least 1 veteran player and among players No. 1... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "37925",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "37926",
"split": "train"
} |
math | [
{
"content": "How many integers between $\\frac{23}{3}$ and $\\frac{65}{2}$ are multiples of $5$ or $3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37927",
"split": "train"
} |
math | [
{
"content": "How many (possibly empty) sets of lattice points $\\{P_1, P_2, ... , P_M\\}$ , where each point $P_i =(x_i, y_i)$ for $x_i\n, y_i \\in \\{0, 1, 2, 3, 4, 5, 6\\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \\le i < j \\le M$ ? An infinite slope, e.g. $P_i$ is ve... | {
"ground_truth": "3432",
"style": "rule"
} | {
"index": "37928",
"split": "train"
} |
math | [
{
"content": "In a pile of apples, the ratio of large apples to small apples is $9:1$. Now, a fruit sorting machine is used for screening, with a probability of $5\\%$ that a large apple is sorted as a small apple and a probability of $2\\%$ that a small apple is sorted as a large apple. Calculate the probabili... | {
"ground_truth": "\\frac{855}{857}",
"style": "rule"
} | {
"index": "37929",
"split": "train"
} |
math | [
{
"content": "Divide the sides of a unit square \\(ABCD\\) into 5 equal parts. Let \\(D'\\) denote the second division point from \\(A\\) on side \\(AB\\), and similarly, let the second division points from \\(B\\) on side \\(BC\\), from \\(C\\) on side \\(CD\\), and from \\(D\\) on side \\(DA\\) be \\(A'\\), \... | {
"ground_truth": "\\frac{9}{29}",
"style": "rule"
} | {
"index": "37930",
"split": "train"
} |
math | [
{
"content": "Find the maximum number of different integers that can be selected from the set $ \\{1,2,...,2013\\}$ so that no two exist that their difference equals to $17$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "37931",
"split": "train"
} |
math | [
{
"content": "Complete the following questions: \n$(1)$ Calculate: $(\\sqrt{8}-\\sqrt{\\frac{1}{2}})\\div \\sqrt{2}$. \n$(2)$ Calculate: $2\\sqrt{3}\\times (\\sqrt{12}-3\\sqrt{75}+\\frac{1}{3}\\sqrt{108})$. \n$(3)$ Given $a=3+2\\sqrt{2}$ and $b=3-2\\sqrt{2}$, find the value of the algebraic expression $a^{2}... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "37932",
"split": "train"
} |
math | [
{
"content": "Given the function $$f(x)= \\begin{cases} \\sqrt {x}+3, & x\\geq0 \\\\ ax+b, & x<0\\end{cases}$$ satisfies the condition: $y=f(x)$ is a monotonic function on $\\mathbb{R}$ and $f(a)=-f(b)=4$, then the value of $f(-1)$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "37933",
"split": "train"
} |
math | [
{
"content": "Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "193",
"style": "rule"
} | {
"index": "37934",
"split": "train"
} |
math | [
{
"content": "Give an example of a function that, when \\( x \\) is equal to a known number, takes the form \\( \\frac{0}{0} \\), but as \\( x \\) approaches this number, tends to a certain limit. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{7}",
"style": "rule"
} | {
"index": "37935",
"split": "train"
} |
math | [
{
"content": "Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);\ndot(origin);\ndot(dir(0));\ndot(dir(60));\ndot(dir(120));\ndot(dir(180));\ndot(dir(240));\ndot(dir(300));\n[/asy] Please output th... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "37936",
"split": "train"
} |
math | [
{
"content": "A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "2, 3, \\text{ and } 4",
"style": "rule"
} | {
"index": "37937",
"split": "train"
} |
math | [
{
"content": "Evaluate \\( ((a^b)^a + c) - ((b^a)^b + c) \\) for \\( a = 3 \\), \\( b = 4 \\), and \\( c = 5 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16245775",
"style": "rule"
} | {
"index": "37938",
"split": "train"
} |
math | [
{
"content": "The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\\overrightarrow{MF}⋅\\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find the area of $\\trian... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "37939",
"split": "train"
} |
math | [
{
"content": "The operation $\\star$ is defined as $a \\star b = a + \\frac{a}{b}$. What is the value of $12 \\star 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37940",
"split": "train"
} |
math | [
{
"content": "Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37941",
"split": "train"
} |
math | [
{
"content": "For what base is the representation of $285_{10}$ a four digit number whose final digit is odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "37942",
"split": "train"
} |
math | [
{
"content": "Let \\( a_{1}, a_{2}, \\ldots, a_{2000} \\) be real numbers in the interval \\([0,1]\\). Find the maximum possible value of\n\\[ \n\\sum_{1 \\leq i < j \\leq 2000}(j - i) \\left| a_{j} - a_{i} \\right| \n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1000000000",
"style": "rule"
} | {
"index": "37943",
"split": "train"
} |
math | [
{
"content": "Jarris is a weighted tetrahedral die with faces $F_{1}, F_{2}, F_{3}, F_{4}$. He tosses himself onto a table, so that the probability he lands on a given face is proportional to the area of that face. Let $k$ be the maximum distance any part of Jarris is from the table after he rolls himself. Give... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37944",
"split": "train"
} |
math | [
{
"content": "Determine the coefficient of \\(x^{29}\\) in the expansion of \\(\\left(1 + x^{5} + x^{7} + x^{9}\\right)^{16}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65520",
"style": "rule"
} | {
"index": "37945",
"split": "train"
} |
math | [
{
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9\\pi - 18}{2}",
"style": "rule"
} | {
"index": "37946",
"split": "train"
} |
math | [
{
"content": "What is the value of $x$ in the diagram?\n\n[asy]\n\nimport olympiad;\n\ndraw((0,0)--(sqrt(3),0)--(0,sqrt(3))--cycle);\n\ndraw((0,0)--(-1,0)--(0,sqrt(3))--cycle);\n\nlabel(\"8\",(-1/2,sqrt(3)/2),NW);\n\nlabel(\"$x$\",(sqrt(3)/2,sqrt(3)/2),NE);\n\ndraw(\"$45^{\\circ}$\",(1.5,0),NW);\n\ndraw(\"$60^{... | {
"ground_truth": "4\\sqrt{6}",
"style": "rule"
} | {
"index": "37947",
"split": "train"
} |
math | [
{
"content": "Calculate the distance between the foci of the ellipse defined by the equation\n\\[\\frac{x^2}{36} + \\frac{y^2}{9} = 9.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "37948",
"split": "train"
} |
math | [
{
"content": "Let $N = 99999$. Then $N^3 = \\ $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "999970000299999",
"style": "rule"
} | {
"index": "37949",
"split": "train"
} |
math | [
{
"content": "Stuart has drawn a pair of concentric circles, as shown. He draws chords $\\overline{AB}$, $\\overline{BC}, \\ldots$ of the large circle, each tangent to the small one. If $m\\angle ABC=75^\\circ$, then how many segments will he draw before returning to his starting point at $A$? [asy]\nsize(100... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37950",
"split": "train"
} |
math | [
{
"content": "The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. The number of complete rotations that the wheel made during this time is closest to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1020",
"style": "rule"
} | {
"index": "37951",
"split": "train"
} |
math | [
{
"content": "Find the sum\\[1+11+111+\\cdots+\\underbrace{111\\ldots111}_{n\\text{ digits}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10^{n+1} - 10 - 9n}{81}",
"style": "rule"
} | {
"index": "37952",
"split": "train"
} |
math | [
{
"content": "Given an ellipse and a hyperbola,\\[\\frac{x^2}{16} - \\frac{y^2}{25} = 1\\]and \\[\\frac{x^2}{K} + \\frac{y^2}{25} = 1\\], have the same asymptotes. Find the value of $K$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37953",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ ... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "37954",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH^{}_{}$ are tangent to $\\overline{CH}$. If $AB = 1995\\,$, $AC = 1994\\,$, and $BC = 1993\\,$, then $RS\\,$ can be expressed as $m/n\\,$, wher... | {
"ground_truth": "997",
"style": "rule"
} | {
"index": "37955",
"split": "train"
} |
math | [
{
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "37956",
"split": "train"
} |
math | [
{
"content": "Define $\\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 that satisfies:\n$$\n\\begin{array}{l}\n\\operatorname{gcd}(63, n+120) = 21, \\\\\n\\operatorname{gcd}(n+63, 120) = 60\n\\end{array}\n$$\n\nT... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "37957",
"split": "train"
} |
math | [
{
"content": "Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{28}\\left(10-\\sqrt{2}\\right)",
"style": "rule"
} | {
"index": "37958",
"split": "train"
} |
math | [
{
"content": "The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\\triangle A P F$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2} + \\sqrt{10}",
"style": "rule"
} | {
"index": "37959",
"split": "train"
} |
math | [
{
"content": "The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\\natural$ , $\\sharp$ , $\\flat$ such that for all $x, y\\in \\mathbb{R}$ , we have \\[x\\mathbin\\natural y = x + y,\\quad x\\mathbin\\sharp y = \\max\\{x, y\\},\\quad x\\mathbin\\flat y = \\m... | {
"ground_truth": "419",
"style": "rule"
} | {
"index": "37960",
"split": "train"
} |
math | [
{
"content": "Fill in the blanks with unique digits in the following equation: \n\\[ \\square \\times(\\square+\\square \\square) \\times(\\square+\\square+\\square+\\square \\square) = 2014 \\]\nThe maximum sum of the five one-digit numbers among the choices is: Please output the final answer within \\boxed{}.... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "37961",
"split": "train"
} |
math | [
{
"content": "Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \\(d\\), \\(e\\), and \\(f\\) are integers and \\(d > 0\\). What are the values of \\(d + e + f\\)? Please output th... | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "37962",
"split": "train"
} |
math | [
{
"content": "What is the smallest result that can be obtained from the following process?\n\nChoose three different numbers from the set $\\{3,5,7,11,13,17\\}$.\nAdd two of these numbers.\nMultiply their sum by the third number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "37963",
"split": "train"
} |
math | [
{
"content": "Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two he... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "37964",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = 2\\sqrt{3}\\sin x \\cos x + 2\\cos^2x - 1$,\n\n(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;\n\n(2) In $\\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, $B$, and $C$ respect... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "37965",
"split": "train"
} |
math | [
{
"content": "Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "37966",
"split": "train"
} |
math | [
{
"content": "Given $a, b, c > 0$ and $(a+b)bc = 5$, find the minimum value of $2a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "37967",
"split": "train"
} |
math | [
{
"content": "How many ways can 8 teaching positions be allocated to three schools, given that each school receives at least one position, and School A receives at least two positions? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "37968",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "495",
"style": "rule"
} | {
"index": "37969",
"split": "train"
} |
math | [
{
"content": "Let $\\bigtriangleup PQR$ be a triangle with $\\angle P = 75^\\circ$ and $\\angle Q = 60^\\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\\triangle PQR$ so that side $\\overline{AB}$ lies on $\\overline{PQ}$, side $\\overline{CD}$ lies on $\\overline{QR}$, and one of the re... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37970",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ shares a common vertex with the hyperbola $\\frac{x^{2}}{16} - \\frac{y^{2}}{9} = 1$ and passes through the point $A(6, \\sqrt{5})$.\n1. Find the equation of the hyperbola $C$ and write out the equations of i... | {
"ground_truth": "\\frac{28\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "37971",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\\overline{BC}$ with $CD=6.$ Point $E$ is on $\\overline{BC}$ such that $\\angle BAE\\cong \\angle CAD.$ Given that $BE=\\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$ Please output the final answer wit... | {
"ground_truth": "463",
"style": "rule"
} | {
"index": "37972",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\left\\lceil{\\frac54}\\right\\rceil+\\left\\lfloor{-\\frac54}\\right\\rfloor$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "37973",
"split": "train"
} |
math | [
{
"content": "The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \\(\\begin{tabular}[t]{lllllllll} 89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\\\ 63, & 84, & 78, & 71, & 80, & 90. & & & \\\\ \\end{tabular}\\)\nIn Mr. Freeman's class, what percent of ... | {
"ground_truth": "33\\frac{1}{3}\\%",
"style": "rule"
} | {
"index": "37974",
"split": "train"
} |
math | [
{
"content": "When numbers are represented in \"base fourteen\", the digit before it becomes full is fourteen. If in \"base fourteen\", the fourteen digits are sequentially noted as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ten, J, Q, K, then convert the three-digit number JQK in \"base fourteen\" into a \"binary\" number ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37975",
"split": "train"
} |
math | [
{
"content": "Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Al... | {
"ground_truth": "\\frac{5}{18}",
"style": "rule"
} | {
"index": "37976",
"split": "train"
} |
math | [
{
"content": "Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4769280",
"style": "rule"
} | {
"index": "37977",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\), \\( b \\), and \\( c \\) be the roots of the polynomial \\( x^3 - 15x^2 + 25x - 10 = 0 \\). Compute \n\\[\n(a-b)^2 + (b-c)^2 + (c-a)^2.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "37978",
"split": "train"
} |
math | [
{
"content": "Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misinformed, how many to... | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "37979",
"split": "train"
} |
math | [
{
"content": "Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 times. Afterward, he po... | {
"ground_truth": "0.05",
"style": "rule"
} | {
"index": "37980",
"split": "train"
} |
math | [
{
"content": "Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1\\le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "n_1n_2",
"style": "rule"
} | {
"index": "37981",
"split": "train"
} |
math | [
{
"content": "A telephone number has the form \\text{ABC-DEF-GHIJ}, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, an... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "37982",
"split": "train"
} |
math | [
{
"content": "If $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ are vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{a} + \\mathbf{b}\\| = \\sqrt{3},$ and\n\\[\\mathbf{c} - \\mathbf{a} - 2 \\mathbf{b} = 3 (\\mathbf{a} \\times \\mathbf{b}),\\]then find $\\mathbf{b} \\cdot \\mathbf{c}.... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "37983",
"split": "train"
} |
math | [
{
"content": "The graph of \\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]has its foci at $(0,\\pm 4),$ while the graph of \\[\\frac{x^2}{a^2}-\\frac{y^2}{b^2} = 1\\]has its foci at $(\\pm 6,0).$ Compute the value of $|ab|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{65}",
"style": "rule"
} | {
"index": "37984",
"split": "train"
} |
math | [
{
"content": "Given $sin2θ=-\\frac{1}{3}$, if $\\frac{π}{4}＜θ＜\\frac{3π}{4}$, then $\\tan \\theta =$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3-2\\sqrt{2}",
"style": "rule"
} | {
"index": "37985",
"split": "train"
} |
math | [
{
"content": "For an integer $n>2$, the tuple $(1, 2, \\ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\\{1, 2,\\ldots, n\\}$ can ... | {
"ground_truth": "n \\geq 14",
"style": "rule"
} | {
"index": "37986",
"split": "train"
} |
math | [
{
"content": "Let $w, x, y$, and $z$ be positive real numbers such that $0 \\neq \\cos w \\cos x \\cos y \\cos z$, $2 \\pi =w+x+y+z$, $3 \\tan w =k(1+\\sec w)$, $4 \\tan x =k(1+\\sec x)$, $5 \\tan y =k(1+\\sec y)$, $6 \\tan z =k(1+\\sec z)$. Find $k$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\sqrt{19}",
"style": "rule"
} | {
"index": "37987",
"split": "train"
} |
math | [
{
"content": "Suppose a number $m$ is randomly selected from the set $\\{7, 9, 12, 18, 21\\}$, and a number $n$ is randomly selected from $\\{2005, 2006, 2007, \\ldots, 2025\\}$. Calculate the probability that the last digit of $m^n$ is $6$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{8}{105}",
"style": "rule"
} | {
"index": "37988",
"split": "train"
} |
math | [
{
"content": "Let $T$ be a triangle with side lengths $1, 1, \\sqrt{2}$. Two points are chosen independently at random on the sides of $T$. The probability that the straight-line distance between the points is at least $\\dfrac{\\sqrt{2}}{2}$ is $\\dfrac{d-e\\pi}{f}$, where $d$, $e$, and $f$ are positive intege... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "37989",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{1}+2a_{2}+\\cdots +na_{n}=n$, and a sequence $\\{b_{n}\\}$ such that ${b_{m-1}}+{b_m}=\\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>\n$(1)$ The general formula for $\\{a_{n}\\}$;<br/>\n$(2)$ The sum of the first $20$ terms of $\\{b_{n}\\}$. Please output th... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "37990",
"split": "train"
} |
math | [
{
"content": "The average of five distinct natural numbers is 15, and the median is 18. What is the maximum possible value of the largest number among these five numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "37991",
"split": "train"
} |
math | [
{
"content": "Find the distance between the vertices of the hyperbola\n\\[\\frac{y^2}{27} - \\frac{x^2}{11} = 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6 \\sqrt{3}",
"style": "rule"
} | {
"index": "37992",
"split": "train"
} |
math | [
{
"content": "A point is randomly dropped onto the interval $[8, 13]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-2k-35\\right)x^{2}+(3k-9)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "0.6",
"style": "rule"
} | {
"index": "37993",
"split": "train"
} |
math | [
{
"content": "Solve in positive integers the following equation:\n\\[{1\\over n^2}-{3\\over 2n^3}={1\\over m^2}\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(m, n) = (4, 2)",
"style": "rule"
} | {
"index": "37994",
"split": "train"
} |
math | [
{
"content": "Kelvin the Frog is trying to hop across a river. The river has 10 lilypads on it, and he must hop on them in a specific order (the order is unknown to Kelvin). If Kelvin hops to the wrong lilypad at any point, he will be thrown back to the wrong side of the river and will have to start over. Assum... | {
"ground_truth": "176",
"style": "rule"
} | {
"index": "37995",
"split": "train"
} |
math | [
{
"content": "Given that $a, b, c$ are integers with $a b c=60$, and that complex number $\\omega \\neq 1$ satisfies $\\omega^{3}=1$, find the minimum possible value of $\\left|a+b \\omega+c \\omega^{2}\\right|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "37996",
"split": "train"
} |
math | [
{
"content": "Given the complex numbers $z\\_1=a^2-2-3ai$ and $z\\_2=a+(a^2+2)i$, if $z\\_1+z\\_2$ is a purely imaginary number, determine the value of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "37997",
"split": "train"
} |
math | [
{
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{9}{16}",
"style": "rule"
} | {
"index": "37998",
"split": "train"
} |
math | [
{
"content": "Among the following four propositions: \n(1) If line $a$ is parallel to line $b$, and line $a$ is parallel to plane $\\alpha$, then line $b$ is parallel to plane $\\alpha$. \n(2) If line $a$ is parallel to plane $\\alpha$, and line $b$ is contained in plane $\\alpha$, then plane $\\alpha$ is par... | {
"ground_truth": "(4)",
"style": "rule"
} | {
"index": "37999",
"split": "train"
} |
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