data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to the angles \\( A \\), \\( B \\), and \\( C \\) are of lengths \\( a \\), \\( b \\), and \\( c \\) respectively. Point \\( G \\) satisfies\n$$\n\\overrightarrow{GA} + \\overrightarrow{GB} + \\overrightarrow{GC} = \\mathbf{0}, \\quad \\overri... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39200",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, how far is point $F$ from the midpoint of segment $DE$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "39201",
"split": "train"
} |
math | [
{
"content": "The operation $*$ is defined by\n\\[a * b = \\frac{a - b}{1 - ab}.\\]Compute\n\\[1 * (2 * (3 * (\\dotsb (999 * 1000) \\dotsb))).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39202",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sqrt5-\\sqrt{20}+\\sqrt{45}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt5",
"style": "rule"
} | {
"index": "39203",
"split": "train"
} |
math | [
{
"content": "In triangle \\(PQR\\) with side \\(PQ = 3\\), a median \\(PM = \\sqrt{14}\\) and an altitude \\(PH = \\sqrt{5}\\) are drawn from vertex \\(P\\) to side \\(QR\\).\n\nFind side \\(PR\\), given that \\(\\angle QPR + \\angle PRQ < 90^\\circ\\). Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\sqrt{21}",
"style": "rule"
} | {
"index": "39204",
"split": "train"
} |
math | [
{
"content": "A travel agency conducted a promotion: \"Buy a trip to Egypt, bring four friends who also buy trips, and get your trip cost refunded.\" During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of these customers each brought exactly four new clients, while t... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "39205",
"split": "train"
} |
math | [
{
"content": "What is the largest integer value of $x$ for which $5-4x>17$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "39206",
"split": "train"
} |
math | [
{
"content": "Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\\ldots,A_n$, there exists $1 \\leq i < j < k \\leq n$ such that\n$$A_i \\subseteq A_j \\subseteq A_k \\text{ or } A_i \\supseteq A_j \\supseteq A... | {
"ground_truth": "2 \\binom{100}{50} + 2 \\binom{100}{49} + 1",
"style": "rule"
} | {
"index": "39207",
"split": "train"
} |
math | [
{
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\n[list=]\n... | {
"ground_truth": "3030",
"style": "rule"
} | {
"index": "39208",
"split": "train"
} |
math | [
{
"content": "We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and\n\n$f(b)=3b+1$ if $b$ is odd\n\n$f(b)=\\frac{b}{2}$ if $b$ is even.\n\nWhat is the smallest possible number of integers in the domain of $f$? Please output the final ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "39209",
"split": "train"
} |
math | [
{
"content": "How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39210",
"split": "train"
} |
math | [
{
"content": "Let ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ have left and right foci $F_{1}$, $F_{2}$, and line $l$ passing through point $F_{1}$. If the point $P$, which is the symmetric point of $F_{2}$ with respect to line $l$, lies exactly on ellipse $C$, and $\\overrightarrow{F_1P} \\cdot \... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "39211",
"split": "train"
} |
math | [
{
"content": "Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$ Please output the final answer within \\bo... | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "39212",
"split": "train"
} |
math | [
{
"content": "Determine the largest natural number \\( n \\) so that\n\n\\[ 4^{995} + 4^{1500} + 4^{n} \\]\n\nis a square number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "39213",
"split": "train"
} |
math | [
{
"content": "How many four-digit positive integers are multiples of 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1286",
"style": "rule"
} | {
"index": "39214",
"split": "train"
} |
math | [
{
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\\frac{S}{b^{2}+2c^{2}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{14}}{24}",
"style": "rule"
} | {
"index": "39215",
"split": "train"
} |
math | [
{
"content": "Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39216",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the portion of the graph of $$ y=\\frac{6x+1}{32x+8} - \\frac{2x-1}{32x-8} $$ located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$ . Find $\\lfloor 1000d \\rfloor$ .\n\n*Proposed b... | {
"ground_truth": "433",
"style": "rule"
} | {
"index": "39217",
"split": "train"
} |
math | [
{
"content": "The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine ... | {
"ground_truth": "175",
"style": "rule"
} | {
"index": "39218",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the altitude and the median from vertex $C$ each divide the angle $ACB$ into three equal parts. Determine the ratio of the sides of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 : \\sqrt{3} : 1",
"style": "rule"
} | {
"index": "39219",
"split": "train"
} |
math | [
{
"content": "The number $2013$ is expressed in the form $2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? Please output the final answer w... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "39220",
"split": "train"
} |
math | [
{
"content": "Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie? Please output the f... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "39221",
"split": "train"
} |
math | [
{
"content": "The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \\( (\\ \\ \\ ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "39222",
"split": "train"
} |
math | [
{
"content": "A right circular cone has for its base a circle having the same radius as a given sphere.\nThe volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2/1",
"style": "rule"
} | {
"index": "39223",
"split": "train"
} |
math | [
{
"content": "Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that\n$a + b + c = 3k + 1$,\n$ab + bc + ca = 3k^2 + 2k$.\n\nSlovakia Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(k+1, k, k)",
"style": "rule"
} | {
"index": "39224",
"split": "train"
} |
math | [
{
"content": "A positive number $x$ satisfies the inequality $\\sqrt{x} < 2x$ if and only if Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "39225",
"split": "train"
} |
math | [
{
"content": "Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "39226",
"split": "train"
} |
math | [
{
"content": "Petrov booked an apartment in a newly built house, which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment number was 636. Later, the developer changed the numbering to the opposite direction (right to left, as shown in the diagram). Th... | {
"ground_truth": "985",
"style": "rule"
} | {
"index": "39227",
"split": "train"
} |
math | [
{
"content": "A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how many chairs should be ... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "39228",
"split": "train"
} |
math | [
{
"content": "John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "39229",
"split": "train"
} |
math | [
{
"content": "It is known that the vertex of angle $\\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \\frac {1}{2}x$ ($x\\leq0$). \n(Ⅰ) Find the value of $\\cos\\left( \\frac {\\pi}{2}+\\theta \\right)$; ... | {
"ground_truth": "- \\frac { \\sqrt {2}}{10}",
"style": "rule"
} | {
"index": "39230",
"split": "train"
} |
math | [
{
"content": "What is the number of square units in the area of the octagon below?\n\n[asy]\nunitsize(0.5cm);\ndefaultpen(linewidth(0.7)+fontsize(10));\ndotfactor = 4;\nint i,j;\nfor(i=0;i<=5;++i)\n\n{\n\nfor(j=-4;j<=4;++j)\n\n{\n\ndot((i,j));\n\n}\n\n}\n\nfor(i=1;i<=5;++i)\n\n{\n\ndraw((i,-1/3)--(i,1/3));\n\n}... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "39231",
"split": "train"
} |
math | [
{
"content": "The diagram shows a square with sides of length \\(4 \\text{ cm}\\). Four identical semicircles are drawn with their centers at the midpoints of the square’s sides. Each semicircle touches two other semicircles. What is the shaded area, in \\(\\text{cm}^2\\)? \n\nA) \\(8 - \\pi\\)\n\nB) \\(\\pi\\)... | {
"ground_truth": "8 - 2\\pi",
"style": "rule"
} | {
"index": "39232",
"split": "train"
} |
math | [
{
"content": "Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \\frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine the eccentricity of... | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "39233",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "39234",
"split": "train"
} |
math | [
{
"content": "In the triangular prism \\( P-ABC \\),\n\\[\n\\begin{array}{l}\n\\angle APB = \\angle BPC = \\angle CPA = 90^{\\circ}, \\\\\nPA = 4, \\, PB = PC = 3.\n\\end{array}\n\\]\n\nFind the minimum sum of the squares of the distances from any point on the base \\( ABC \\) to the three lateral faces. Please... | {
"ground_truth": "\\frac{144}{41}",
"style": "rule"
} | {
"index": "39235",
"split": "train"
} |
math | [
{
"content": "A ball thrown vertically upwards has a height above the ground that is a quadratic function of its travel time. Xiaohong throws two balls vertically upwards at intervals of 1 second. Assuming the height above the ground is the same at the moment of release for both balls, and both balls reach the ... | {
"ground_truth": "1.6",
"style": "rule"
} | {
"index": "39236",
"split": "train"
} |
math | [
{
"content": "Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?\n\n(M. V. Karlukova) Please output the final answer within \\bo... | {
"ground_truth": "1234.4321",
"style": "rule"
} | {
"index": "39237",
"split": "train"
} |
math | [
{
"content": "Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from warehouse B to wareho... | {
"ground_truth": "153",
"style": "rule"
} | {
"index": "39238",
"split": "train"
} |
math | [
{
"content": "Increase Grisha's yield by $40\\%$ and Vasya's yield by $20\\%$.\n\nGrisha, the cleverest among them, calculated that their combined yield in each of the cases would increase by 5 kg, 10 kg, and 9 kg, respectively. What was the total yield of the friends (in kilograms) before meeting Hottabych? Pl... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "39239",
"split": "train"
} |
math | [
{
"content": "Find the modular inverse of $4$, modulo $21$.\n\nExpress your answer as an integer from $0$ to $20$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "39240",
"split": "train"
} |
math | [
{
"content": "A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately: Please output the final answer w... | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "39241",
"split": "train"
} |
math | [
{
"content": "$(1)$ Given real numbers $x \\gt 0$, $y \\gt 0$, and $\\frac{2}{x}+y=1$, find the minimum value of $2x+\\frac{1}{3y}$;<br/>$(2)$ Given real numbers $x$ and $y$ satisfying $x+y=1$, $x \\gt 0$, $y \\gt 0$, find the minimum value of $\\frac{1}{2x}+\\frac{x}{y+1}$. Please output the final answer withi... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "39242",
"split": "train"
} |
math | [
{
"content": "Given that \\( b\\left[\\frac{1}{1 \\times 3}+\\frac{1}{3 \\times 5}+\\cdots+\\frac{1}{1999 \\times 2001}\\right]=2 \\times\\left[\\frac{1^{2}}{1 \\times 3}+\\frac{2^{2}}{3 \\times 5}+\\cdots+\\frac{1000^{2}}{1999 \\times 2001}\\right] \\), find the value of \\( b \\). Please output the final answ... | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "39243",
"split": "train"
} |
math | [
{
"content": "How can 50 cities be connected with the smallest number of airline routes so that it is possible to travel from any city to any other city with no more than two layovers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "39244",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AC}\\perp\\overline{CD}$. If $CD = 15$, $\\tan D = 2$, and $\\tan B = 3$, then what is $BC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\sqrt{10}",
"style": "rule"
} | {
"index": "39245",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\\overline{AB},$ $\\overline{BC},$ and $\\overline{AC},$ respectively. Angle $\\angle BAC = 60^\\circ$ and $\\overline{DE}$ and $\\overline{EF}$ are parallel to $\\overline{AC}$ and $\\overline{AB},$ respective... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "39246",
"split": "train"
} |
math | [
{
"content": "Given the set\n$$\nT=\\left\\{n \\mid n=5^{a}+5^{b}, 0 \\leqslant a \\leqslant b \\leqslant 30, a, b \\in \\mathbf{Z}\\right\\},\n$$\nif a number is randomly selected from set $T$, what is the probability that the number is a multiple of 9? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "5/31",
"style": "rule"
} | {
"index": "39247",
"split": "train"
} |
math | [
{
"content": "Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1,2,3,4,5,6)$ that satisfy\n\\[\\frac{a_1 + 1}{2} \\cdot \\frac{a_2 + 2}{2} \\cdot \\frac{a_3 + 3}{2} \\cdot \\frac{a_4 + 4}{2} \\cdot \\frac{a_5 + 5}{2} \\cdot \\frac{a_6 + 6}{2} > 6!.\\] Please output the final answer within \... | {
"ground_truth": "719",
"style": "rule"
} | {
"index": "39248",
"split": "train"
} |
math | [
{
"content": "The minimum positive period of the function $y=\\sin x \\cdot |\\cos x|$ is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "39249",
"split": "train"
} |
math | [
{
"content": "Below is the graph of an ellipse. (Assume that tick marks are placed every $1$ unit along the axes.)\n[asy]\nsize(8cm);\n\nint x, y;\nfor (y = -4; y <= 2; ++y) {\n draw((-1,y)--(7,y),gray(0.7));\n}\nfor (x = -1; x <= 7; ++x) {\n draw((x,-4)--(x,2),gray(0.7));\n}\n\ndraw(shift((3,-1))*xscale(1.5)... | {
"ground_truth": "(3+\\sqrt{5},-1)",
"style": "rule"
} | {
"index": "39250",
"split": "train"
} |
math | [
{
"content": "Given the function \\( y = y_1 + y_2 \\), where \\( y_1 \\) is directly proportional to \\( x^2 \\) and \\( y_2 \\) is inversely proportional to \\( x^2 \\). When \\( x = 1 \\), \\( y = 5 \\); when \\( x = \\sqrt{3} \\), \\( y = 7 \\). Find the value of \\( x \\) when \\( y \\) is minimized. Pleas... | {
"ground_truth": "\\sqrt[4]{\\frac{3}{2}}",
"style": "rule"
} | {
"index": "39251",
"split": "train"
} |
math | [
{
"content": "If physical education is not the first class, and Chinese class is not adjacent to physics class, calculate the total number of different scheduling arrangements for five subjects - mathematics, physics, history, Chinese, and physical education - on Tuesday morning. Please output the final answer ... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "39252",
"split": "train"
} |
math | [
{
"content": "Express the sum as a common fraction: $.1 + .02 + .003 + .0004 + .00005.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2469}{20,\\!000}",
"style": "rule"
} | {
"index": "39253",
"split": "train"
} |
math | [
{
"content": "Given the line $l: ax+y+b=0$ intersects with the circle $O: x^{2}+y^{2}=4$ at points $A$ and $B$, and $M(\\sqrt{3},-1)$, and $\\overrightarrow{OA}+ \\overrightarrow{OB}= \\frac{2}{3} \\overrightarrow{OM}$, calculate the value of $\\sqrt{3}ab$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "39254",
"split": "train"
} |
math | [
{
"content": "Suppose that $g(x)$ is a function such that\n\\[ g(xy) + x = xg(y) + g(x) \\] for all real numbers $x$ and $y$. If $g(-2) = 4$, compute $g(-1002)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "39255",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, let $\\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\\phi$ counter-clockwise, and let $\\mathbf{T}$ be a translation matrix that we will ignore in the computation as translations do ... | {
"ground_truth": "\\frac{12}{5}",
"style": "rule"
} | {
"index": "39256",
"split": "train"
} |
math | [
{
"content": "Find the infinite sum of \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\).\n\n求 \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\) 無限項之和。 Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{33}{8}",
"style": "rule"
} | {
"index": "39257",
"split": "train"
} |
math | [
{
"content": "The side length of an equilateral triangle ABC is 2. Calculate the area of the orthographic (isometric) projection of triangle ABC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{4}",
"style": "rule"
} | {
"index": "39258",
"split": "train"
} |
math | [
{
"content": "The first three terms of an arithmetic sequence are 1, 10 and 19, respectively. What is the value of the 21st term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "181",
"style": "rule"
} | {
"index": "39259",
"split": "train"
} |
math | [
{
"content": "Given that in triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\\sin A + \\sin B = 2\\sin C\\cos B$ and the area of $\\triangle ABC$ is $S = \\frac{\\sqrt{3}}{2}c$, find the minimum value of $ab$. Please output the final answer within... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "39260",
"split": "train"
} |
math | [
{
"content": "How many 5-digit numbers beginning with $2$ are there that have exactly three identical digits which are not $2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "324",
"style": "rule"
} | {
"index": "39261",
"split": "train"
} |
math | [
{
"content": "Plane M is parallel to plane N. There are 3 different points on plane M and 4 different points on plane N. The maximum number of tetrahedrons with different volumes that can be determined by these 7 points is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "39262",
"split": "train"
} |
math | [
{
"content": "What is the sum of the different prime factors of $210630$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "39263",
"split": "train"
} |
math | [
{
"content": "Find the largest natural number from which it is impossible to obtain a number divisible by 11 by deleting some of its digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "987654321",
"style": "rule"
} | {
"index": "39264",
"split": "train"
} |
math | [
{
"content": "A rectangular prism has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \\le i \\le 4$. The dimensions of the prism are given by lengths 2 along the x-axis, 3 along the... | {
"ground_truth": "\\frac{\\sqrt{14}}{2}",
"style": "rule"
} | {
"index": "39265",
"split": "train"
} |
math | [
{
"content": "In a mathematics competition, there are four problems carrying 1, 2, 3, and 4 marks respectively. For each question, full score is awarded if the answer is correct; otherwise, 0 mark will be given. The total score obtained by a contestant is multiplied by a time bonus of 4, 3, 2, or 1 according to... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "39266",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is $\\begin{cases}x=1+t\\cdot\\cos \\alpha \\\\ y=2+t\\cdot\\sin \\alpha \\end{cases} (t \\text{ is a parameter})$, and in the polar coordinate system (with the same length unit as the rectangular coordinate system $... | {
"ground_truth": "2\\sqrt{7}",
"style": "rule"
} | {
"index": "39267",
"split": "train"
} |
math | [
{
"content": "Suppose there is an octahedral die with the numbers 1, 2, 3, 4, 5, 6, 7, and 8 written on its eight faces. Each time the die is rolled, the chance of any of these numbers appearing is the same. If the die is rolled three times, and the numbers appearing on the top face are recorded in sequence, le... | {
"ground_truth": "\\frac{45}{256}",
"style": "rule"
} | {
"index": "39268",
"split": "train"
} |
math | [
{
"content": "The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "39269",
"split": "train"
} |
math | [
{
"content": "A right circular cone is inscribed in a right rectangular prism as shown. The base of the prism has dimensions, where one side is exactly twice the length of the other ($a$ and $2a$). The cone's base fits perfectly into the base of the prism making one side of the rectangle the diameter of the con... | {
"ground_truth": "\\frac{\\pi}{24}",
"style": "rule"
} | {
"index": "39270",
"split": "train"
} |
math | [
{
"content": "Let $m \\in \\mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$. Please output the final answer within \\boxed{... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "39271",
"split": "train"
} |
math | [
{
"content": "Arrange the numbers 1 to 9 in a row from left to right such that every three consecutive numbers form a three-digit number that is a multiple of 3. How many arrangements are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1296",
"style": "rule"
} | {
"index": "39272",
"split": "train"
} |
math | [
{
"content": "Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n$. Please output the final ans... | {
"ground_truth": "118",
"style": "rule"
} | {
"index": "39273",
"split": "train"
} |
math | [
{
"content": "What is the greatest prime factor of $15! + 18!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "39274",
"split": "train"
} |
math | [
{
"content": "Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "634",
"style": "rule"
} | {
"index": "39275",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin({ωx+φ})$ $({ω＞0,|φ|≤\\frac{π}{2}})$, $f(0)=\\frac{{\\sqrt{2}}}{2}$, and the function $f\\left(x\\right)$ is monotonically decreasing on the interval $({\\frac{π}{{16}},\\frac{π}{8}})$, then the maximum value of $\\omega$ is ______. Please output the final answer with... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "39276",
"split": "train"
} |
math | [
{
"content": "Four distinct integers $a, b, c$, and $d$ are chosen from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$. What is the greatest possible value of $ac+bd-ad-bc$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "39277",
"split": "train"
} |
math | [
{
"content": "The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "39278",
"split": "train"
} |
math | [
{
"content": "Let $a,b$ be real numbers such that $|a| \\neq |b|$ and $\\frac{a+b}{a-b}+\\frac{a-b}{a+b}=6.$ Find the value of the expression $\\frac{a^3+b^3}{a^3-b^3}+\\frac{a^3-b^3}{a^3+b^3}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18/7",
"style": "rule"
} | {
"index": "39279",
"split": "train"
} |
math | [
{
"content": "The Rotokas alphabet, consisting of letters A, E, G, I, K, O, P, R, T, U, and V (note S is removed from the original alphabet), is used to design four-letter license plates. How many license plates are possible that start with E, end with O, cannot contain I, and must not have repeated letters? Pl... | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "39280",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39281",
"split": "train"
} |
math | [
{
"content": "Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "39282",
"split": "train"
} |
math | [
{
"content": "The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39283",
"split": "train"
} |
math | [
{
"content": "Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "39284",
"split": "train"
} |
math | [
{
"content": "In her chemistry class, Samantha now needs to prepare a special solution that consists of chemical A, water, and a new substance, chemical B. The proportions in her initial test mix are 40% chemical A, 50% water, and 10% chemical B. She needs to prepare 1.20 liters of this mixture for an experimen... | {
"ground_truth": "0.60",
"style": "rule"
} | {
"index": "39285",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial whose roots are all negative integers. If $a + b + c + d = 2009,$ find $d.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "528",
"style": "rule"
} | {
"index": "39286",
"split": "train"
} |
math | [
{
"content": "Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "39287",
"split": "train"
} |
math | [
{
"content": "What is the measure of $\\angle X Z Y$ if $M$ is the midpoint of $Y Z$, $\\angle X M Z=30^{\\circ}$, and $\\angle X Y Z=15^{\\circ}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75^{\\circ}",
"style": "rule"
} | {
"index": "39288",
"split": "train"
} |
math | [
{
"content": "Given $\\cos (\\alpha - \\pi) = -\\frac{5}{13}$, and $\\alpha$ is an angle in the fourth quadrant, find the value of $\\sin (-2\\pi + \\alpha)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{12}{13}",
"style": "rule"
} | {
"index": "39289",
"split": "train"
} |
math | [
{
"content": "Let $a=\\sqrt{17}$ and $b=i \\sqrt{19}$, where $i=\\sqrt{-1}$. Find the maximum possible value of the ratio $|a-z| /|b-z|$ over all complex numbers $z$ of magnitude 1 (i.e. over the unit circle $|z|=1$ ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "39290",
"split": "train"
} |
math | [
{
"content": "Given $ \\sin \\alpha = \\frac{1}{3} $, and $ 0 < \\alpha < \\pi $, then $ \\tan \\alpha = $_____, and $ \\sin \\frac{\\alpha}{2} + \\cos \\frac{\\alpha}{2} = $_____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "39291",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. P... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "39292",
"split": "train"
} |
math | [
{
"content": "For the specific example $M=5$, find a value of $k$, not necessarily the smallest, such that $\\sum_{n=1}^{k} \\frac{1}{n}>M$. Justify your answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "39293",
"split": "train"
} |
math | [
{
"content": "The value of $\\tan {75}^{{o}}$ is $\\dfrac{\\sqrt{6}+\\sqrt{2}}{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2+\\sqrt{3}",
"style": "rule"
} | {
"index": "39294",
"split": "train"
} |
math | [
{
"content": "Given the vectors $\\overrightarrow{m}=(\\cos x,\\sin x)$ and $\\overrightarrow{n}=(2 \\sqrt {2}+\\sin x,2 \\sqrt {2}-\\cos x)$, and the function $f(x)= \\overrightarrow{m}\\cdot \\overrightarrow{n}$, where $x\\in R$.\n(I) Find the maximum value of the function $f(x)$;\n(II) If $x\\in(-\\frac {3π}... | {
"ground_truth": "-\\frac {3 \\sqrt {5}+1}{8}",
"style": "rule"
} | {
"index": "39295",
"split": "train"
} |
math | [
{
"content": "Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum possible value of $N$ ?\n\n*2019 CCA Math Bonanza Lightning Round #3.1* Please output the final answer within \\bo... | {
"ground_truth": "154",
"style": "rule"
} | {
"index": "39296",
"split": "train"
} |
math | [
{
"content": "What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "39297",
"split": "train"
} |
math | [
{
"content": "For each even positive integer $x$, let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square. Please outpu... | {
"ground_truth": "899",
"style": "rule"
} | {
"index": "39298",
"split": "train"
} |
math | [
{
"content": "How many perfect squares are factors of 180? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "39299",
"split": "train"
} |
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