data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "31400",
"split": "train"
} |
math | [
{
"content": "For real numbers $x$ and $y$, define $x \\spadesuit y = (x+y)(x-y)$. What is $3 \\spadesuit (4 \\spadesuit 5)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-72",
"style": "rule"
} | {
"index": "31401",
"split": "train"
} |
math | [
{
"content": "Find the largest real \\( k \\) such that if \\( a, b, c, d \\) are positive integers such that \\( a + b = c + d \\), \\( 2ab = cd \\) and \\( a \\geq b \\), then \\(\\frac{a}{b} \\geq k\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "31402",
"split": "train"
} |
math | [
{
"content": "A bag has 4 red marbles, 5 white marbles, and 6 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that they are all the same color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{34}{455}",
"style": "rule"
} | {
"index": "31403",
"split": "train"
} |
math | [
{
"content": "Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many stude... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "31404",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^{2}+y^{2}=16$, and the parameter equation of the line is $\\left\\{\\begin{array}{l}x=1+t\\cos \\theta \\\\y=2+t\\sin \\theta \\end{array}\\right.$ (where $t$ is the parameter). If the line $l$ intersects curve $C$ at points... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "31405",
"split": "train"
} |
math | [
{
"content": "The diagonals of a trapezoid are mutually perpendicular, and one of them is 13. Find the area of the trapezoid if its height is 12. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1014/5",
"style": "rule"
} | {
"index": "31406",
"split": "train"
} |
math | [
{
"content": "In a math test, the scores of 6 students are as follows: 98, 88, 90, 92, 90, 94. The mode of this set of data is ______; the median is ______; the average is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "31407",
"split": "train"
} |
math | [
{
"content": "Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\\left(\\sin C+\\sin B\\right)\\left(c-b\\right)=a\\left(\\sin A-\\sin B\\right)$. \n$(1)$ Find the measure of angle $C$. \n$(2)$ If the angle bisecto... | {
"ground_truth": "\\frac{3\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "31408",
"split": "train"
} |
math | [
{
"content": "What is the greatest 3-digit base 8 positive integer that is divisible by 5? (Express your answer in base 8.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "776_8",
"style": "rule"
} | {
"index": "31409",
"split": "train"
} |
math | [
{
"content": "Given the parametric equation of line $l$ as $\\begin{cases} x=m+\\frac{\\sqrt{2}}{2}t \\\\ y=\\frac{\\sqrt{2}}{2}t \\end{cases} (t \\text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31410",
"split": "train"
} |
math | [
{
"content": "The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31411",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(1-\\frac{y}{x})(x+y)^{8}$, the coefficient of $x^{2}y^{6}$ is ____ (provide your answer as a number). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-28",
"style": "rule"
} | {
"index": "31412",
"split": "train"
} |
math | [
{
"content": "The area of the closed shape formed by the graph of the function $y=2\\sin x$, where $x \\in \\left[\\frac{\\pi}{2}, \\frac{5\\pi}{2}\\right]$, and the lines $y=\\pm2$ is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "31413",
"split": "train"
} |
math | [
{
"content": "Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{30}",
"style": "rule"
} | {
"index": "31414",
"split": "train"
} |
math | [
{
"content": "Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and it is then returned to the jar. This process is repeated three times, with each ball having an equal chance of being drawn in each trial. If the sum of the recorded numbers is 6, what is ... | {
"ground_truth": "$\\frac{1}{7}$",
"style": "rule"
} | {
"index": "31415",
"split": "train"
} |
math | [
{
"content": "Consider $n$ disks $C_{1}, C_{2}, \\ldots, C_{n}$ in a plane such that for each $1 \\leq i<n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the score of such an arrangement of $n$ disks to be the number of pairs $(... | {
"ground_truth": "(n-1)(n-2)/2",
"style": "rule"
} | {
"index": "31416",
"split": "train"
} |
math | [
{
"content": "On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts the number of vehic... | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "31417",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c$ be positive real numbers such that $a + b + c = 2.$ Find the maximum value of $a^2 b^3 c^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{128}{2187}",
"style": "rule"
} | {
"index": "31418",
"split": "train"
} |
math | [
{
"content": "Samson derives utility according the relation $$\\text{Utility} = \\text{hours of math done} \\times \\text{hours of frisbee played}.$$On Monday he plays $t$ hours of frisbee and spends $8 - t$ hours doing math. On Tuesday, he ends up getting the same amount of utility as Monday while spending $... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "31419",
"split": "train"
} |
math | [
{
"content": "If $y=(\\log_23)(\\log_34)\\cdots(\\log_n[n+1])\\cdots(\\log_{31}32)$ then Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y=5",
"style": "rule"
} | {
"index": "31420",
"split": "train"
} |
math | [
{
"content": "How many digits are in the product $4^5 \\cdot 5^{10}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31421",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\frac{x^{2}}{4} + \\frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $\\frac{\\sqrt{3}}{2}$, find the slope of the line that intersects the ellipse at points $A$ and $B$, where the midpoint of segment $AB$ is $M(-2, 1)$. Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31422",
"split": "train"
} |
math | [
{
"content": "How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?\n\nA sequence of three numbers \\( a, b, c \\) is said to form an arithmetic progression if \\( a + c = 2b \\). \n\nA correct numerical answer without justification will e... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "31423",
"split": "train"
} |
math | [
{
"content": "Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "31424",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{v} = \\begin{pmatrix} 2 \\\\ 1 \\\\ -1 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 3 \\end{pmatrix}.$ The columns of a matrix are $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w},$ where $\\mathbf{u}$ is a unit vector. Find the largest possible determinant of th... | {
"ground_truth": "\\sqrt{59}",
"style": "rule"
} | {
"index": "31425",
"split": "train"
} |
math | [
{
"content": "Dr. Math's house number $WXYZ$ is a four-digit number where each digit $W$, $X$, $Y$, and $Z$ is non-zero, and the two portions of the house number, $WX$ and $YZ$, form two-digit primes. Every prime number selected must be less than 50. Additionally, the sum of the digits in $YZ$ must be even, and... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "31426",
"split": "train"
} |
math | [
{
"content": "Team A has a probability of $$\\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{19}{27}",
"style": "rule"
} | {
"index": "31427",
"split": "train"
} |
math | [
{
"content": "Convert $5214_8$ to a base 10 integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2700",
"style": "rule"
} | {
"index": "31428",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $\\angle Y = 90^\\circ$, $YZ = 4$, and $XY = 5$. What is $\\tan X$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "31429",
"split": "train"
} |
math | [
{
"content": "If in the expansion of the binomial \\((x- \\frac {2}{ \\sqrt {x}})^{n}\\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \\(x^{2}\\) is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1120",
"style": "rule"
} | {
"index": "31430",
"split": "train"
} |
math | [
{
"content": "On Marika's $8^{\\text{th}}$ birthday, in 2004, her father said, ''My age is now four times your age.'' In what year will Marika's father be able to say, ''My age is now three times your age,'' on Marika's birthday? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2008",
"style": "rule"
} | {
"index": "31431",
"split": "train"
} |
math | [
{
"content": "Let \\( p, q, r, s \\) be distinct real numbers such that the roots of \\( x^2 - 12px - 13q = 0 \\) are \\( r \\) and \\( s \\), and the roots of \\( x^2 - 12rx - 13s = 0 \\) are \\( p \\) and \\( q \\). Find the value of \\( p + q + r + s \\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "31432",
"split": "train"
} |
math | [
{
"content": "A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60\\text{ inches}",
"style": "rule"
} | {
"index": "31433",
"split": "train"
} |
math | [
{
"content": "In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30.$ Find the sum of the four terms. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "129",
"style": "rule"
} | {
"index": "31434",
"split": "train"
} |
math | [
{
"content": "Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "31435",
"split": "train"
} |
math | [
{
"content": "Calculate the volumes of solids formed by rotating the region bounded by the function graphs about the \\( O y \\) (y-axis).\n\n$$\ny = x^{2} + 1, \\quad y = x, \\quad x = 0, \\quad x = 1\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "31436",
"split": "train"
} |
math | [
{
"content": "Let \\( x, y, z \\in [0, 1] \\). The maximum value of \\( M = \\sqrt{|x-y|} + \\sqrt{|y-z|} + \\sqrt{|z-x|} \\) is ______ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "31437",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\\infty)$, find the minimum value of $f(x)$ on the interval $(-\\infty, 0)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "31438",
"split": "train"
} |
math | [
{
"content": "Among the two-digit numbers less than 20, the largest prime number is ____, and the largest composite number is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31439",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is denoted as $S_n$. Given $a_1= \\frac{1}{2}$ and $2a_{n+1}+S_n=0$ for $n=1, 2, 3, \\ldots$. If $S_n \\leqslant k$ always holds, then the minimum value of $k$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31440",
"split": "train"
} |
math | [
{
"content": "On a line, 5 points \\( P, Q, R, S, T \\) are marked in that order. It is known that the sum of the distances from \\( P \\) to the other 4 points is 67, and the sum of the distances from \\( Q \\) to the other 4 points is 34. Find the length of segment \\( PQ \\). Please output the final answer w... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31441",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the expression $(8 \\times 6)-(4 \\div 2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "31442",
"split": "train"
} |
math | [
{
"content": "Five doctors A, B, C, D, and E are assigned to four different service positions located in the Sichuan disaster zone, labeled A, B, C, and D. Each position must be filled by at least one doctor. Calculate the total number of ways doctors A and B can serve independently in different positions. Plea... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "31443",
"split": "train"
} |
math | [
{
"content": "Inside a square with side length 8, four congruent equilateral triangles are drawn such that each triangle shares one side with a side of the square and each has a vertex at one of the square's vertices. What is the side length of the largest square that can be inscribed in the space inside the sq... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "31444",
"split": "train"
} |
math | [
{
"content": "A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31445",
"split": "train"
} |
math | [
{
"content": "Given the circle \\\\({{x}^{2}}+{{y}^{2}}-6x-2y+3=0\\\\) and the distance from its center to the line \\\\(x+ay-1=0\\\\) is \\\\(1,\\\\) then the real number \\\\(a=\\\\)_______ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\dfrac{3}{4}",
"style": "rule"
} | {
"index": "31446",
"split": "train"
} |
math | [
{
"content": "Given the graph of the power function $y=f(x)$ passes through the point $\\left( \\frac{1}{2}, \\frac{\\sqrt{2}}{2} \\right)$, then $\\lg [f(2)]+\\lg [f(5)]=$ \\_\\_\\_\\_\\_\\_ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31447",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $\\angle{A}=90^{\\circ}$ , $AB=2$ , and $AC=4$ . Circle $\\omega_1$ has center $C$ and radius $CA$ , while circle $\\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\\omega_2$ and in th... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31448",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = x^4 + 20x^3 + 150x^2 + 500x + 625$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a^2 + z_b^2 + z_cz_d|$ where $\\{a, b, c, d\\} = \\{1, 2, 3, 4\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1875",
"style": "rule"
} | {
"index": "31449",
"split": "train"
} |
math | [
{
"content": "Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$ . Let $N$ be the smallest positive integer such that $S(N) = 2013$ . What is the value of $S(5N + 2013)$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "31450",
"split": "train"
} |
math | [
{
"content": "A tree grows in a rather peculiar manner. Lateral cross-sections of the trunk, leaves, branches, twigs, and so forth are circles. The trunk is 1 meter in diameter to a height of 1 meter, at which point it splits into two sections, each with diameter .5 meter. These sections are each one meter long... | {
"ground_truth": "\\pi / 2",
"style": "rule"
} | {
"index": "31451",
"split": "train"
} |
math | [
{
"content": "The lines $-2x + y = k$ and $0.5x + y = 14$ intersect when $x = -8.4$. What is the value of $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "31452",
"split": "train"
} |
math | [
{
"content": "Suppose $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 4y^2 = 12$. Find the maximum possible value of $x^2 + 3xy + 4y^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "31453",
"split": "train"
} |
math | [
{
"content": "Compute \\(\\frac{x^{10} + 15x^5 + 125}{x^5 + 5}\\) when \\( x=3 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248.4032",
"style": "rule"
} | {
"index": "31454",
"split": "train"
} |
math | [
{
"content": "A student needs to provide his waist size in centimeters for a customized lab coat, based on the measurements in inches. If there are $10$ inches in a foot and $25$ centimeters in a foot, then what size should the student specify in centimeters if his waist size is $40$ inches? Please output the f... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "31455",
"split": "train"
} |
math | [
{
"content": "How many squares can be formed by joining four of the twelve points marked on a rectangular grid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31456",
"split": "train"
} |
math | [
{
"content": "Suppose \\( N \\) is a 6-digit number having base-10 representation \\( \\underline{a} \\underline{b} \\underline{c} \\underline{d} \\underline{e} \\underline{f} \\). If \\( N \\) is \\( \\frac{6}{7} \\) of the number having base-10 representation \\( \\underline{d} \\underline{e} \\underline{f} \... | {
"ground_truth": "461538",
"style": "rule"
} | {
"index": "31457",
"split": "train"
} |
math | [
{
"content": "How many zeros are at the end of the product \\( s(1) \\cdot s(2) \\cdot \\ldots \\cdot s(100) \\), where \\( s(n) \\) denotes the sum of the digits of the natural number \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "31458",
"split": "train"
} |
math | [
{
"content": "We define $N$ as the set of natural numbers $n<10^6$ with the following property:\n\nThere exists an integer exponent $k$ with $1\\le k \\le 43$ , such that $2012|n^k-1$ .\n\nFind $|N|$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1988",
"style": "rule"
} | {
"index": "31459",
"split": "train"
} |
math | [
{
"content": "A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule thes... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31460",
"split": "train"
} |
math | [
{
"content": "Alice is once again very bored in class. On a whim, she chooses three primes $p, q, r$ independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of $p x^{2}+q x+r$. What is the probability that at least one of her roots is an integer? Please output t... | {
"ground_truth": "\\frac{3}{200}",
"style": "rule"
} | {
"index": "31461",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that $\\frac{n}{n+150}$ is equal to a terminating decimal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "31462",
"split": "train"
} |
math | [
{
"content": "Find all $c$ which satisfy $$\\frac{c}{3} \\le 2+c < -2(1+c).$$Express your answer in interval notation, simplifying any fractions which occur in your answer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left[-3,-\\frac{4}{3}\\right)",
"style": "rule"
} | {
"index": "31463",
"split": "train"
} |
math | [
{
"content": "Simplify and write the result as a common fraction: $$\\sqrt{\\sqrt[3]{\\sqrt{\\frac{1}{4096}}}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31464",
"split": "train"
} |
math | [
{
"content": "What is the value of the sum $\\left[\\log _{2} 1\\right]+\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} 2002\\right]$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17984",
"style": "rule"
} | {
"index": "31465",
"split": "train"
} |
math | [
{
"content": "Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "31466",
"split": "train"
} |
math | [
{
"content": "Inside the circle \\(\\omega\\), there is a circle \\(\\omega_{1}\\) that touches it at point \\(K\\). Circle \\(\\omega_{2}\\) touches circle \\(\\omega_{1}\\) at point \\(L\\) and intersects circle \\(\\omega\\) at points \\(M\\) and \\(N\\). It turns out that points \\(K\\), \\(L\\), and \\(M\\... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31467",
"split": "train"
} |
math | [
{
"content": "Given that $f(n) = \\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise,}\\end{matrix}\\right.$, evaluate the sum $\\sum_{n = 1}^{1023}{f(n)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22.5",
"style": "rule"
} | {
"index": "31468",
"split": "train"
} |
math | [
{
"content": "A rectangle and a regular pentagon have the same perimeter. Let $A$ be the area of the circle circumscribed about the rectangle, with the rectangle having side lengths in the ratio 3:1, and $B$ the area of the circle circumscribed around the pentagon. Find $A/B$.\nA) $\\frac{125(\\sqrt{5}-1)}{256}... | {
"ground_truth": "\\frac{5(5-\\sqrt{5})}{64}",
"style": "rule"
} | {
"index": "31469",
"split": "train"
} |
math | [
{
"content": "Determine the number of relatively prime dates in the month with the second fewest relatively prime dates. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31470",
"split": "train"
} |
math | [
{
"content": "How many non-congruent triangles can be formed by selecting vertices from the ten points in the triangular array, where the bottom row has four points, the next row has three points directly above the gaps of the previous row, followed by two points, and finally one point at the top? Please output... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31471",
"split": "train"
} |
math | [
{
"content": "Calculate: $$\\frac {1}{2}\\log_{2}3 \\cdot \\frac {1}{2}\\log_{9}8 = \\_\\_\\_\\_\\_\\_ ．$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {3}{8}",
"style": "rule"
} | {
"index": "31472",
"split": "train"
} |
math | [
{
"content": "After viewing the John Harvard statue, a group of tourists decides to estimate the distances of nearby locations on a map by drawing a circle, centered at the statue, of radius $\\sqrt{n}$ inches for each integer $2020 \\leq n \\leq 10000$, so that they draw 7981 circles altogether. Given that, on... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "31473",
"split": "train"
} |
math | [
{
"content": "An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$ . Now, she pulls her arms in and her moment of inertia is now $\\frac{7}{10}I$ . What is the ratio... | {
"ground_truth": "$\\dfrac{10}{7}$",
"style": "rule"
} | {
"index": "31474",
"split": "train"
} |
math | [
{
"content": "What is the greatest number of consecutive non-negative integers whose sum is $120$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31475",
"split": "train"
} |
math | [
{
"content": "Compute $1-2+3-4+ \\dots -100+101$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "31476",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{7}{3+\\sqrt{8}}$. The answer can be expressed as $\\frac{P\\sqrt{Q}+R}{S}$, where $P$, $Q$, $R$, and $S$ are integers, $S$ is positive, and $Q$ is not divisible by the square of any prime. If the greatest common divisor of $P$, $R$, and $S$ is 1, find $P+Q+R+... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "31477",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \\max\\{f(x),g(x),h(x)\\},$$$$k(x) = \\min\\{f(x),g(x),h(x)\\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)... | {
"ground_truth": "245",
"style": "rule"
} | {
"index": "31478",
"split": "train"
} |
math | [
{
"content": "The number of right-angled triangles with integer side lengths $a$ and $b$ (where $b < 2011$), and hypotenuse length $b + 1$ is . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "31479",
"split": "train"
} |
math | [
{
"content": "At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "31480",
"split": "train"
} |
math | [
{
"content": "A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\\pi$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144\\pi",
"style": "rule"
} | {
"index": "31481",
"split": "train"
} |
math | [
{
"content": "A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\\spadesuit$ and $\\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are p... | {
"ground_truth": "2652",
"style": "rule"
} | {
"index": "31482",
"split": "train"
} |
math | [
{
"content": "Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the problem above. If $AB = 34$ units and $CD = 66$ units, what is the area of square $BCFE$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2244",
"style": "rule"
} | {
"index": "31483",
"split": "train"
} |
math | [
{
"content": "Six six-sided dice are rolled. We are told there are no four-of-a-kind, but there are two different pairs of dice showing the same numbers. These four dice (two pairs) are set aside, and the other two dice are re-rolled. What is the probability that after re-rolling these two dice, at least three ... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "31484",
"split": "train"
} |
math | [
{
"content": "A car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minut... | {
"ground_truth": "198",
"style": "rule"
} | {
"index": "31485",
"split": "train"
} |
math | [
{
"content": "Given the sequence 2008, 2009, 1, -2008, -2009, ..., where each term from the second term onward is equal to the sum of the term before it and the term after it, calculate the sum of the first 2013 terms of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4018",
"style": "rule"
} | {
"index": "31486",
"split": "train"
} |
math | [
{
"content": "$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "138600",
"style": "rule"
} | {
"index": "31487",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{2 \\sqrt{2}} \\frac{x^{4} \\, dx}{\\left(16-x^{2}\\right) \\sqrt{16-x^{2}}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20 - 6\\pi",
"style": "rule"
} | {
"index": "31488",
"split": "train"
} |
math | [
{
"content": "Find all values of \\( a \\) for which the quadratic trinomials \\( x^{2} - 6x + 4a \\) and \\( x^{2} + ax + 6 \\) each have two roots, with the sum of the squares of the roots of the first trinomial equal to the sum of the squares of the roots of the second trinomial. Please output the final answ... | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "31489",
"split": "train"
} |
math | [
{
"content": "Calculate the harmonic mean of the numbers $1999$ and $2001$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": "31490",
"split": "train"
} |
math | [
{
"content": "A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "97",
"style": "rule"
} | {
"index": "31491",
"split": "train"
} |
math | [
{
"content": "A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue mov... | {
"ground_truth": "10:40",
"style": "rule"
} | {
"index": "31492",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_{1}=10$, $a_{n+1}=a_{n}+18n+10$ for $n \\in \\mathbb{N}^*$, find $\\lim_{n \\to \\infty} (\\sqrt{a_{n}} - [\\sqrt{a_{n}}])$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "31493",
"split": "train"
} |
math | [
{
"content": "Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \\frac{a^{2}+1}{ab} - 2) \\cdot c + \\frac{\\sqrt{2}}{c - 1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "31494",
"split": "train"
} |
math | [
{
"content": "In rectangle $ADEH$, points $B$ and $C$ trisect $\\overline{AD}$, and points $G$ and $F$ trisect $\\overline{HE}$. In addition, $AH=AC=2$. What is the area of quadrilateral $WXYZ$ shown in the figure?\n\n[asy]\nunitsize(1cm);\npair A,B,C,D,I,F,G,H,U,Z,Y,X;\nA=(0,0);\nB=(1,0);\nC=(2,0);\nD=(3,... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "31495",
"split": "train"
} |
math | [
{
"content": "Decompose $\\frac{1}{4}$ into unit fractions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}+\\frac{1}{12}+\\frac{1}{24}",
"style": "rule"
} | {
"index": "31496",
"split": "train"
} |
math | [
{
"content": "During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item $x$ (yuan) (where $8... | {
"ground_truth": "525",
"style": "rule"
} | {
"index": "31497",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word MOREMOM. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "31498",
"split": "train"
} |
math | [
{
"content": "Find the square root of $\\dfrac{9!}{210}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "216\\sqrt{3}",
"style": "rule"
} | {
"index": "31499",
"split": "train"
} |
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