data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "On the Cartesian plane, the midpoint between two points $A(a,b)$ and $B(c,d)$ is $M(m,n)$. If $A$ is moved vertically upwards 8 units and horizontally to the right 2 units, and $B$ is moved vertically downwards 2 units and horizontally to the left 10 units, then the new midpoint between $A$ and $B... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "4700",
"split": "train"
} |
math | [
{
"content": "Compute the number of digits is $2015!$ . Your score will be given by $\\max\\{\\lfloor125(\\min\\{\\tfrac{A}{C},\\tfrac{C}{A}\\}-\\tfrac{1}{5})\\rfloor,0\\}$ , where $A$ is your answer and $C$ is the actual answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5787",
"style": "rule"
} | {
"index": "4701",
"split": "train"
} |
math | [
{
"content": "The angles of quadrilateral $PQRS$ satisfy $\\angle P = 3\\angle Q = 4\\angle R = 6\\angle S$. What is the degree measure of $\\angle P$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "206",
"style": "rule"
} | {
"index": "4702",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a convex quadrilateral inscribed in a circle with shortest side $A B$. The ratio $[B C D] /[A B D]$ is an integer (where $[X Y Z]$ denotes the area of triangle $X Y Z$.) If the lengths of $A B, B C, C D$, and $D A$ are distinct integers no greater than 10, find the largest possibl... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "4703",
"split": "train"
} |
math | [
{
"content": "How many positive 3-digit numbers are multiples of 30, but not of 75? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "4704",
"split": "train"
} |
math | [
{
"content": "There are 10 sheikhs each with a harem of 100 wives standing on the bank of a river along with a yacht that can hold $n$ passengers. According to the law, a woman must not be on the same bank, on the yacht, or at any stopover point with a man unless her husband is present. What is the smallest val... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4705",
"split": "train"
} |
math | [
{
"content": "A given sequence $r_1, r_2, \\dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more \"bubble passes\". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller... | {
"ground_truth": "931",
"style": "rule"
} | {
"index": "4706",
"split": "train"
} |
math | [
{
"content": "Given the sequence elements \\( a_{n} \\) such that \\( a_{1}=1337 \\) and \\( a_{2n+1}=a_{2n}=n-a_{n} \\) for all positive integers \\( n \\). Determine the value of \\( a_{2004} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": "4707",
"split": "train"
} |
math | [
{
"content": "The NASA Space Shuttle transports material to the International Space Station at a cost of $\\$22,\\!000$ per kilogram. What is the number of dollars in the cost of transporting a 250 g control module? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5500",
"style": "rule"
} | {
"index": "4708",
"split": "train"
} |
math | [
{
"content": "Given a fair coin is tossed, the probability of heads or tails is both $\\frac{1}{2}$. Construct a sequence $\\{a_n\\}$, where $a_n = \\begin{cases} 1, \\text{if heads on the nth toss} \\\\ -1, \\text{if tails on the nth toss} \\end{cases}$. Let $S_n = a_1 + a_2 + ... + a_n$. Find the probability ... | {
"ground_truth": "\\frac{13}{128}",
"style": "rule"
} | {
"index": "4709",
"split": "train"
} |
math | [
{
"content": "Find the minimum distance between point $P$ on the line $y = 2x + 1$ and point $Q$ on the curve $y = x + \\ln{x}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "4710",
"split": "train"
} |
math | [
{
"content": "A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the... | {
"ground_truth": "372",
"style": "rule"
} | {
"index": "4711",
"split": "train"
} |
math | [
{
"content": "The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin. \nOne of these numbers is the reciprocal of $F$. Which one? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "C",
"style": "rule"
} | {
"index": "4712",
"split": "train"
} |
math | [
{
"content": "Given $\\sin\\left( \\frac{\\pi}{3} + a \\right) = \\frac{5}{13}$, and $a \\in \\left( \\frac{\\pi}{6}, \\frac{2\\pi}{3} \\right)$, find the value of $\\sin\\left( \\frac{\\pi}{12} + a \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17\\sqrt{2}}{26}",
"style": "rule"
} | {
"index": "4713",
"split": "train"
} |
math | [
{
"content": "Find the largest prime divisor of $25^2+72^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "157",
"style": "rule"
} | {
"index": "4714",
"split": "train"
} |
math | [
{
"content": "Given \\(\\alpha \\in (0, \\pi)\\), if \\(\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{3}}{3}\\), then \\(\\cos^2 \\alpha - \\sin^2 \\alpha = \\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "4715",
"split": "train"
} |
math | [
{
"content": "A two-digit integer $AB$ equals $\\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "4716",
"split": "train"
} |
math | [
{
"content": "1. In the Cartesian coordinate system, given three points A(5, 4), B(k, 10), and C(12, -2), for what value of k are the vectors $$\\overrightarrow {AB}$$ and $$\\overrightarrow {BC}$$ collinear?\n2. In the Cartesian coordinate system, given O as the origin, $$\\overrightarrow {OA}=(-7, 6)$$, $$\\o... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4717",
"split": "train"
} |
math | [
{
"content": "Let's say you have three numbers A, B, and C, which satisfy the equations $2002C + 4004A = 8008$, and $3003B - 5005A = 7007$. What is the average of A, B, and C?\n**A)** $\\frac{7}{3}$ \n**B)** $7$ \n**C)** $\\frac{26}{9}$ \n**D)** $\\frac{22}{9}$ \n**E)** $0$ Please output the final answer w... | {
"ground_truth": "\\frac{22}{9}",
"style": "rule"
} | {
"index": "4718",
"split": "train"
} |
math | [
{
"content": "A pirate finds three chests on the wrecked ship S.S. Triumph, recorded in base 7. The chests contain $3214_7$ dollars worth of silver, $1652_7$ dollars worth of precious stones, $2431_7$ dollars worth of pearls, and $654_7$ dollars worth of ancient coins. Calculate the total value of these treasur... | {
"ground_truth": "3049",
"style": "rule"
} | {
"index": "4719",
"split": "train"
} |
math | [
{
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $6$ and are placed in the plane so that each circle is externally tangent to the other two. Points $Q_1$, $Q_2$, and $Q_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that triangle $\\triangle Q_1Q_2Q_3$ is a... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "4720",
"split": "train"
} |
math | [
{
"content": "In the equation $\\frac{1}{7} + \\frac{7}{x} = \\frac{16}{x} + \\frac{1}{16}$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "4721",
"split": "train"
} |
math | [
{
"content": "Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an \"Audi\" and a \"Jetta\". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determine the number of diff... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "4722",
"split": "train"
} |
math | [
{
"content": "The height of a right triangle $ABC$ dropped to the hypotenuse is 9.6. From the vertex $C$ of the right angle, a perpendicular $CM$ is raised to the plane of the triangle $ABC$, with $CM = 28$. Find the distance from point $M$ to the hypotenuse $AB$. Please output the final answer within \\boxed{}... | {
"ground_truth": "29.6",
"style": "rule"
} | {
"index": "4723",
"split": "train"
} |
math | [
{
"content": "Compute $\\sqrt[4]{256000000}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40\\sqrt{10}",
"style": "rule"
} | {
"index": "4724",
"split": "train"
} |
math | [
{
"content": "A hexagonal prism has a regular hexagonal base, and its lateral edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism are on the same spherical surface, and the volume of the hexagonal prism is $\\frac{9}{8}$, with a base perimeter of 3. The volume of this s... | {
"ground_truth": "\\frac{4\\pi}{3}",
"style": "rule"
} | {
"index": "4725",
"split": "train"
} |
math | [
{
"content": "Starting from $2$, the sum of consecutive even numbers is shown in the table below:\n\n| Number of terms $n$ | $S$ |\n|----------------------|-----------|\n| $1$ | $2=1\\times 2$ |\n| $2$ | $2+4=6=2\\times 3$ |\n| $3$ | $2+4+6=12=3\\times 4$... | {
"ground_truth": "8792",
"style": "rule"
} | {
"index": "4726",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\dfrac{13! - 12! - 2 \\times 11!}{10!}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1430",
"style": "rule"
} | {
"index": "4727",
"split": "train"
} |
math | [
{
"content": "Let $x, y$ be two positive integers, with $x> y$ , such that $2n = x + y$ , where n is a number two-digit integer. If $\\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "4728",
"split": "train"
} |
math | [
{
"content": "Harvard has recently built a new house for its students consisting of $n$ levels, where the $k$ th level from the top can be modeled as a 1-meter-tall cylinder with radius $k$ meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4729",
"split": "train"
} |
math | [
{
"content": "There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=3$, $e=-a-c$, and the sum of the numbers is $2i$, find $d+f$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "4730",
"split": "train"
} |
math | [
{
"content": "Two circles have the same center O. Point X is the midpoint of segment OP. What is the ratio of the area of the circle with radius OX to the area of the circle with radius OP? Express your answer as a common fraction.\n\n[asy]\nimport graph;\ndraw(Circle((0,0),20));\ndraw(Circle((0,0),12));\ndot((... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "4731",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{AB}$ have a length of 8 units and $\\overline{A'B'}$ have a length of 6 units. $D$ is located 3 units away from $A$ on $\\overline{AB}$ and $D'$ is located 1 unit away from $A'$ on $\\overline{A'B'}$. If $P$ is a point on $\\overline{AB}$ such that $x$ (the distance from $P$ to $D$... | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "4732",
"split": "train"
} |
math | [
{
"content": "Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into t... | {
"ground_truth": "18480",
"style": "rule"
} | {
"index": "4733",
"split": "train"
} |
math | [
{
"content": "In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena. \n\nOn a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Gött... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "4734",
"split": "train"
} |
math | [
{
"content": "Jerry has ten distinguishable coins, each of which currently has heads facing up. He chooses one coin and flips it over, so it now has tails facing up. Then he picks another coin (possibly the same one as before) and flips it over. How many configurations of heads and tails are possible after thes... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "4735",
"split": "train"
} |
math | [
{
"content": "$174 S-A B C D-S^{\\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\\prime} A B$. Find the radius $r$ of sphere $O$. Please output the final answer within \\bo... | {
"ground_truth": "\\frac{1}{4} \\sqrt{\\frac{2}{3}}",
"style": "rule"
} | {
"index": "4736",
"split": "train"
} |
math | [
{
"content": "A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.\n\n(1) If the shop buys 16 roses in one day, find the profit function \\( y \\) (in yuan) with... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4737",
"split": "train"
} |
math | [
{
"content": "The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 42, $x$ is twice $y$. What is the value of $y$ when $x=-8$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-49",
"style": "rule"
} | {
"index": "4738",
"split": "train"
} |
math | [
{
"content": "Suppose the polar coordinate equation of circle $C$ is $ρ=2\\cos θ$, the parametric equation of line $l$ is $\\begin{cases}x=\\frac{1}{2}+\\frac{\\sqrt{3}}{2}t\\\\y=\\frac{1}{2}+\\frac{1}{2}t\\end{cases}$ ($t$ is the parameter), and the polar coordinates of point $A$ are $(\\frac{\\sqrt{2}}{2},\\f... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4739",
"split": "train"
} |
math | [
{
"content": "Given that $17^{-1} \\equiv 26 \\pmod{53}$, find $36^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "4740",
"split": "train"
} |
math | [
{
"content": "Given the complex numbers \\( z_1 \\) and \\( z_2 \\) such that \\( \\left| z_1 + z_2 \\right| = 20 \\) and \\( \\left| z_1^2 + z_2^2 \\right| = 16 \\), find the minimum value of \\( \\left| z_1^3 + z_2^3 \\right| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3520",
"style": "rule"
} | {
"index": "4741",
"split": "train"
} |
math | [
{
"content": "Given that $a > 0$, if the area enclosed by the curve $y=\\sqrt{x}$ and the lines $x=a$, $y=0$ is equal to $a$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "4742",
"split": "train"
} |
math | [
{
"content": "The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{460}{14}",
"style": "rule"
} | {
"index": "4743",
"split": "train"
} |
math | [
{
"content": "The area of the base of a hemisphere is $100\\pi$. What is the total surface area of the hemisphere? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300\\pi",
"style": "rule"
} | {
"index": "4744",
"split": "train"
} |
math | [
{
"content": "The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{53}{3}",
"style": "rule"
} | {
"index": "4745",
"split": "train"
} |
math | [
{
"content": "On a circular track, Alphonse is at point \\( A \\) and Beryl is diametrically opposite at point \\( B \\). Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the... | {
"ground_truth": "2:1",
"style": "rule"
} | {
"index": "4746",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c$ be real numbers such that $1 \\le a \\le b \\le c \\le 4.$ Find the minimum value of\n\\[(a - 1)^2 + \\left( \\frac{b}{a} - 1 \\right)^2 + \\left( \\frac{c}{b} - 1 \\right)^2 + \\left( \\frac{4}{c} - 1 \\right)^2.\\] Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "12 - 8 \\sqrt{2}",
"style": "rule"
} | {
"index": "4747",
"split": "train"
} |
math | [
{
"content": "Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination repr... | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "4748",
"split": "train"
} |
math | [
{
"content": "Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \\frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function. Please output the final answer... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "4749",
"split": "train"
} |
math | [
{
"content": "If $x+\\sqrt{81}=25$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4750",
"split": "train"
} |
math | [
{
"content": "Mário completed 30 hours in an extra math course. On days when he had class, it lasted only 1 hour, and it took place exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.\n\na) On how many days were there n... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "4751",
"split": "train"
} |
math | [
{
"content": "On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: \"Three Pavilions, Two Gardens, One Belt, and One Valley.\" The \"Three Pavilions\" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smar... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "4752",
"split": "train"
} |
math | [
{
"content": "Suppose that $m$ and $n$ are positive integers with $m<n$ such that the interval $[m, n)$ contains more multiples of 2021 than multiples of 2000. Compute the maximum possible value of $n-m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "191999",
"style": "rule"
} | {
"index": "4753",
"split": "train"
} |
math | [
{
"content": "Given the function f(x) = 2cos^2(x) - 2$\\sqrt{3}$sin(x)cos(x).\n(I) Find the monotonically decreasing interval of the function f(x);\n(II) Find the sum of all the real roots of the equation f(x) = $- \\frac{1}{3}$ in the interval [0, $\\frac{\\pi}{2}$]. Please output the final answer within \\box... | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": "4754",
"split": "train"
} |
math | [
{
"content": "Given that the rhombus has diagonals of length $8$ and $30$, calculate the radius of the circle inscribed in the rhombus. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{30}{\\sqrt{241}}",
"style": "rule"
} | {
"index": "4755",
"split": "train"
} |
math | [
{
"content": "Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell.... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "4756",
"split": "train"
} |
math | [
{
"content": "Given $cos({α-\\frac{π}{6}})=\\frac{1}{3}$, find $sin({2α+\\frac{π}{6}})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7}{9}",
"style": "rule"
} | {
"index": "4757",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a quadrilateral inscribed in a circle with diameter $\\overline{A D}$. If $A B=5, A C=6$, and $B D=7$, find $C D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{38}",
"style": "rule"
} | {
"index": "4758",
"split": "train"
} |
math | [
{
"content": "The sum of the first \\( n \\) terms of a geometric sequence \\(\\left\\{a_{n}\\right\\}\\) with all real numbers is \\( S_{n} \\). If \\( S_{10} = 10 \\) and \\( S_{30} = 70 \\), then \\( S_{40} \\) is equal to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "4759",
"split": "train"
} |
math | [
{
"content": "Determine the monotonicity of the function $f(x) = \\frac{x}{x^2 + 1}$ on the interval $(1, +\\infty)$, and find the maximum and minimum values of the function when $x \\in [2, 3]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "4760",
"split": "train"
} |
math | [
{
"content": "The sum of 20 consecutive integers is a triangular number. What is the smallest such sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "190",
"style": "rule"
} | {
"index": "4761",
"split": "train"
} |
math | [
{
"content": "A conference lasted for 2 days. On the first day, the sessions lasted for 7 hours and 15 minutes, and on the second day, they lasted for 8 hours and 45 minutes. Calculate the total number of minutes the conference sessions lasted. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "960",
"style": "rule"
} | {
"index": "4762",
"split": "train"
} |
math | [
{
"content": "In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\\frac{1}... | {
"ground_truth": "429",
"style": "rule"
} | {
"index": "4763",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a trapezoid such that $|AC|=8$ , $|BD|=6$ , and $AD \\parallel BC$ . Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$ , respectively. If $|PS|=5$ , find the area of the trapezoid $ABCD$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "4764",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, $\\angle B$ is a right angle, diagonal $\\overline{AC}$ is perpendicular to $\\overline{CD}$, $AB=18$, $BC=21$, and $CD=14$. Find the perimeter of $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "4765",
"split": "train"
} |
math | [
{
"content": "In $\\Delta ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $c=\\frac{7}{2}$, the area of $\\Delta ABC$ is $\\frac{3\\sqrt{3}}{2}$, and $\\tan A+\\tan B=\\sqrt{3}(\\tan A\\tan B-1)$,\n\n(1) Find the measure of angle $C$;\n\n(2) Find the value of $a+b$. P... | {
"ground_truth": "\\frac{11}{2}",
"style": "rule"
} | {
"index": "4766",
"split": "train"
} |
math | [
{
"content": "The edge of the cube \\(E F G H E_1 F_1 G_1 H_1\\) is equal to 2. Points \\(A\\) and \\(B\\) are taken on the edges \\(E H\\) and \\(H H_1\\) such that \\(\\frac{E A}{A H} = 2\\) and \\(\\frac{B H}{B H_1} = \\frac{1}{2}\\). A plane is drawn through the points \\(A\\), \\(B\\), and \\(G_1\\). Find ... | {
"ground_truth": "2 \\sqrt{\\frac{2}{11}}",
"style": "rule"
} | {
"index": "4767",
"split": "train"
} |
math | [
{
"content": "A boy named Vasya wrote down the nonzero coefficients of a tenth-degree polynomial \\( P(x) \\) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, which he also wrote dow... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4768",
"split": "train"
} |
math | [
{
"content": "How many triangles exist such that the lengths of the sides are integers not greater than 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "4769",
"split": "train"
} |
math | [
{
"content": "Distribute five students, namely A, B, C, D, and E, to Peking University, Tsinghua University, and Renmin University of China for recommendation, with the condition that each university gets at least one student and student A cannot be recommended to Peking University. How many different recommend... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "4770",
"split": "train"
} |
math | [
{
"content": "A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3\\sqrt{2}+\\sqrt{6}$",
"style": "rule"
} | {
"index": "4771",
"split": "train"
} |
math | [
{
"content": "Quadrilateral $ABCD$ has $AB = BC = CD$, $m\\angle ABC = 70^\\circ$ and $m\\angle BCD = 170^\\circ$. What is the degree measure of $\\angle BAD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "4772",
"split": "train"
} |
math | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.293",
"style": "rule"
} | {
"index": "4773",
"split": "train"
} |
math | [
{
"content": "Given that $x$ and $y$ are real numbers, and they satisfy $xy + x + y = 17$, $x^2y + xy^2 = 66$, find the value of $x^4 + x^3y + x^2y^2 + xy^3 + y^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12499",
"style": "rule"
} | {
"index": "4774",
"split": "train"
} |
math | [
{
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "4775",
"split": "train"
} |
math | [
{
"content": "A high school basketball team has 16 players, including a set of twins named Bob and Bill, and another set of twins named Tom and Tim. How many ways can we choose 5 starters if no more than one player from each set of twins can be chosen? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "3652",
"style": "rule"
} | {
"index": "4776",
"split": "train"
} |
math | [
{
"content": "In a geometric sequence with positive terms $\\{a_n\\}$, where $a_5= \\frac {1}{2}$ and $a_6 + a_7 = 3$, find the maximum positive integer value of $n$ such that $a_1 + a_2 + \\ldots + a_n > a_1 a_2 \\ldots a_n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4777",
"split": "train"
} |
math | [
{
"content": "What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16.67\\%",
"style": "rule"
} | {
"index": "4778",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the repeating decimals $0.\\overline{2}$, $0.\\overline{02}$, and $0.\\overline{0002}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{224422}{9999}",
"style": "rule"
} | {
"index": "4779",
"split": "train"
} |
math | [
{
"content": "George has a quadratic of the form $x^2+bx+\\frac13$, where $b$ is a specific negative number. Using his knowledge of how to complete the square, George is able to rewrite this quadratic in the form $(x+m)^2+\\frac{1}{12}$. What is $b$? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "4780",
"split": "train"
} |
math | [
{
"content": "The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "182",
"style": "rule"
} | {
"index": "4781",
"split": "train"
} |
math | [
{
"content": "Count the number of functions $f: \\mathbb{Z} \\rightarrow\\{$ 'green', 'blue' $\\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39601",
"style": "rule"
} | {
"index": "4782",
"split": "train"
} |
math | [
{
"content": "Given that $40\\%$ of the birds were pigeons, $20\\%$ were sparrows, $15\\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18.75\\%",
"style": "rule"
} | {
"index": "4783",
"split": "train"
} |
math | [
{
"content": "If $\\displaystyle\\frac{m}{n} = 15$, $\\displaystyle\\frac{p}{n} = 3$, and $\\displaystyle \\frac{p}{q} = \\frac{1}{10}$, then what is $\\displaystyle\\frac{m}{q}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4784",
"split": "train"
} |
math | [
{
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "495",
"style": "rule"
} | {
"index": "4785",
"split": "train"
} |
math | [
{
"content": "A deck of 100 cards is numbered from 1 to 100, each card having the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. He then examines ... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "4786",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $50$ have an odd number of positive integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "4787",
"split": "train"
} |
math | [
{
"content": "Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\\{1,2, \\cdots, 1000\\}$, there exist two subsets that have at least $r$ common elements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "4788",
"split": "train"
} |
math | [
{
"content": "Calculate $8 \\cdot 9\\frac{2}{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75\\frac{1}{5}",
"style": "rule"
} | {
"index": "4789",
"split": "train"
} |
math | [
{
"content": "The inclination angle of the line given by the parametric equations \\[\n\\begin{cases} \nx=1+t \\\\\ny=1-t\n\\end{cases}\n\\]\ncalculate the inclination angle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {3\\pi }{4}",
"style": "rule"
} | {
"index": "4790",
"split": "train"
} |
math | [
{
"content": "The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "4791",
"split": "train"
} |
math | [
{
"content": "Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\\left(f^{-1}(5)+f^{-1}(4)\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "4792",
"split": "train"
} |
math | [
{
"content": "If $\\mathbf{a} \\times \\mathbf{b} = \\begin{pmatrix} 5 \\\\ 4 \\\\ -7 \\end{pmatrix},$ then compute $\\mathbf{a} \\times (3 \\mathbf{b}).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 15 \\\\ 12 \\\\ -21 \\end{pmatrix}",
"style": "rule"
} | {
"index": "4793",
"split": "train"
} |
math | [
{
"content": "Find the area of the shaded region. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\dfrac{1}{2}",
"style": "rule"
} | {
"index": "4794",
"split": "train"
} |
math | [
{
"content": "A $4 \\times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "4795",
"split": "train"
} |
math | [
{
"content": "A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$. Please output the final answer wit... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "4796",
"split": "train"
} |
math | [
{
"content": "Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105263157894736842",
"style": "rule"
} | {
"index": "4797",
"split": "train"
} |
math | [
{
"content": "Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11250",
"style": "rule"
} | {
"index": "4798",
"split": "train"
} |
math | [
{
"content": "A fair die, numbered 1, 2, 3, 4, 5, 6, is thrown three times. The numbers obtained are recorded sequentially as $a$, $b$, and $c$. The probability that $a+bi$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}-2x+c=0$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \... | {
"ground_truth": "\\frac{1}{108}",
"style": "rule"
} | {
"index": "4799",
"split": "train"
} |
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