data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Triangle $DEF$ is isosceles with angle $E$ congruent to angle $F$. The measure of angle $F$ is three times the measure of angle $D$. What is the number of degrees in the measure of angle $E$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{540}{7}",
"style": "rule"
} | {
"index": "1100",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\\triangle ABC$. If the length of segment $AX$ can be written as $\\frac{a}{b}$, where $a$ and $b$ are relati... | {
"ground_truth": "6710",
"style": "rule"
} | {
"index": "1101",
"split": "train"
} |
math | [
{
"content": "Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "396",
"style": "rule"
} | {
"index": "1102",
"split": "train"
} |
math | [
{
"content": "If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1103",
"split": "train"
} |
math | [
{
"content": "If the probability that a baby born in a certain hospital will speak in the next day is 1/4, what is the probability that at least 2 babies out of a cluster of 5 babies will speak tomorrow? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{47}{128}",
"style": "rule"
} | {
"index": "1104",
"split": "train"
} |
math | [
{
"content": "Ten points are spaced around at equal intervals on the circumference of a regular pentagon, each side being further divided into two equal segments. Two of the 10 points are chosen at random. What is the probability that the two points are exactly one side of the pentagon apart?\nA) $\\frac{1}{5}$... | {
"ground_truth": "\\frac{2}{9}",
"style": "rule"
} | {
"index": "1105",
"split": "train"
} |
math | [
{
"content": "Find \\( x_{1000} \\) if \\( x_{1} = 4 \\), \\( x_{2} = 6 \\), and for any natural \\( n \\geq 3 \\), \\( x_{n} \\) is the smallest composite number greater than \\( 2 x_{n-1} - x_{n-2} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2002",
"style": "rule"
} | {
"index": "1106",
"split": "train"
} |
math | [
{
"content": "How many different positive values of \\( x \\) will make this statement true: there are exactly 3 three-digit multiples of \\( x \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "1107",
"split": "train"
} |
math | [
{
"content": "Given an infinite geometric sequence $\\{a_n\\}$, the product of its first $n$ terms is $T_n$, and $a_1 > 1$, $a_{2008}a_{2009} > 1$, $(a_{2008} - 1)(a_{2009} - 1) < 0$, determine the maximum positive integer $n$ for which $T_n > 1$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "4016",
"style": "rule"
} | {
"index": "1108",
"split": "train"
} |
math | [
{
"content": "In a regular tetrahedron ABCD with an edge length of 2, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. The surface area of the circumscribed sphere of the tetrahedron M-BCD is __________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\pi",
"style": "rule"
} | {
"index": "1109",
"split": "train"
} |
math | [
{
"content": "Convert $1729_{10}$ to base 6. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12001_6",
"style": "rule"
} | {
"index": "1110",
"split": "train"
} |
math | [
{
"content": "In a circle, parallel chords of lengths 5, 12, and 13 determine central angles of $\\theta$, $\\phi$, and $\\theta + \\phi$ radians, respectively, where $\\theta + \\phi < \\pi$. If $\\sin \\theta$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "1111",
"split": "train"
} |
math | [
{
"content": "What is the area enclosed by the graph of $|x| + |3y| = 9$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "1112",
"split": "train"
} |
math | [
{
"content": "The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1-\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "1113",
"split": "train"
} |
math | [
{
"content": "The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1114",
"split": "train"
} |
math | [
{
"content": "Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket? Please output the final answer withi... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1115",
"split": "train"
} |
math | [
{
"content": "In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes, each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are considered the same... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "1116",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xoy$, the parametric equations of the curve $C$ are $x=3\\cos \\alpha$ and $y=\\sin \\alpha$ ($\\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of the... | {
"ground_truth": "\\frac{18\\sqrt{2}}{5}",
"style": "rule"
} | {
"index": "1117",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin (\\omega x+\\varphi)$ $(0 < \\omega < 3,0 < \\varphi < \\pi)$, if $x=-\\frac{\\pi}{4}$ is a zero of the function $f(x)$, and $x=\\frac{\\pi}{3}$ is an axis of symmetry for the graph of the function $f(x)$, then the value of $\\omega$ is \\_\\_\\_\\_. Please output th... | {
"ground_truth": "\\frac{6}{7}",
"style": "rule"
} | {
"index": "1118",
"split": "train"
} |
math | [
{
"content": "The quantity $\\tan 7.5^\\circ$ can be expressed in the form\n\\[\\tan 7.5^\\circ = \\sqrt{a} - \\sqrt{b} + \\sqrt{c} - d,\\]where $a \\ge b \\ge c \\ge d$ are positive integers. Find $a + b + c + d.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "1119",
"split": "train"
} |
math | [
{
"content": "Jake will roll two standard six-sided dice and make a two-digit number from the numbers he rolls. If he rolls a 4 and a 2, he can form either 42 or 24. What is the probability that he will be able to make an integer between 30 and 40, inclusive? Express your answer as a common fraction. Please out... | {
"ground_truth": "\\frac{11}{36}",
"style": "rule"
} | {
"index": "1120",
"split": "train"
} |
math | [
{
"content": "Consider a 9x9 chessboard where the squares are labelled from a starting square at the bottom left (1,1) increasing incrementally across each row to the top right (9,9). Each square at position $(i,j)$ is labelled with $\\frac{1}{i+j-1}$. Nine squares are chosen such that there is exactly one chos... | {
"ground_truth": "\\frac{1}{362880}",
"style": "rule"
} | {
"index": "1121",
"split": "train"
} |
math | [
{
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40\\pi",
"style": "rule"
} | {
"index": "1122",
"split": "train"
} |
math | [
{
"content": "In the number \\(2 * 0 * 1 * 6 * 0 * 2 *\\), each of the 6 asterisks needs to be replaced with any of the digits \\(0, 2, 4, 5, 7, 9\\) (digits can be repeated) so that the resulting 12-digit number is divisible by 75. How many ways can this be done? Please output the final answer within \\boxed{}... | {
"ground_truth": "2592",
"style": "rule"
} | {
"index": "1123",
"split": "train"
} |
math | [
{
"content": "Given quadrilateral ABCD, ∠A = 120∘, and ∠B and ∠D are right angles. Given AB = 13 and AD = 46, find the length of AC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "1124",
"split": "train"
} |
math | [
{
"content": "You have four tiles marked X and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that no two X tiles are adjacent to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{35}",
"style": "rule"
} | {
"index": "1125",
"split": "train"
} |
math | [
{
"content": "I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "1126",
"split": "train"
} |
math | [
{
"content": "Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{3+\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "1127",
"split": "train"
} |
math | [
{
"content": "If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\\sin(\\pi x)$ ($0<x<2$), find the smallest positive period for $y=\\tan\\left(\\frac{(a+b)x}{2}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "1128",
"split": "train"
} |
math | [
{
"content": "For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$. What is the $x$-coordinate of the other $x$-intercept? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "1129",
"split": "train"
} |
math | [
{
"content": "If an integer $a$ ($a \\neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1130",
"split": "train"
} |
math | [
{
"content": "While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\\%$ of his bill and Joe tipped $20\\%$ of his bill. What was the difference, in dollars between their bills? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "1131",
"split": "train"
} |
math | [
{
"content": "Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\\left(A|B\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "1132",
"split": "train"
} |
math | [
{
"content": "The diagram shows a right-angled triangle \\( ACD \\) with a point \\( B \\) on the side \\( AC \\). The sides of triangle \\( ABD \\) have lengths 3, 7, and 8. What is the area of triangle \\( BCD \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "1133",
"split": "train"
} |
math | [
{
"content": "For certain ordered pairs $(a,b)\\,$ of real numbers, the system of equations\n\\[\\begin{aligned} ax+by&=1 \\\\ x^2 + y^2 &= 50 \\end{aligned}\\]has at least one solution, and each solution is an ordered pair $(x,y)\\,$ of integers. How many such ordered pairs $(a,b)\\,$ are there? Please output... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "1134",
"split": "train"
} |
math | [
{
"content": "Let \\(a, b, c, d\\) be positive integers such that \\(a^5 =\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "757",
"style": "rule"
} | {
"index": "1135",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/2000$. Find $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "1136",
"split": "train"
} |
math | [
{
"content": "What is the degree measure of angle $LOQ$ when polygon $\\allowbreak LMNOPQ$ is a regular hexagon? [asy]\ndraw((-2,0)--(-1,1.73205081)--(1,1.73205081)--(2,0)--(1,-1.73205081)--(-1,-1.73205081)--cycle);\ndraw((-1,-1.73205081)--(1,1.73205081)--(1,-1.73205081)--cycle);\nlabel(\"L\",(-1,-1.73205081),S... | {
"ground_truth": "30^\\circ",
"style": "rule"
} | {
"index": "1137",
"split": "train"
} |
math | [
{
"content": "In triangle $DEF$, we have $\\angle D = 90^\\circ$ and $\\sin E = \\frac{3}{5}$. Find $\\cos F$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "1138",
"split": "train"
} |
math | [
{
"content": "How many four-digit integers are divisible by both 7 and 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "257",
"style": "rule"
} | {
"index": "1139",
"split": "train"
} |
math | [
{
"content": "Through the vertices \\( A, C, D \\) of the parallelogram \\( ABCD \\) with sides \\( AB = 7 \\) and \\( AD = 4 \\), a circle is drawn that intersects the line \\( BD \\) at point \\( E \\), and \\( DE = 13 \\). Find the length of diagonal \\( BD \\). Please output the final answer within \\boxed{... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1140",
"split": "train"
} |
math | [
{
"content": "Given that $α$ is an angle in the third quadrant and $\\cos 2α=-\\frac{3}{5}$, find $\\tan (\\frac{π}{4}+2α)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{7}",
"style": "rule"
} | {
"index": "1141",
"split": "train"
} |
math | [
{
"content": "In what ratio does the angle bisector of the acute angle divide the area of a right trapezoid inscribed in a circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1:1",
"style": "rule"
} | {
"index": "1142",
"split": "train"
} |
math | [
{
"content": "A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form\n\\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ Please output the final answ... | {
"ground_truth": "49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0",
"style": "rule"
} | {
"index": "1143",
"split": "train"
} |
math | [
{
"content": "Let $x$ be a real number selected uniformly at random between 100 and 200. If $\\lfloor {\\sqrt{x}} \\rfloor = 12$, find the probability that $\\lfloor {\\sqrt{100x}} \\rfloor = 120$. ($\\lfloor {v} \\rfloor$ means the greatest integer less than or equal to $v$.) Please output the final answer wit... | {
"ground_truth": "\\frac{241}{2500}",
"style": "rule"
} | {
"index": "1144",
"split": "train"
} |
math | [
{
"content": "There are 54 students in a class, and there are 4 tickets for the Shanghai World Expo. Now, according to the students' ID numbers, the tickets are distributed to 4 students through systematic sampling. If it is known that students with ID numbers 3, 29, and 42 have been selected, then the ID numbe... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1145",
"split": "train"
} |
math | [
{
"content": "Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simulta... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "1146",
"split": "train"
} |
math | [
{
"content": "There exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ unique integers $a_k$ ($1\\le k\\le r$) with each $a_k$ either $1$ or $- 1$ such that \\[a_13^{n_1} + a_23^{n_2} + \\cdots + a_r3^{n_r} = 2008.\\] Find $n_1 + n_2 + \\cdots + n_r$. Please output the final answer within... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "1147",
"split": "train"
} |
math | [
{
"content": "The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum\n\\[ \\frac{1}{1-p} + \\frac{1}{1-q} +\\frac{1}{1-r} ?\\] Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1148",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "1149",
"split": "train"
} |
math | [
{
"content": "Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "1150",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi-2}{4}",
"style": "rule"
} | {
"index": "1151",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $\\angle BAC=40^\\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\\parallel OG$ , find $\\angle ACB$ . Please output the final answer... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "1152",
"split": "train"
} |
math | [
{
"content": "Twenty switches in an office computer network are to be connected so that each switch has a direct connection to exactly three other switches. How many connections will be necessary? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1153",
"split": "train"
} |
math | [
{
"content": "Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \\frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\\triangle DEF$ to the area of quadrilateral $ABEF$ is: Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "1: 5",
"style": "rule"
} | {
"index": "1154",
"split": "train"
} |
math | [
{
"content": "Simplify $2(3-i)+i(2+i)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "1155",
"split": "train"
} |
math | [
{
"content": "Find the next two smallest juicy numbers after 6, and show a decomposition of 1 into unit fractions for each of these numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12, 15",
"style": "rule"
} | {
"index": "1156",
"split": "train"
} |
math | [
{
"content": "Given that $P$ is a moving point on the curve $y= \\frac {1}{4}x^{2}- \\frac {1}{2}\\ln x$, and $Q$ is a moving point on the line $y= \\frac {3}{4}x-1$, then the minimum value of $PQ$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {2-2\\ln 2}{5}",
"style": "rule"
} | {
"index": "1157",
"split": "train"
} |
math | [
{
"content": "In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks must be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 10-digit number is divisible by 18. How many ways can this be done? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "3645",
"style": "rule"
} | {
"index": "1158",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the maximum possible value of $x_1x_2 + x_2x_3 + \\cdots + x_6x_1$ where $x_1, x_2, \\dots, x_6$ is a permutation of $(1,2,3,4,5,6)$ and let $Q$ be the number of permutations for which this maximum is achieved, given the additional condition that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 21$... | {
"ground_truth": "83",
"style": "rule"
} | {
"index": "1159",
"split": "train"
} |
math | [
{
"content": "How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "1160",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system xOy, the parametric equation of line l is $$\\begin{cases} x=4+ \\frac { \\sqrt {2}}{2}t \\\\ y=3+ \\frac { \\sqrt {2}}{2}t\\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positiv... | {
"ground_truth": "\\frac{86}{7}",
"style": "rule"
} | {
"index": "1161",
"split": "train"
} |
math | [
{
"content": "Find the number of integers $n$ with $1 \\leq n \\leq 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of 1001. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "1162",
"split": "train"
} |
math | [
{
"content": "The equation $z^6+z^3+1=0$ has complex roots with argument $\\theta$ between $90^\\circ$ and $180^\\circ$ in the complex plane. Determine the degree measure of $\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "1163",
"split": "train"
} |
math | [
{
"content": "Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.\n\nOne day, Uncle Chernomor asked each warrior four questions:\n- Is your favorite weapon a sword?\n- Is your favorite weapon a s... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1164",
"split": "train"
} |
math | [
{
"content": "Inside a square with side length 12, two congruent equilateral triangles are drawn such that each has one vertex touching two adjacent vertices of the square and they share one side. What is the side length of the largest square that can be inscribed in the space inside the square and outside of t... | {
"ground_truth": "12 - 4\\sqrt{3}",
"style": "rule"
} | {
"index": "1165",
"split": "train"
} |
math | [
{
"content": "We have an equilateral triangle with circumradius $1$ . We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$ , is maximum.\n(The total distance of the point P from the... | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "1166",
"split": "train"
} |
math | [
{
"content": "The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$ , where $n$ and $k$ are integers and $0\\leq k<2013$ . What is $k$ ? Recall $2013=3\\cdot 11\\cdot 61$ . Please output the final answer within \\bo... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "1167",
"split": "train"
} |
math | [
{
"content": "In a class at school, all students are the same age, except seven of them who are 1 year younger and two of them who are 2 years older. The sum of the ages of all the students in this class is 330. How many students are in this class? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "1168",
"split": "train"
} |
math | [
{
"content": "Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given in usual (x, y) notat... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "1169",
"split": "train"
} |
math | [
{
"content": "A rectangular table of dimensions \\( x \\) cm \\(\\times 80\\) cm is covered with identical sheets of paper of size \\( 5 \\) cm \\(\\times 8 \\) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of t... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "1170",
"split": "train"
} |
math | [
{
"content": "How many natural numbers greater than 6 but less than 60 are relatively prime to 15? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "1171",
"split": "train"
} |
math | [
{
"content": "The diagram shows a square, its two diagonals, and two line segments, each of which connects two midpoints of the sides of the square. What fraction of the area of the square is shaded?\n\nA) $\\frac{1}{8}$ \nB) $\\frac{1}{10}$ \nC) $\\frac{1}{12}$ \nD) $\\frac{1}{16}$ \nE) $\\frac{1}{24}$ Ple... | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "1172",
"split": "train"
} |
math | [
{
"content": "Given the square of an integer $x$ is 1521, what is the value of $(x+1)(x-1)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1520",
"style": "rule"
} | {
"index": "1173",
"split": "train"
} |
math | [
{
"content": "Given the numbers \\( x, y, z \\in [0, \\pi] \\), find the minimum value of the expression\n\n$$\nA = \\cos (x - y) + \\cos (y - z) + \\cos (z - x)\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "1174",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $sin(A+\\frac{π}{4})sin(B+\\frac{π}{4})=cosAcosB$. Find:<br/>\n$(1)$ the value of angle $C$;<br/>\n$(2)$ if $AB=\\sqrt{2}$, find the minimum value of $\\overrightarrow{CA}•\\overrightarrow{CB}$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "-\\sqrt{2}+1",
"style": "rule"
} | {
"index": "1175",
"split": "train"
} |
math | [
{
"content": "The attached figure is an undirected graph. The circled numbers represent the nodes, and the numbers along the edges are their lengths (symmetrical in both directions). An Alibaba Hema Xiansheng carrier starts at point A and will pick up three orders from merchants B_{1}, B_{2}, B_{3} and deliver ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1176",
"split": "train"
} |
math | [
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n\\frac{x}{21} & \\text{ if }x\\text{ is a multiple of 3 and 7}, \\\\\n3x & \\text{ if }x\\text{ is only a multiple of 7}, \\\\\n7x & \\text{ if }x\\text{ is only a multiple of 3}, \\\\\nx+3 & \\text{ if }x\\text{ is not a multiple of 3 or 7}.\n\\end{arr... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "1177",
"split": "train"
} |
math | [
{
"content": "In \\\\(\\triangle ABC\\\\), the sides opposite to angles \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) are \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\) respectively. It is given that \\\\(a+b=5\\\\), \\\\(c=\\sqrt{7}\\\\), and \\\\(4{{\\left( \\sin \\frac{A+B}{2} \\right)}^{2}}-\\cos 2C=\\frac{7}{2}\\\\... | {
"ground_truth": "\\frac {3 \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "1178",
"split": "train"
} |
math | [
{
"content": "In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \\le n \\le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original starting position afte... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "1179",
"split": "train"
} |
math | [
{
"content": "The equations of the asymptotes of the hyperbola $\\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "1180",
"split": "train"
} |
math | [
{
"content": "Determine the value of $x$ for which $10^x \\cdot 500^{x} = 1000000^{3}$.\nA) $\\frac{9}{1.699}$\nB) $6$\nC) $\\frac{18}{3.699}$\nD) $5$\nE) $20$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{18}{3.699}",
"style": "rule"
} | {
"index": "1181",
"split": "train"
} |
math | [
{
"content": "Fran writes the numbers \\(1,2,3, \\ldots, 20\\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \\(n\\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \\(n\\) that are... | {
"ground_truth": "\\frac{131}{10}",
"style": "rule"
} | {
"index": "1182",
"split": "train"
} |
math | [
{
"content": "Find the number of subsets of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ that are subsets of neither $\\{1, 2, 3, 4, 5\\}$ nor $\\{4, 5, 6, 7, 8\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "196",
"style": "rule"
} | {
"index": "1183",
"split": "train"
} |
math | [
{
"content": "What is the value of $\\frac13\\cdot\\frac92\\cdot\\frac1{27}\\cdot\\frac{54}{1}\\cdot\\frac{1}{81}\\cdot\\frac{162}{1}\\cdot\\frac{1}{243}\\cdot\\frac{486}{1}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1184",
"split": "train"
} |
math | [
{
"content": "Let $x_1$ , $x_2$ , …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "171",
"style": "rule"
} | {
"index": "1185",
"split": "train"
} |
math | [
{
"content": "A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "1186",
"split": "train"
} |
math | [
{
"content": "Given a function defined on the set of positive integers as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{if } n \\geq 1000 \\\\\nf[f(n + 7)], & \\text{if } n < 1000 \n\\end{cases} \\]\nFind the value of \\( f(90) \\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "999",
"style": "rule"
} | {
"index": "1187",
"split": "train"
} |
math | [
{
"content": "Given that point \\( P \\) lies on the hyperbola \\(\\frac{x^{2}}{16} - \\frac{y^{2}}{9} = 1\\), and the distance from \\( P \\) to the right directrix of this hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of this hyperbola, find the x-coordinate of \\( P \\). Pl... | {
"ground_truth": "-\\frac{64}{5}",
"style": "rule"
} | {
"index": "1188",
"split": "train"
} |
math | [
{
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{2, 3, 5, 5, 5, 6, 6\\}$ if no digit may be used more times than it appears in the given set of available digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "1189",
"split": "train"
} |
math | [
{
"content": "A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18\\%",
"style": "rule"
} | {
"index": "1190",
"split": "train"
} |
math | [
{
"content": "In the diagram, the number line between 0 and 2 is divided into 8 equal parts. The numbers 1 and \\(S\\) are marked on the line. What is the value of \\(S\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.25",
"style": "rule"
} | {
"index": "1191",
"split": "train"
} |
math | [
{
"content": "A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "1192",
"split": "train"
} |
math | [
{
"content": "Define the operation \"\" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \\lg|x + 2|$ (where $x \\neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$. Please output the final answer within \\boxe... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "1193",
"split": "train"
} |
math | [
{
"content": "Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points? Please output the fi... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1194",
"split": "train"
} |
math | [
{
"content": "Find, with proof, the number of positive integers whose base- $n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest f... | {
"ground_truth": "\\[ 2^{n+1} - 2(n+1) \\]",
"style": "rule"
} | {
"index": "1195",
"split": "train"
} |
math | [
{
"content": "In the diagram, $D$ and $E$ are the midpoints of $\\overline{AB}$ and $\\overline{BC}$ respectively. Find the sum of the slope and $y$-intercept of the line passing through the points $C$ and $D.$ [asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsize(10pt));\npair A, B, C, D, E, F;\nA=(0,6);\nB=(0... | {
"ground_truth": "\\frac{21}{8}",
"style": "rule"
} | {
"index": "1196",
"split": "train"
} |
math | [
{
"content": "For any positive integers $a$ and $b$ with $b>1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\\sum_{i=1}^{\\left\\lfloor\\log _{23} n\\right\\rfloor} s_{20}\\left(\\left\\lfloor\\frac{n}{23^{i}}\\right\\rfloor\\right... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "1197",
"split": "train"
} |
math | [
{
"content": "Regular octagonal pyramid $\\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyramid. Please output th... | {
"ground_truth": "\\frac{250\\sqrt{3}(1 + \\sqrt{2})}{3}",
"style": "rule"
} | {
"index": "1198",
"split": "train"
} |
math | [
{
"content": "On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "1199",
"split": "train"
} |
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