data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A rectangular piece of paper with vertices $A B C D$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $D A B$ until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times the area of the oth... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "35000",
"split": "train"
} |
math | [
{
"content": "Six students sign up for three different intellectual competition events. How many different registration methods are there under the following conditions? (Not all six students must participate)\n\n(1) Each person participates in exactly one event, with no limit on the number of people per event;... | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "35001",
"split": "train"
} |
math | [
{
"content": "Find the product of $0.5$ and $0.8$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.4",
"style": "rule"
} | {
"index": "35002",
"split": "train"
} |
math | [
{
"content": "For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\\alpha$ satisfying the inequality $-1988 \\leqslant \\alpha \\leqslant 1988$, and making $\\frac{3}{5} \\alpha$ an integer. How many possible values are there for $m$? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "2385",
"style": "rule"
} | {
"index": "35003",
"split": "train"
} |
math | [
{
"content": "A biased coin with the probability of landing heads as 1/3 is flipped 12 times. What is the probability of getting exactly 9 heads in the 12 flips? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1760}{531441}",
"style": "rule"
} | {
"index": "35004",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to select 4 representatives from a group of 5 male students and 4 female students to participate in an activity, ensuring that there are at least two males and at least one female among the representatives. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "35005",
"split": "train"
} |
math | [
{
"content": "Given $m \\gt 0$, $n \\gt 0$, and $m+2n=1$, find the minimum value of $\\frac{(m+1)(n+1)}{mn}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8+4\\sqrt{3}",
"style": "rule"
} | {
"index": "35006",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a_n\\}$, if $a_2a_5= -\\frac{3}{4}$ and $a_2+a_3+a_4+a_5= \\frac{5}{4}$, calculate $\\frac{1}{a_2}+ \\frac{1}{a_3}+ \\frac{1}{a_4}+ \\frac{1}{a_5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{3}",
"style": "rule"
} | {
"index": "35007",
"split": "train"
} |
math | [
{
"content": "The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 years. How old is th... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "35008",
"split": "train"
} |
math | [
{
"content": "The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan. Please output the f... | {
"ground_truth": "4536",
"style": "rule"
} | {
"index": "35009",
"split": "train"
} |
math | [
{
"content": "Consider a circular cone with vertex $V$, and let $ABC$ be a triangle inscribed in the base of the cone, such that $AB$ is a diameter and $AC=BC$. Let $L$ be a point on $BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ABCL$. Find the value of $BL/LV$. Please output t... | {
"ground_truth": "\\frac{\\pi}{4-\\pi}",
"style": "rule"
} | {
"index": "35010",
"split": "train"
} |
math | [
{
"content": "Let the set \\(M = \\{1,2,\\cdots, 1000\\}\\). For any non-empty subset \\(X\\) of \\(M\\), let \\(\\alpha_X\\) denote the sum of the largest and smallest numbers in \\(X\\). Find the arithmetic mean of all such \\(\\alpha_X\\). Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "35011",
"split": "train"
} |
math | [
{
"content": "Given an ant crawling inside an equilateral triangle with side length $4$, calculate the probability that the distance from the ant to all three vertices of the triangle is more than $1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1- \\dfrac { \\sqrt {3}\\pi}{24}",
"style": "rule"
} | {
"index": "35012",
"split": "train"
} |
math | [
{
"content": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$, $b$, and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$, $(bca)$, $(bac)$, $(cab)$, and $(cba)$, to add these fi... | {
"ground_truth": "358",
"style": "rule"
} | {
"index": "35013",
"split": "train"
} |
math | [
{
"content": "Given points \\(A(4,5)\\), \\(B(4,0)\\) and \\(C(0,5)\\), compute the line integral of the second kind \\(\\int_{L}(4 x+8 y+5) d x+(9 x+8) d y\\) where \\(L:\\)\na) the line segment \\(OA\\);\nb) the broken line \\(OCA\\);\nc) the parabola \\(y=k x^{2}\\) passing through the points \\(O\\) and \\(... | {
"ground_truth": "\\frac{796}{3}",
"style": "rule"
} | {
"index": "35014",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sqrt{3}\\sin x\\cos x + \\cos^2 x + a$.\n\n(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;\n\n(2) If the sum of the maximum and minimum values of $f(x)$ in the interval $[-\\frac{\\pi}{6}, \\frac{\\pi}{3}]$ is $1$, find the va... | {
"ground_truth": "a = -\\frac{1}{4}",
"style": "rule"
} | {
"index": "35015",
"split": "train"
} |
math | [
{
"content": "There exist constants $c_1$ and $c_2$ such that\n\\[c_1 \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} + c_2 \\begin{pmatrix} -2 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -1 \\\\ 4 \\end{pmatrix}.\\]Enter the ordered pair $(c_1,c_2).$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\left( \\frac{3}{16}, \\frac{11}{16} \\right)",
"style": "rule"
} | {
"index": "35016",
"split": "train"
} |
math | [
{
"content": "In the list where each integer $n$ appears $n$ times for $1 \\leq n \\leq 300$, find the median of the numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "212",
"style": "rule"
} | {
"index": "35017",
"split": "train"
} |
math | [
{
"content": "A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "35018",
"split": "train"
} |
math | [
{
"content": "Let $n$ be a positive integer. A pair of $n$-tuples \\left(a_{1}, \\ldots, a_{n}\\right)$ and \\left(b_{1}, \\ldots, b_{n}\\right)$ with integer entries is called an exquisite pair if $$\\left|a_{1} b_{1}+\\cdots+a_{n} b_{n}\\right| \\leq 1$$ Determine the maximum number of distinct $n$-tuples wit... | {
"ground_truth": "n^{2}+n+1",
"style": "rule"
} | {
"index": "35019",
"split": "train"
} |
math | [
{
"content": "Let $z$ and $w$ be complex numbers such that $|z| = 1$ and $|w| = 3$. If $|z+w| = 2$, what is $ \\left | \\frac{1}{z} + \\frac{1}{w} \\right|$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "35020",
"split": "train"
} |
math | [
{
"content": "$\\frac{9}{7 \\times 53} =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{0.9}{0.7 \\times 53}$",
"style": "rule"
} | {
"index": "35021",
"split": "train"
} |
math | [
{
"content": "A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "35022",
"split": "train"
} |
math | [
{
"content": "As $x$ ranges over all real numbers, find the range of\n\\[f(x) = \\sin^4 x + \\cos ^2 x.\\]Enter your answer using interval notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left[ \\frac{3}{4}, 1 \\right]",
"style": "rule"
} | {
"index": "35023",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word \"PERCEPTION\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "907200",
"style": "rule"
} | {
"index": "35024",
"split": "train"
} |
math | [
{
"content": "**Compute the sum of all four-digit numbers where every digit is distinct and then find the remainder when this sum is divided by 1000.** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "35025",
"split": "train"
} |
math | [
{
"content": "The steamboat \"Rarity\" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in ... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "35026",
"split": "train"
} |
math | [
{
"content": "Let $A, B, C, D, E, F$ be 6 points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX=3, BX=2, CY=4, DY=10, ... | {
"ground_truth": "\\frac{77}{6}",
"style": "rule"
} | {
"index": "35027",
"split": "train"
} |
math | [
{
"content": "Point $M(4,4)$ is the midpoint of $\\overline{AB}$. If point $A$ has coordinates $(8,4)$, what is the sum of the coordinates of point $B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "35028",
"split": "train"
} |
math | [
{
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 45^\\circ$, and $AE=28$. Find the length of $CE$, given that $CE$ forms the diagonal of a square $CDEF$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "35029",
"split": "train"
} |
math | [
{
"content": "To prepare for her half-marathon, Mary ran a total of $2847_9$ miles. How many miles (in base ten) did Mary run? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2149",
"style": "rule"
} | {
"index": "35030",
"split": "train"
} |
math | [
{
"content": "The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first $\\$28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her annual income. What was her annual income? Pleas... | {
"ground_truth": "32000",
"style": "rule"
} | {
"index": "35031",
"split": "train"
} |
math | [
{
"content": "In the arithmetic sequence $\\{a_n\\}$, $S_{10} = 4$, $S_{20} = 20$. What is $S_{30}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "35032",
"split": "train"
} |
math | [
{
"content": "If $A,B$ and $C$ are non-zero distinct digits in base $6$ such that $\\overline{ABC}_6 + \\overline{BCA}_6+ \\overline{CAB}_6 = \\overline{AAA0}_6$, find $B+C$ in base $6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "35033",
"split": "train"
} |
math | [
{
"content": "For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a \"naive number.\" For example, the four-digit number $7311$ is a \"naive nu... | {
"ground_truth": "9313",
"style": "rule"
} | {
"index": "35034",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $1000$ are either a perfect cube or a perfect square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "35035",
"split": "train"
} |
math | [
{
"content": "Estimate $A$, the number of times an 8-digit number appears in Pascal's triangle. An estimate of $E$ earns $\\max (0,\\lfloor 20-|A-E| / 200\\rfloor)$ points. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180020660",
"style": "rule"
} | {
"index": "35036",
"split": "train"
} |
math | [
{
"content": "Consider the ellipse $\\frac{x^2}{16} + \\frac{y^2}{12} = 1$ whose left and right intersection points are $F_1$ and $F_2$, respectively. Let point $P$ be on the ellipse and satisfy $\\vec{PF_1} \\cdot \\vec{PF_2} = 9$. Find the value of $|\\vec{PF_1}| \\cdot |\\vec{PF_2}|$. Please output the final... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "35037",
"split": "train"
} |
math | [
{
"content": "The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\\frac{1992}{n} \\cdot \\sum_{k=0}^{n-1} x_k$ for $n \\geq 1$ .\nFind $\\sum_{n=0}^{1992} 2^nx_n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1992",
"style": "rule"
} | {
"index": "35038",
"split": "train"
} |
math | [
{
"content": "Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner post... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "35039",
"split": "train"
} |
math | [
{
"content": "What is the sum of the digits of all numbers from one to one billion? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40500000001",
"style": "rule"
} | {
"index": "35040",
"split": "train"
} |
math | [
{
"content": "Given that Mary is 30% older than Sally, and Sally is 50% younger than Danielle, and the sum of their ages is 45 years, determine Mary's age on her next birthday. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "35041",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the integer closest to $\\sqrt[4]{n}.$ Find $\\sum_{k=1}^{1995}\\frac 1{f(k)}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "35042",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane, the number of integer points (points where both the x-coordinate and y-coordinate are integers) that satisfy the system of inequalities \n\\[\n\\begin{cases}\ny \\leq 3x, \\\\\ny \\geq \\frac{1}{3}x, \\\\\nx + y \\leq 100\n\\end{cases}\n\\]\nis ___. Please output... | {
"ground_truth": "2551",
"style": "rule"
} | {
"index": "35043",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $x^{2}+4y^{2}=16$, and the line $AB$ passes through point $P(2,-1)$ and intersects the ellipse at points $A$ and $B$. If the slope of line $AB$ is $\\frac{1}{2}$, then the value of $|AB|$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "35044",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ passing through the point $E(\\sqrt{3}, 1)$, with an eccentricity of $\\frac{\\sqrt{6}}{3}$, and $O$ as the coordinate origin.\n(I) Find the equation of the ellipse $C$;\n(II) If point $P$ is a moving point on the el... | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "35045",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ is an acute angle, $\\beta$ is an obtuse angle, $\\cos\\alpha=\\frac{3}{5}$, $\\sin\\beta=\\frac{5}{13}$, \n$(1)$ Find $\\sin(\\alpha - \\beta)$ and $\\cos(\\alpha - \\beta)$; \n$(2)$ Find the value of $\\tan 2\\alpha$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "-\\frac{24}{7}",
"style": "rule"
} | {
"index": "35046",
"split": "train"
} |
math | [
{
"content": "A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$ . What is the value of $x+y$ ?\n\n*2019 CCA Math Bonanza Team Round #2* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "35047",
"split": "train"
} |
math | [
{
"content": "Mafia is a game where there are two sides: The village and the Mafia. Every night, the Mafia kills a person who is sided with the village. Every day, the village tries to hunt down the Mafia through communication, and at the end of every day, they vote on who they think the mafia are.**p6.** Patri... | {
"ground_truth": "319/512",
"style": "rule"
} | {
"index": "35048",
"split": "train"
} |
math | [
{
"content": "If $\\lceil{\\sqrt{x}}\\rceil=20$, how many possible integer values of $x$ are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "35049",
"split": "train"
} |
math | [
{
"content": "Let $P A B C$ be a tetrahedron such that $\\angle A P B=\\angle A P C=\\angle B P C=90^{\\circ}, \\angle A B C=30^{\\circ}$, and $A P^{2}$ equals the area of triangle $A B C$. Compute $\\tan \\angle A C B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8+5 \\sqrt{3}",
"style": "rule"
} | {
"index": "35050",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer that is divisible by the primes 7, 11, and 13? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "35051",
"split": "train"
} |
math | [
{
"content": "Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer su... | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "35052",
"split": "train"
} |
math | [
{
"content": "**How many positive factors does 72 have, and what is their sum?** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "195",
"style": "rule"
} | {
"index": "35053",
"split": "train"
} |
math | [
{
"content": "The three roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $1,$ $2,$ and $3.$ Compute $\\frac{c}{d}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{11}{6}",
"style": "rule"
} | {
"index": "35054",
"split": "train"
} |
math | [
{
"content": "The value of \\(\\frac{1}{1+\\frac{1}{1+\\frac{1}{2}}}\\) can be expressed as a simplified fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "35055",
"split": "train"
} |
math | [
{
"content": "Find the flux of the vector field\n\n$$\n\\vec{a}=-x \\vec{i}+2 y \\vec{j}+z \\vec{k}\n$$\n\nthrough the portion of the plane\n\n$$\nx+2 y+3 z=1\n$$\n\nlocated in the first octant (the normal forms an acute angle with the $OZ$ axis). Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{1}{18}",
"style": "rule"
} | {
"index": "35056",
"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{}.",
"role": "user"
}
] | {
"ground_truth": "1716",
"style": "rule"
} | {
"index": "35057",
"split": "train"
} |
math | [
{
"content": "A particle moves in a straight line inside a square of side 1. It is reflected from the sides, but absorbed by the four corners. It starts from an arbitrary point \\( P \\) inside the square. Let \\( c(k) \\) be the number of possible starting directions from which it reaches a corner after travel... | {
"ground_truth": "\\pi",
"style": "rule"
} | {
"index": "35058",
"split": "train"
} |
math | [
{
"content": "Piravena must make a trip from $A$ to $B,$ then from $B$ to $C,$ then from $C$ to $A.$ Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000\\text{ km}$ from $A$ and with $B$ a distance ... | {
"ground_truth": "\\$425",
"style": "rule"
} | {
"index": "35059",
"split": "train"
} |
math | [
{
"content": "Let \\( ABC \\) be any triangle. Let \\( D \\) and \\( E \\) be points on \\( AB \\) and \\( BC \\) respectively such that \\( AD = 7DB \\) and \\( BE = 10EC \\). Assume that \\( AE \\) and \\( CD \\) meet at a point \\( F \\). Determine \\( \\lfloor k \\rfloor \\), where \\( k \\) is the real num... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "35060",
"split": "train"
} |
math | [
{
"content": "Let $q(x) = 2x^6 - 3x^4 + Dx^2 + 6$ be a polynomial. When $q(x)$ is divided by $x - 2$, the remainder is 14. Find the remainder when $q(x)$ is divided by $x + 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "158",
"style": "rule"
} | {
"index": "35061",
"split": "train"
} |
math | [
{
"content": "Let $S=\\{1,2, \\ldots, 2021\\}$, and let $\\mathcal{F}$ denote the set of functions $f: S \\rightarrow S$. For a function $f \\in \\mathcal{F}$, let $$T_{f}=\\left\\{f^{2021}(s): s \\in S\\right\\}$$ where $f^{2021}(s)$ denotes $f(f(\\cdots(f(s)) \\cdots))$ with 2021 copies of $f$. Compute the re... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "35062",
"split": "train"
} |
math | [
{
"content": "Calculate the limit of the function:\n\n$$\\lim _{x \\rightarrow 0}\\left(\\frac{\\arcsin ^{2} x}{\\arcsin ^{2} 4 x}\\right)^{2 x+1}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/16",
"style": "rule"
} | {
"index": "35063",
"split": "train"
} |
math | [
{
"content": "Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with a radius of 10. The sphere intersects the face $A A_{1} D_{1} D$ by a circle with a radius of 1, the face $A_{1} B_{1} C_{1} D_{1}$ by a circle with a radius of 1, and the face $C D D_{1} C_{1}$ by a circle with a... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35064",
"split": "train"
} |
math | [
{
"content": "What is the three-digit number that is one less than twice the number formed by switching its outermost digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "793",
"style": "rule"
} | {
"index": "35065",
"split": "train"
} |
math | [
{
"content": "A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "35066",
"split": "train"
} |
math | [
{
"content": "Fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. His leash is exactly long enough to reach the midpoint of each side of his yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest ra... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "35067",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c,$ $d,$ $e,$ $f,$ $g,$ and $h$ be real numbers such that $abcd = 4$ and $efgh = 9.$ Find the minimum value of\n\\[(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "35068",
"split": "train"
} |
math | [
{
"content": "If $x = -3$, what is the value of $(x-3)^{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "35069",
"split": "train"
} |
math | [
{
"content": "Given Lara ate $\\frac{1}{4}$ of a pie and Ryan ate $\\frac{3}{10}$ of the same pie, then Cassie ate $\\frac{2}{3}$ of the pie that was left. Calculate the fraction of the original pie that was not eaten. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{20}",
"style": "rule"
} | {
"index": "35070",
"split": "train"
} |
math | [
{
"content": "In how many ways can five girls and five boys be seated around a circular table such that no two people of the same gender sit next to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28800",
"style": "rule"
} | {
"index": "35071",
"split": "train"
} |
math | [
{
"content": "Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1052}{8192}",
"style": "rule"
} | {
"index": "35072",
"split": "train"
} |
math | [
{
"content": "Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1, 2, 3, 5",
"style": "rule"
} | {
"index": "35073",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively. It is given that $c \\cos B = (2a - b) \\cos C$.\n1. Find the magnitude of angle $C$.\n2. If $AB = 4$, find the maximum value of the area $S$ of $\\triangle ABC$. Please output... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "35074",
"split": "train"
} |
math | [
{
"content": "Solve for $r$: $\\frac{r+9}{r-3} = \\frac{r-2}{r+5}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{39}{19}",
"style": "rule"
} | {
"index": "35075",
"split": "train"
} |
math | [
{
"content": "Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60718293",
"style": "rule"
} | {
"index": "35076",
"split": "train"
} |
math | [
{
"content": "My friend and I both have the same math homework one day. I work at a rate of $p$ problems per hour and it takes me $t$ hours to finish my homework. My friend works at a rate of $2p-4$ problems per hour and it only takes him $t-2$ hours to finish his homework. Given that $p$ and $t$ are positive w... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "35077",
"split": "train"
} |
math | [
{
"content": "Real numbers $x_{1}, x_{2}, \\cdots, x_{2001}$ satisfy $\\sum_{k=1}^{2000}\\left|x_{k}-x_{k+1}\\right|=2001$. Let $y_{k}=\\frac{1}{k} \\sum_{i=1}^{k} x_{i}$ for $k=1,2, \\cdots, 2001$. Find the maximum possible value of $\\sum_{k=1}^{2000}\\left|y_{k}-y_{k+1}\\right|$. Please output the final answ... | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": "35078",
"split": "train"
} |
math | [
{
"content": "Given the plane vectors $\\overrightarrow{a}=(1,0)$ and $\\overrightarrow{b}=\\left( -\\frac{1}{2}, \\frac{\\sqrt{3}}{2} \\right)$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{a}+ \\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "35079",
"split": "train"
} |
math | [
{
"content": "How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable, the boxes are distinguishable, and each box must contain at least one ball? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35080",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "35081",
"split": "train"
} |
math | [
{
"content": "A track and field team has three sprinters. According to the statistics from their usual training sessions, the probabilities of sprinters A, B, and C finishing a 100m dash within 13 seconds (considered qualified) are $\\frac{2}{5}$, $\\frac{3}{4}$, and $\\frac{1}{3}$, respectively. If the perform... | {
"ground_truth": "\\frac{9}{10}",
"style": "rule"
} | {
"index": "35082",
"split": "train"
} |
math | [
{
"content": "A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-37",
"style": "rule"
} | {
"index": "35083",
"split": "train"
} |
math | [
{
"content": "Let $S = \\{1, 22, 333, \\dots , 999999999\\}$ . For how many pairs of integers $(a, b)$ where $a, b \\in S$ and $a < b$ is it the case that $a$ divides $b$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "35084",
"split": "train"
} |
math | [
{
"content": "I have 7 books, three of which are identical copies of the same novel, and the others are distinct. If a particular book among these must always be placed at the start of the shelf, in how many ways can I arrange the rest of the books? Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "35085",
"split": "train"
} |
math | [
{
"content": "Calculate 8 divided by $\\frac{1}{8}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "35086",
"split": "train"
} |
math | [
{
"content": "Find the perimeter of an equilateral triangle inscribed in a circle, given that a chord of this circle, equal to 2, is at a distance of 3 from its center. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 \\sqrt{30}",
"style": "rule"
} | {
"index": "35087",
"split": "train"
} |
math | [
{
"content": "In a certain exam with 50 questions, each correct answer earns 3 points, each incorrect answer deducts 1 point, and unanswered questions neither add nor deduct points. Xiaolong scored 120 points. How many questions did Xiaolong answer correctly at most? Please output the final answer within \\boxe... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "35088",
"split": "train"
} |
math | [
{
"content": "Express $213_{8}-142_{7}$ as a base 10 integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "35089",
"split": "train"
} |
math | [
{
"content": "Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\\triangle AFD$ to the area of square $ABCE$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "35090",
"split": "train"
} |
math | [
{
"content": "Given $a \\gt 0$, $b \\gt 0$, if ${a}^{2}+{b}^{2}-\\sqrt{3}ab=1$, determine the maximum value of $\\sqrt{3}{a}^{2}-ab$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 + \\sqrt{3}",
"style": "rule"
} | {
"index": "35091",
"split": "train"
} |
math | [
{
"content": "Express this sum as a common fraction: $.\\overline{8} + .\\overline{2}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{9}",
"style": "rule"
} | {
"index": "35092",
"split": "train"
} |
math | [
{
"content": "What is the value of $x$ in the equation $16^{16}+16^{16}+16^{16}+16^{16}=2^x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "35093",
"split": "train"
} |
math | [
{
"content": "Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "35094",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{AB}$ be a diameter in a circle of radius $5\\sqrt{2}.$ Let $\\overline{CD}$ be a chord in the circle that intersects $\\overline{AB}$ at a point $E$ such that $BE=2\\sqrt{5}$ and $\\angle AEC = 45^{\\circ}.$ What is $CE^2+DE^2?$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "35095",
"split": "train"
} |
math | [
{
"content": "Higher Secondary P4\n\nIf the fraction $\\dfrac{a}{b}$ is greater than $\\dfrac{31}{17}$ in the least amount while $b<17$ , find $\\dfrac{a}{b}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{6}",
"style": "rule"
} | {
"index": "35096",
"split": "train"
} |
math | [
{
"content": "Vivek is painting three doors numbered 1, 2, and 3. Each door is to be painted either black or gold. How many different ways can the three doors be painted? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "35097",
"split": "train"
} |
math | [
{
"content": "Given two fixed points $A(-2,0)$ and $B(2,0)$, a moving point $P(x,y)$ is located on the line $l:y=x+3$. An ellipse $c$ has foci at points $A$ and $B$ and passes through point $P$. Determine the maximum eccentricity of the ellipse $c$. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{2\\sqrt{26}}{13}",
"style": "rule"
} | {
"index": "35098",
"split": "train"
} |
math | [
{
"content": "Given the function $f\\left(x\\right)=\\left(x+1\\right)e^{x}$. \n$(1)$ Find the intervals where the function $f\\left(x\\right)$ is monotonic. \n$(2)$ Find the maximum and minimum values of $f\\left(x\\right)$ on the interval $\\left[-4,0\\right]$. Please output the final answer within \\boxed{... | {
"ground_truth": "-\\frac{1}{e^2}",
"style": "rule"
} | {
"index": "35099",
"split": "train"
} |
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