data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the integer $n$, $0 \\le n \\le 11$, such that \\[n \\equiv 10389 \\pmod{12}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "35100",
"split": "train"
} |
math | [
{
"content": "If set $A=\\{x\\in N\\left|\\right.-1 \\lt x\\leqslant 2\\}$, $B=\\{x\\left|\\right.x=ab,a,b\\in A\\}$, then the number of non-empty proper subsets of set $B$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "35101",
"split": "train"
} |
math | [
{
"content": "Let the real numbers \\(a_1, a_2, \\cdots, a_{100}\\) satisfy the following conditions: (i) \\(a_1 \\geq a_2 \\geq \\cdots \\geq a_{100} \\geq 0\\); (ii) \\(a_1 + a_2 \\leq 100\\); (iii) \\(a_3 + a_4 + \\cdots + a_{100} \\leq 100\\). Find the maximum value of \\(a_1^2 + a_2^2 + \\cdots + a_{100}^2... | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "35102",
"split": "train"
} |
math | [
{
"content": "In tetrahedron \\( SABC \\), the circumradius of triangles \\( \\triangle SAB \\), \\( \\triangle SBC \\), and \\( \\triangle SCA \\) are all 108. The center of the inscribed sphere is \\( I \\) and \\( SI = 125 \\). Let \\( R \\) be the circumradius of \\( \\triangle ABC \\). If \\( R \\) can be ... | {
"ground_truth": "11665",
"style": "rule"
} | {
"index": "35103",
"split": "train"
} |
math | [
{
"content": "In \\(\\triangle ABC\\), \\(D\\) is the midpoint of \\(BC\\), \\(E\\) is the foot of the perpendicular from \\(A\\) to \\(BC\\), and \\(F\\) is the foot of the perpendicular from \\(D\\) to \\(AC\\). Given that \\(BE = 5\\), \\(EC = 9\\), and the area of \\(\\triangle ABC\\) is 84, compute \\(|EF|... | {
"ground_truth": "\\frac{6 \\sqrt{37}}{5}",
"style": "rule"
} | {
"index": "35104",
"split": "train"
} |
math | [
{
"content": "In what ratio does the point \\( P \\) divide the perpendicular segment dropped from vertex \\( A \\) of a regular tetrahedron \\( ABCD \\) to the face \\( BCD \\), given that the lines \\( PB \\), \\( PC \\), and \\( PD \\) are mutually perpendicular to each other? Please output the final answer ... | {
"ground_truth": "1:1",
"style": "rule"
} | {
"index": "35105",
"split": "train"
} |
math | [
{
"content": "Compute the smallest positive integer $k$ such that 49 divides $\\binom{2 k}{k}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "35106",
"split": "train"
} |
math | [
{
"content": "If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \\ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \\ldots + a_{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "364",
"style": "rule"
} | {
"index": "35107",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ be a triangle with a right angle $\\angle ABC$ . Let $D$ be the midpoint of $\\overline{BC}$ , let $E$ be the midpoint of $\\overline{AC}$ , and let $F$ be the midpoint of $\\overline{AB}$ . Let $G$ be the midpoint of $\\overline{EC}$ . One of the angles of ... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "35108",
"split": "train"
} |
math | [
{
"content": "A psychic is faced with a deck of 36 cards placed face down (four suits, with nine cards of each suit). He names the suit of the top card, after which the card is revealed and shown to him. Then the psychic names the suit of the next card, and so on. The goal is for the psychic to guess as many su... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "35109",
"split": "train"
} |
math | [
{
"content": "Given that $7^{-1} \\equiv 55 \\pmod{78}$, find $49^{-1} \\pmod{78}$, as a residue modulo 78. (Give an answer between 0 and 77, inclusive.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "35110",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The length of the base $BC$ is 10 units, and the height from $AB$ to $CD$ is 5 units. The base $AD$ is 22 units. Calculate the perimeter of $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{61} + 32",
"style": "rule"
} | {
"index": "35111",
"split": "train"
} |
math | [
{
"content": "An economist-cryptographer received a cryptogram from the ruler that contained a secret decree to introduce a per-unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that it was not possible to collect a higher amount of tax rev... | {
"ground_truth": "8640",
"style": "rule"
} | {
"index": "35112",
"split": "train"
} |
math | [
{
"content": "Given triangle $ABC$ with midpoint $D$ on side $BC$, and point $G$ satisfies $\\overrightarrow{GA}+ \\overrightarrow{BG}+ \\overrightarrow{CG}= \\overrightarrow{0}$, and $\\overrightarrow{AG}=\\lambda \\overrightarrow{GD}$, determine the value of $\\lambda$. Please output the final answer within \... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "35113",
"split": "train"
} |
math | [
{
"content": "If $\\displaystyle\\frac{a}{b} = 4$, $\\displaystyle\\frac{b}{c} = \\frac{1}{3}$, and $\\displaystyle \\frac{c}{d} = 6$, then what is $\\displaystyle\\frac{d}{a}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "35114",
"split": "train"
} |
math | [
{
"content": "The sum of all roots of the equation $2x^2-3x-5=0$ and the equation $x^2-6x+2=0$ is equal to $\\boxed{\\text{blank}}$, and the product of all roots is equal to $\\boxed{\\text{blank}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7\\frac{1}{2}",
"style": "rule"
} | {
"index": "35115",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ such that it is the mean, median, and mode of the $9$ data values $-10, -5, x, x, 0, 15, 20, 25, 30$.\nA) $\\frac{70}{7}$\nB) $\\frac{75}{7}$\nC) $\\frac{80}{7}$\nD) $10$\nE) $15$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{75}{7}",
"style": "rule"
} | {
"index": "35116",
"split": "train"
} |
math | [
{
"content": "After the implementation of the \"double reduction\" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badmin... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "35117",
"split": "train"
} |
math | [
{
"content": "What is the value of \n1-(-2)-3-(-4)-5-(-6)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "35118",
"split": "train"
} |
math | [
{
"content": "Let $a_n = n(2n+1)$ . Evaluate\n\\[\n \\biggl | \\sum_{1 \\le j < k \\le 36} \\sin\\bigl( \\frac{\\pi}{6}(a_k-a_j) \\bigr) \\biggr |.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35119",
"split": "train"
} |
math | [
{
"content": "In the octagon COMPUTER exhibited below, all interior angles are either $90^{\\circ}$ or $270^{\\circ}$ and we have $C O=O M=M P=P U=U T=T E=1$. Point $D$ (not to scale in the diagram) is selected on segment $R E$ so that polygons COMPUTED and $C D R$ have the same area. Find $D R$. Please output ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "35120",
"split": "train"
} |
math | [
{
"content": "A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.\n\nWhat is the value of $8a-4b+2c-d$?\n\n[asy]\nimport graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;\n\npen cqcqcq=rgb(0.75,0.75,0.7... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "35121",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 3$ and $PT = AB - PA,$ then what is $PB$? Please output the final answer within \\boxe... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35122",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, the ordinary equation of curve $C_1$ is $x^2+y^2-2x=0$. Establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C_2$ is $\\rho^{2}= \\frac {3}{1+2\\sin^{2}\\the... | {
"ground_truth": "\\frac { \\sqrt {30}}{5}-1",
"style": "rule"
} | {
"index": "35123",
"split": "train"
} |
math | [
{
"content": "In a circle, chords $A B$ and $C D$, which are not diameters, are drawn perpendicular to each other. Chord $C D$ divides chord $A B$ in the ratio $1:5$, and it divides the longer arc of $A B$ in the ratio $1:2$. In what ratio does chord $A B$ divide chord $C D$? Please output the final answer with... | {
"ground_truth": "1 : 3",
"style": "rule"
} | {
"index": "35124",
"split": "train"
} |
math | [
{
"content": "The side length of square $A$ is 36 cm. The side length of square $B$ is 42 cm. What is the ratio of the area of square $A$ to the area of square $B$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{36}{49}",
"style": "rule"
} | {
"index": "35125",
"split": "train"
} |
math | [
{
"content": "Given that point $P$ is an intersection point of the ellipse $\\frac{x^{2}}{a_{1}^{2}} + \\frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\\frac{x^{2}}{a_{2}^{2}} - \\frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyp... | {
"ground_truth": "\\frac{4 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "35126",
"split": "train"
} |
math | [
{
"content": "The kite \\( ABCD \\) is symmetric with respect to diagonal \\( AC \\). The length of \\( AC \\) is 12 cm, the length of \\( BC \\) is 6 cm, and the internal angle at vertex \\( B \\) is a right angle. Points \\( E \\) and \\( F \\) are given on sides \\( AB \\) and \\( AD \\) respectively, such t... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "35127",
"split": "train"
} |
math | [
{
"content": "Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number ... | {
"ground_truth": "831",
"style": "rule"
} | {
"index": "35128",
"split": "train"
} |
math | [
{
"content": "If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3x^2+4$",
"style": "rule"
} | {
"index": "35129",
"split": "train"
} |
math | [
{
"content": "If I have a $5\\times5$ chess board, in how many ways can I place five 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": "14400",
"style": "rule"
} | {
"index": "35130",
"split": "train"
} |
math | [
{
"content": "In a round-robin hockey tournament, 2016 teams participated. According to the tournament rules, 3 points are awarded for a win. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6093360",
"style": "rule"
} | {
"index": "35131",
"split": "train"
} |
math | [
{
"content": "Calculate the sum: $\\sum_{n=1}^{99} \\left(n^{3}+3n^{2}+3n\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25502400",
"style": "rule"
} | {
"index": "35132",
"split": "train"
} |
math | [
{
"content": "If we divide $2020$ by a prime $p$ , the remainder is $6$ . Determine the largest\npossible value of $p$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "35133",
"split": "train"
} |
math | [
{
"content": "Calculate: $1.23 \\times 67 + 8.2 \\times 12.3 - 90 \\times 0.123$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "172.2",
"style": "rule"
} | {
"index": "35134",
"split": "train"
} |
math | [
{
"content": "The vertex of the parabola $y^2 = 4x$ is $O$, and the coordinates of point $A$ are $(5, 0)$. A line $l$ with an inclination angle of $\\frac{\\pi}{4}$ intersects the line segment $OA$ (but does not pass through points $O$ and $A$) and intersects the parabola at points $M$ and $N$. The maximum area... | {
"ground_truth": "8\\sqrt{2}",
"style": "rule"
} | {
"index": "35135",
"split": "train"
} |
math | [
{
"content": "On the Cartesian plane, the midpoint between two points $P(p,q)$ and $Q(r,s)$ is $N(x,y)$. If $P$ is moved vertically upwards 10 units and horizontally to the right 5 units, and $Q$ is moved vertically downwards 5 units and horizontally to the left 15 units, find the new midpoint $N'$ between $P$ ... | {
"ground_truth": "5.59",
"style": "rule"
} | {
"index": "35136",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) lies on the parabola \\( y = 2x^2 - 3x + 4 \\), and point \\( F \\) lies on the line \\( y = 3x - 4 \\).\n\nFind the minimum value of \\( MF \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7 \\sqrt{10}}{20}",
"style": "rule"
} | {
"index": "35137",
"split": "train"
} |
math | [
{
"content": "Inside the cube \\(ABCD A_{1}B_{1}C_{1}D_{1}\\) there is a sphere with center \\(O\\) and radius 10. The sphere intersects the face \\(AA_{1}D_{1}D\\) in a circle of radius 1, the face \\(A_{1}B_{1}C_{1}D_{1}\\) in a circle of radius 1, and the face \\(CD D_{1}C_{1}\\) in a circle of radius 3. Fin... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35138",
"split": "train"
} |
math | [
{
"content": "Given a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1, what is the maximum number of roots it can have within the interval \\( (0,1) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "35139",
"split": "train"
} |
math | [
{
"content": "Given a circle with center \\(O\\) and radius \\(OD\\) perpendicular to chord \\(AB\\), intersecting \\(AB\\) at point \\(C\\). Line segment \\(AO\\) is extended to intersect the circle at point \\(E\\). If \\(AB = 8\\) and \\(CD = 2\\), what is the area of \\(\\triangle BCE\\)? Please output the ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35140",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xoy$, given point $A(0,-2)$, point $B(1,-1)$, and $P$ is a moving point on the circle $x^{2}+y^{2}=2$, then the maximum value of $\\dfrac{|\\overrightarrow{PB}|}{|\\overrightarrow{PA}|}$ is ______. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "35141",
"split": "train"
} |
math | [
{
"content": "Randomly split 2.5 into the sum of two non-negative numbers, and round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "35142",
"split": "train"
} |
math | [
{
"content": "All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "35143",
"split": "train"
} |
math | [
{
"content": "Part of an \\(n\\)-pointed regular star is shown. It is a simple closed polygon in which all \\(2n\\) edges are congruent, angles \\(A_1,A_2,\\cdots,A_n\\) are congruent, and angles \\(B_1,B_2,\\cdots,B_n\\) are congruent. If the acute angle at \\(A_1\\) is \\(10^\\circ\\) less than the acute angl... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35144",
"split": "train"
} |
math | [
{
"content": "What is the smallest eight-digit positive integer that has exactly four digits which are 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10004444",
"style": "rule"
} | {
"index": "35145",
"split": "train"
} |
math | [
{
"content": "If for any real number $x$, the equation $(1-2x)^{10} = a_0 + a_1x + a_2x^2 + a_3x^3 + \\ldots + a_{10}x^{10}$ holds, then the value of $(a_0 + a_1) + (a_0 + a_2) + (a_0 + a_3) + \\ldots + (a_0 + a_{10})$ is _____. (Answer with a number) Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35146",
"split": "train"
} |
math | [
{
"content": "Translate the graph of the function y = 2sin( $$\\frac {π}{3}$$ - x) - cos( $$\\frac {π}{6}$$ + x) by shifting it to the right by $$\\frac {π}{4}$$ units. Determine the minimum value of the corresponding function. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "35147",
"split": "train"
} |
math | [
{
"content": "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the fro... | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "35148",
"split": "train"
} |
math | [
{
"content": "Two lines passing through point \\( M \\), which lies outside the circle with center \\( O \\), touch the circle at points \\( A \\) and \\( B \\). Segment \\( OM \\) is divided in half by the circle. In what ratio is segment \\( OM \\) divided by line \\( AB \\)? Please output the final answer wi... | {
"ground_truth": "1:3",
"style": "rule"
} | {
"index": "35149",
"split": "train"
} |
math | [
{
"content": "Given four distinct points P, A, B, C on a plane satisfying $\\overrightarrow{PA} + \\overrightarrow{PB} + \\overrightarrow{PC} = \\overrightarrow{0}$ and $\\overrightarrow{AB} + \\overrightarrow{AC} + m\\overrightarrow{AP} = \\overrightarrow{0}$, determine the value of the real number $m$. Please... | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "35150",
"split": "train"
} |
math | [
{
"content": "The equation\n\\[\\frac{1}{x} + \\frac{1}{x + 2} - \\frac{1}{x + 4} - \\frac{1}{x + 6} - \\frac{1}{x + 8} - \\frac{1}{x + 10} + \\frac{1}{x + 12} + \\frac{1}{x + 14} = 0\\]has four roots of the form $-a \\pm \\sqrt{b \\pm c \\sqrt{d}},$ where $a,$ $b,$ $c,$ $d$ are positive integers, and $d$ is no... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "35151",
"split": "train"
} |
math | [
{
"content": "How many of the integers \\(1, 2, \\ldots, 2004\\) can be represented as \\(\\frac{mn+1}{m+n}\\) for positive integers \\(m\\) and \\(n\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "35152",
"split": "train"
} |
math | [
{
"content": "What is the last digit of the decimal expansion of $\\frac{1}{2^{10}}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "35153",
"split": "train"
} |
math | [
{
"content": "Given that $α,β$ satisfy $\\frac{\\sin α}{\\sin (α +2β)}=\\frac{2018}{2019}$, find the value of $\\frac{\\tan (α +β)}{\\tan β}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4037",
"style": "rule"
} | {
"index": "35154",
"split": "train"
} |
math | [
{
"content": "Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express your answer as a common... | {
"ground_truth": "\\frac{5}{18}",
"style": "rule"
} | {
"index": "35155",
"split": "train"
} |
math | [
{
"content": "Simplify\n\\[\\left( \\frac{1 + i}{1 - i} \\right)^{1000}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "35156",
"split": "train"
} |
math | [
{
"content": "A positive integer is called jubilant if the number of 1 's in its binary representation is even. For example, $6=110_{2}$ is a jubilant number. What is the 2009 th smallest jubilant number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4018",
"style": "rule"
} | {
"index": "35157",
"split": "train"
} |
math | [
{
"content": "Compute $\\frac{1-2+4-8+16-32+64-128+256}{2-4+8-16+32-64+128-256+512}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "35158",
"split": "train"
} |
math | [
{
"content": "A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94.5",
"style": "rule"
} | {
"index": "35159",
"split": "train"
} |
math | [
{
"content": "How many times must a die be rolled so that the probability of the inequality\n\n\\[ \\left| \\frac{m}{n} - \\frac{1}{6} \\right| \\leq 0.01 \\]\n\nis at least as great as the probability of the opposite inequality, where \\( m \\) is the number of times a specific face appears in \\( n \\) rolls ... | {
"ground_truth": "632",
"style": "rule"
} | {
"index": "35160",
"split": "train"
} |
math | [
{
"content": "At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form \"sectors\" broken up by the taken slices, e.g. if they take five consecutive slices then the... | {
"ground_truth": "\\frac{11}{3}",
"style": "rule"
} | {
"index": "35161",
"split": "train"
} |
math | [
{
"content": "Define binary operations $\\diamondsuit$ and $\\heartsuit$ by $a \\diamondsuit b = a^{\\log_{7}(b)}$ and $a \\heartsuit b = a^{\\frac{1}{\\log_{7}(b)}}$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \\heartsuit 2$ ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "35162",
"split": "train"
} |
math | [
{
"content": "For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "35163",
"split": "train"
} |
math | [
{
"content": "Compute 1 $2+2 \\cdot 3+\\cdots+(n-1) n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{(n-1) n(n+1)}{3}",
"style": "rule"
} | {
"index": "35164",
"split": "train"
} |
math | [
{
"content": "Given the function\n$$\nf(x) = \\left|8x^3 - 12x - a\\right| + a\n$$\n\nThe maximum value of this function on the interval \\([0, 1]\\) is 0. Find the maximum value of the real number \\(a\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2\\sqrt{2}",
"style": "rule"
} | {
"index": "35165",
"split": "train"
} |
math | [
{
"content": "What is the greatest integer less than 200 for which the greatest common factor of that integer and 72 is 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "189",
"style": "rule"
} | {
"index": "35166",
"split": "train"
} |
math | [
{
"content": "Given real numbers $a_1$, $a_2$, $a_3$ are not all zero, and positive numbers $x$, $y$ satisfy $x+y=2$. Let the maximum value of $$\\frac {xa_{1}a_{2}+ya_{2}a_{3}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ be $M=f(x,y)$, then the minimum value of $M$ is \\_\\_\\_\\_\\_\\_. Please output the final answer wi... | {
"ground_truth": "\\frac { \\sqrt {2}}{2}",
"style": "rule"
} | {
"index": "35167",
"split": "train"
} |
math | [
{
"content": "Find the minimum of the expression \\(6 t^{2} + 3 s^{2} - 4 s t - 8 t + 6 s + 5\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{7}",
"style": "rule"
} | {
"index": "35168",
"split": "train"
} |
math | [
{
"content": "Find all values of \\( a \\) such that the roots \\( x_1, x_2, x_3 \\) of the polynomial\n\\[ x^3 - 6x^2 + ax + a \\]\nsatisfy\n\\[ \\left(x_1 - 3\\right)^3 + \\left(x_2 - 3\\right)^3 + \\left(x_3 - 3\\right)^3 = 0. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "35169",
"split": "train"
} |
math | [
{
"content": "When a number is divided by 7, the remainder is 2. What is the remainder when three times the number minus 7 is divided by 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "35170",
"split": "train"
} |
math | [
{
"content": "The sum of the interior numbers in the sixth row of Pascal's Triangle is 30. What is the sum of the interior numbers of the eighth row? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "35171",
"split": "train"
} |
math | [
{
"content": "Let \\( d_{1}, d_{2}, \\cdots, d_{k} \\) be all the divisors of the positive integer \\( n \\), denoted as:\n\\[ 1 = d_{1} < d_{2} < d_{3} < \\cdots < d_{k} = n. \\]\n\nFind all \\( n \\) such that \\( k \\geq 4 \\) and \\( d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + d_{4}^{2} = n \\). Please output the f... | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "35172",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^3+ax^2+bx+1$, its derivative $f'(x)$ satisfies $f'(1)=2a$, $f'(2)=-b$, where $a$ and $b$ are constants.\n\n(I) Find the values of $a$ and $b$;\n\n(II) Let $g(x)=f'(x)e^{-x}$, find the extreme values of the function $g(x)$. Please output the final answer within \\boxed{}.... | {
"ground_truth": "15e^{-3}",
"style": "rule"
} | {
"index": "35173",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a\\_n\\}$, it is known that $a\\_1+a\\_2+a\\_3=1$, $a\\_4+a\\_5+a\\_6=-2$, find the sum of the first 15 terms of the sequence, denoted as $S\\_{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "35174",
"split": "train"
} |
math | [
{
"content": "A circular spinner used in a game has a radius of 15 cm. The probability of winning on one spin of this spinner is $\\frac{1}{3}$ for the WIN sector and $\\frac{1}{4}$ for the BONUS sector. What is the area, in square centimeters, of both the WIN sector and the BONUS sector? Express your answers i... | {
"ground_truth": "56.25\\pi",
"style": "rule"
} | {
"index": "35175",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=\\sum_{k=2}^{10}(\\lfloor kx \\rfloor -k \\lfloor x \\rfloor)$, where $\\lfloor r \\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \\ge 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "35176",
"split": "train"
} |
math | [
{
"content": "At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "234",
"style": "rule"
} | {
"index": "35177",
"split": "train"
} |
math | [
{
"content": "Find all pairs $(m,n)$ of nonnegative integers for which \\[m^2 + 2 \\cdot 3^n = m\\left(2^{n+1} - 1\\right).\\]\n\n[i] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(9, 3), (6, 3), (9, 5), (54, 5)",
"style": "rule"
} | {
"index": "35178",
"split": "train"
} |
math | [
{
"content": "What is the least natural number that can be added to 71,382 to create a palindrome? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "35179",
"split": "train"
} |
math | [
{
"content": "Consider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "35180",
"split": "train"
} |
math | [
{
"content": "Given sets $A=\\{2, a-1\\}$, $B=\\{a^2-7, -1\\}$, and $A\\cap B=\\{2\\}$, find the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "35181",
"split": "train"
} |
math | [
{
"content": "Find the area of the triangle (see the diagram) on graph paper. (Each side of a square is 1 unit.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "35182",
"split": "train"
} |
math | [
{
"content": "Consider the expression \\(1 \\ast 2 \\ast 3 \\ast 4 \\ast 5 \\ast 6\\).\nEach star in the expression is to be replaced with either ' + ' or ' \\times '.\n\\(N\\) is the largest possible value of the expression. What is the largest prime factor of \\(N\\)? Please output the final answer within \\b... | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "35183",
"split": "train"
} |
math | [
{
"content": "Factor $58x^5-203x^{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-29x^5(7x^6-2)",
"style": "rule"
} | {
"index": "35184",
"split": "train"
} |
math | [
{
"content": "Evaluate\n\\[\\sum_{m = 1}^\\infty \\sum_{n = 1}^\\infty \\frac{1}{mn(m + n + 1)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "35185",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_n \\in \\mathbb{N}^*$, and $S\\_n = \\frac{1}{8}(a\\_n + 2)^2$. If $b\\_n = \\frac{1}{2}a\\_n - 30$, find the minimum value of the sum of the first $\\_\\_\\_\\_\\_\\_$ terms of the sequence $\\{b\\_n\\}$. Please output the final answer within... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35186",
"split": "train"
} |
math | [
{
"content": "The value of $(\\sqrt{1+\\sqrt{1+\\sqrt{1}}})^{4}$ is:\n(a) $\\sqrt{2}+\\sqrt{3}$;\n(b) $\\frac{1}{2}(7+3 \\sqrt{5})$;\n(c) $1+2 \\sqrt{3}$;\n(d) 3 ;\n(e) $3+2 \\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 2 \\sqrt{2}",
"style": "rule"
} | {
"index": "35187",
"split": "train"
} |
math | [
{
"content": "A school has between 130 and 210 students enrolled. Every afternoon, all the students gather to participate in a singing session. The students are divided into eight distinct groups. If two students are absent from school, the groups can all have the same number of students. What is the sum of all... | {
"ground_truth": "1870",
"style": "rule"
} | {
"index": "35188",
"split": "train"
} |
math | [
{
"content": "In an isosceles triangle \\( \\triangle AMC \\), \\( AM = AC \\), the median \\( MV = CU = 12 \\), and \\( MV \\perp CU \\) at point \\( P \\). What is the area of \\( \\triangle AMC \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "35189",
"split": "train"
} |
math | [
{
"content": "Given a right quadrilateral pyramid \\( V-ABCD \\) with the height equal to half the length of \\( AB \\), \\( M \\) is the midpoint of the lateral edge \\( VB \\), and \\( N \\) is a point on the lateral edge \\( VD \\) such that \\( DN=2VN \\). Find the cosine of the angle between the skew lines... | {
"ground_truth": "\\frac{\\sqrt{11}}{11}",
"style": "rule"
} | {
"index": "35190",
"split": "train"
} |
math | [
{
"content": "An amoeba is placed in a puddle one day, and on that same day it splits into two amoebas. The next day, each new amoeba splits into two new amoebas, and so on, so that each day every living amoeba splits into two new amoebas. After one week, how many amoebas are in the puddle? (Assume the puddle ... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "35191",
"split": "train"
} |
math | [
{
"content": "A hexagonal lattice now consists of 13 points where each point is one unit distance from its nearest neighbors. Points are arranged with one center point, six points forming a hexagon around the center, and an additional outer layer of six points forming a larger hexagon. How many equilateral tria... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35192",
"split": "train"
} |
math | [
{
"content": "If we express $x^2 - 5x$ in the form $a(x - h)^2 + k$, then what is $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{25}{4}",
"style": "rule"
} | {
"index": "35193",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a convex trapezoid such that $\\angle A B C=\\angle B C D=90^{\\circ}, A B=3, B C=6$, and $C D=12$. Among all points $X$ inside the trapezoid satisfying $\\angle X B C=\\angle X D A$, compute the minimum possible value of $C X$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\sqrt{113}-\\sqrt{65}",
"style": "rule"
} | {
"index": "35194",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\sin (C-A)=1$, $\\sin B= \\frac{1}{3}$.\n(I) Find the value of $\\sin A$;\n(II) Given $b= \\sqrt{6}$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "35195",
"split": "train"
} |
math | [
{
"content": "The geometric sequence $\\left\\{a_{n}\\right\\}$ has the first term $a_{1}=1536$, and common ratio $q=-\\frac{1}{2}$. Let $\\Pi_{n}$ denote the product of its first $n$ terms $\\left(n \\in \\mathbf{N}^{*}\\right)$. Find the value of $n$ that maximizes $\\Pi_{n}\\$. Please output the final answer... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "35196",
"split": "train"
} |
math | [
{
"content": "There are 49 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then he eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 49 minutes? Please output the final answer wit... | {
"ground_truth": "1176",
"style": "rule"
} | {
"index": "35197",
"split": "train"
} |
math | [
{
"content": "On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "35198",
"split": "train"
} |
math | [
{
"content": "Kai has begun to list, in ascending order, the positive integers which are not factors of 240. What is the sixth number on Kai's list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35199",
"split": "train"
} |
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