data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "31000",
"split": "train"
} |
math | [
{
"content": "Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 2... | {
"ground_truth": "\\frac{91}{100}",
"style": "rule"
} | {
"index": "31001",
"split": "train"
} |
math | [
{
"content": "Given that \\(a, b, c\\) are positive integers and the quadratic equation \\(a x^{2}+b x+c=0\\) has two real roots whose absolute values are both less than \\(\\frac{1}{3}\\), find the minimum value of \\(a + b + c\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "31002",
"split": "train"
} |
math | [
{
"content": "Sally's salary in 2006 was $\\$ 37,500 $. For 2007 she got a salary increase of $ x $ percent. For 2008 she got another salary increase of $ x $ percent. For 2009 she got a salary decrease of $ 2x $ percent. Her 2009 salary is $ \\ $34,825$ . Suppose instead, Sally had gotten a $2x$ percent sal... | {
"ground_truth": "34825",
"style": "rule"
} | {
"index": "31003",
"split": "train"
} |
math | [
{
"content": "The unknown real numbers $x, y, z$ satisfy the equations $$\\frac{x+y}{1+z}=\\frac{1-z+z^{2}}{x^{2}-x y+y^{2}} ; \\quad \\frac{x-y}{3-z}=\\frac{9+3 z+z^{2}}{x^{2}+x y+y^{2}}$$ Find $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[3]{14}",
"style": "rule"
} | {
"index": "31004",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, the coordinates of point P(a,b) satisfy $a \\neq b$, and both $a$ and $b$ are elements of the set $\\{1,2,3,4,5,6\\}$. Additionally, the distance from point P to the origin, $|OP|$, is greater than or equal to 5. The number of such points P is ______. Please out... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31005",
"split": "train"
} |
math | [
{
"content": "There are 101 people participating in a Secret Santa gift exchange. As usual each person is randomly assigned another person for whom (s)he has to get a gift, such that each person gives and receives exactly one gift and no one gives a gift to themself. What is the probability that the first perso... | {
"ground_truth": "0.96039",
"style": "rule"
} | {
"index": "31006",
"split": "train"
} |
math | [
{
"content": "Given the inequality $x^{2}+ax+1\\geqslant 0$, if this inequality holds for all $x\\in(0, \\frac {1}{2}]$, find the minimum value of the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac {5}{2}",
"style": "rule"
} | {
"index": "31007",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $A=\\frac{\\pi}{3}$, $c=4$, and $a=2\\sqrt{6}$. Find the measure of angle $C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "31008",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31009",
"split": "train"
} |
math | [
{
"content": "How many values of $\\theta$ in the interval $0 < \\theta \\leq 4\\pi$ satisfy the equation $2 + 4\\sin2\\theta - 3\\cos4\\theta + 2\\tan\\theta = 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31010",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$, $AB=9$, $BC=13$, and $\\angle B = 90^\\circ$. Points $D$ and $E$ are midpoints of $\\overline{AB}$ and $\\overline{AC}$ respectively; $\\overline{CD}$ and $\\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ to the area of triang... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "31011",
"split": "train"
} |
math | [
{
"content": "$a$ and $b$ are integers such that $a+\\sqrt{b}=\\sqrt{15+\\sqrt{216}}$. Compute $a / b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "31012",
"split": "train"
} |
math | [
{
"content": "Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? ... | {
"ground_truth": "170",
"style": "rule"
} | {
"index": "31013",
"split": "train"
} |
math | [
{
"content": "It is possible to place positive integers into the vacant twenty-one squares of the $5 \\times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*). Please output the final answer w... | {
"ground_truth": "142",
"style": "rule"
} | {
"index": "31014",
"split": "train"
} |
math | [
{
"content": "Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "31015",
"split": "train"
} |
math | [
{
"content": "Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\\gcd(k, n)=1$ is a prime. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "31016",
"split": "train"
} |
math | [
{
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.5",
"style": "rule"
} | {
"index": "31017",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a quadrilateral with an inscribed circle $\\omega$ that has center $I$. If $I A=5, I B=7, I C=4, I D=9$, find the value of $\\frac{A B}{C D}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{35}{36}",
"style": "rule"
} | {
"index": "31018",
"split": "train"
} |
math | [
{
"content": "Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\\frac{3}{4}$.\n$(1)$ Find the equation of the locus $C$ for point $P$.\n$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at an... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "31019",
"split": "train"
} |
math | [
{
"content": "Let $x_1, x_2, \\ldots, x_7$ be natural numbers, and $x_1 < x_2 < \\ldots < x_6 < x_7$, also $x_1 + x_2 + \\ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "31020",
"split": "train"
} |
math | [
{
"content": "Compute: $104 \\times 96$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9984",
"style": "rule"
} | {
"index": "31021",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31022",
"split": "train"
} |
math | [
{
"content": "Let $\\left\\{a_{n}\\right\\}$ be the number of subsets of the set $\\{1,2, \\ldots, n\\}$ with the following properties:\n- Each subset contains at least two elements.\n- The absolute value of the difference between any two elements in the subset is greater than 1.\n\nFind $\\boldsymbol{a}_{10}$.... | {
"ground_truth": "133",
"style": "rule"
} | {
"index": "31023",
"split": "train"
} |
math | [
{
"content": "Given 4 distinct books that are to be distributed evenly between two students, find the probability that the Chinese language book and the Mathematics book are given to the same student. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "31024",
"split": "train"
} |
math | [
{
"content": "Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combinations need to be c... | {
"ground_truth": "504",
"style": "rule"
} | {
"index": "31025",
"split": "train"
} |
math | [
{
"content": "A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "502",
"style": "rule"
} | {
"index": "31026",
"split": "train"
} |
math | [
{
"content": "Determine the number of ways to arrange the letters of the word PERSEVERANCE. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19,958,400",
"style": "rule"
} | {
"index": "31027",
"split": "train"
} |
math | [
{
"content": "Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the ab... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "31028",
"split": "train"
} |
math | [
{
"content": "If $x$ is a positive multiple of 8 and $x^2>100$, but $x<20$, what is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "31029",
"split": "train"
} |
math | [
{
"content": "Find the remainder when the sum \\[75+76+77+78+79+80+81+82\\]is divided by 16. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "31030",
"split": "train"
} |
math | [
{
"content": "For a four-digit natural number $A$, if the digit in the thousands place is $5$ more than the digit in the tens place, and the digit in the hundreds place is $3$ more than the digit in the units place, then $A$ is called a \"five-three number.\" For example, for the four-digit number $6714$, since... | {
"ground_truth": "5401",
"style": "rule"
} | {
"index": "31031",
"split": "train"
} |
math | [
{
"content": "To make lemonade, I use a ratio of $7$ parts water to $1$ part lemon juice. If I want to make a gallon of lemonade, and there are four quarts in a gallon, how many quarts of water do I need? Write your answer as a proper or improper fraction. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{7}{2}",
"style": "rule"
} | {
"index": "31032",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\frac{\\ln x}{x}$.\n(1) Find the monotonic intervals of the function $f(x)$;\n(2) Let $m > 0$, find the maximum value of $f(x)$ on $[m, 2m]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{e}",
"style": "rule"
} | {
"index": "31033",
"split": "train"
} |
math | [
{
"content": "For any permutation $p$ of set $\\{1, 2, \\ldots, n\\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \\ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \\leqq < j \\leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequa... | {
"ground_truth": "$p=(1 \\; n)$.",
"style": "rule"
} | {
"index": "31034",
"split": "train"
} |
math | [
{
"content": "\\( P \\) is the midpoint of the height \\( V O \\) of the regular quadrilateral pyramid \\( V-ABCD \\). If the distance from point \\( P \\) to the side faces is 3, and the distance to the base is 5, what is the volume of the regular quadrilateral pyramid? Please output the final answer within \\... | {
"ground_truth": "750",
"style": "rule"
} | {
"index": "31035",
"split": "train"
} |
math | [
{
"content": "Six orange candies and four purple candies are available to create different flavors. A flavor is considered different if the percentage of orange candies is different. Combine some or all of these ten candies to determine how many unique flavors can be created based on their ratios. Please output... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "31036",
"split": "train"
} |
math | [
{
"content": "Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\\circ$ left and walk one unit to a... | {
"ground_truth": "(13,-22)",
"style": "rule"
} | {
"index": "31037",
"split": "train"
} |
math | [
{
"content": "Given the sequence $1, 4, 9, -16, 25, -36,\\ldots$, whose $n$th term is $(-1)^{n+1}\\cdot n^2$, find the average of the first $100$ terms of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-50.5",
"style": "rule"
} | {
"index": "31038",
"split": "train"
} |
math | [
{
"content": "If the sum of the second and fourth terms of an arithmetic sequence is $6$, what is the third term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "31039",
"split": "train"
} |
math | [
{
"content": "Given a circle $C: x^2+y^2-2x+4y-4=0$, and a line $l$ with a slope of 1 intersects the circle $C$ at points $A$ and $B$.\n\n(1) Express the equation of the circle in standard form, and identify the center and radius of the circle;\n\n(2) Does there exist a line $l$ such that the circle with diamet... | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "31040",
"split": "train"
} |
math | [
{
"content": "The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is $\\boxed{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "31041",
"split": "train"
} |
math | [
{
"content": "Given a $3 \\times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.\n\nFind the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally. Please output the final answer within \\b... | {
"ground_truth": "8/21",
"style": "rule"
} | {
"index": "31042",
"split": "train"
} |
math | [
{
"content": "What is the period of $y = \\sin 5x $? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\pi}{5}",
"style": "rule"
} | {
"index": "31043",
"split": "train"
} |
math | [
{
"content": "In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed u... | {
"ground_truth": "378",
"style": "rule"
} | {
"index": "31044",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \\le 10$ and $3y - x \\ge 15$. When $\\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the volume of the resulting solid is $\\frac {m\\pi}{n\\sqrt {p}}$... | {
"ground_truth": "365",
"style": "rule"
} | {
"index": "31045",
"split": "train"
} |
math | [
{
"content": "Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\\rho_1$ , $\\rho_2$ , $\\rho_3$ such that $|\\rho_1| < |\\rho_2| < |\\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\\rho_1, 0)$ , $C(\\rho_2, ... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "31046",
"split": "train"
} |
math | [
{
"content": "Solve for $y$: $50^4 = 10^y$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6.79588",
"style": "rule"
} | {
"index": "31047",
"split": "train"
} |
math | [
{
"content": "The matrix for projecting onto a certain line $\\ell,$ which passes through the origin, is given by\n\\[\\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} \\frac{2}{15} & -\\frac{1}{15} & -\\frac{1}{3} \\\\ -\\frac{1}{15} & \\frac{1}{30} & \\frac{1}{6} \\\\ -\\frac{1}{3} & \\frac{1}{6} & \\frac{... | {
"ground_truth": "\\begin{pmatrix} 2 \\\\ -1 \\\\ -5 \\end{pmatrix}",
"style": "rule"
} | {
"index": "31048",
"split": "train"
} |
math | [
{
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "31049",
"split": "train"
} |
math | [
{
"content": "Given a regular tetrahedron of volume 1, obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "31050",
"split": "train"
} |
math | [
{
"content": "Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "31051",
"split": "train"
} |
math | [
{
"content": "Find $\\cot (-60^\\circ).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "31052",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 800$ and $AD = 1800$. Let $\\angle A = 45^\\circ$, $\\angle D = 45^\\circ$, and $P$ and $Q$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $PQ$. Please output the final answer wit... | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "31053",
"split": "train"
} |
math | [
{
"content": "In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.\n\n\\begin{tabular}{|c|c|}\n\\multicolumn{2}{c}{}\\\\\\hline\n\\textbf{Score (\\%)}&\\textbf{Number of Students}\\\\\\hline\n95&12\\\\\\hline\n85&24\\\\\\hline\n75&30\\\\\\... | {
"ground_truth": "69.83",
"style": "rule"
} | {
"index": "31054",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{1}{b}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 2 \\sqrt{2}",
"style": "rule"
} | {
"index": "31055",
"split": "train"
} |
math | [
{
"content": "Let \\(A B C\\) be a triangle with \\(\\angle A=18^{\\circ}, \\angle B=36^{\\circ}\\). Let \\(M\\) be the midpoint of \\(A B, D\\) a point on ray \\(C M\\) such that \\(A B=A D ; E\\) a point on ray \\(B C\\) such that \\(A B=B E\\), and \\(F\\) a point on ray \\(A C\\) such that \\(A B=A F\\). Fi... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "31056",
"split": "train"
} |
math | [
{
"content": "In the parallelepiped $ABCD A_1B_1C_1D_1$, the face $ABCD$ is a square with side length 5, the edge $AA_1$ is also equal to 5, and this edge forms angles of $60^\\circ$ with the edges $AB$ and $AD$. Find the length of the diagonal $BD_1$. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "5\\sqrt{3}",
"style": "rule"
} | {
"index": "31057",
"split": "train"
} |
math | [
{
"content": "Given the pattern of positive odd numbers shown below, find the 6th number from the left in the 21st row. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "811",
"style": "rule"
} | {
"index": "31058",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0)=2$, then $f(4)=$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31059",
"split": "train"
} |
math | [
{
"content": "How many lines pass through exactly two points in the following hexagonal grid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "31060",
"split": "train"
} |
math | [
{
"content": "Suppose $b$ is an integer such that $1 \\le b \\le 30$, and $524123_{81}-b$ is a multiple of $17$. What is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "31061",
"split": "train"
} |
math | [
{
"content": "An exchange point conducts two types of operations:\n\n1) Give 2 euros - receive 3 dollars and a candy as a gift;\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31062",
"split": "train"
} |
math | [
{
"content": "Given that the function $f(x)=\\ln (e^{x}+a+1)$ ($a$ is a constant) is an odd function on the set of real numbers $R$.\n(1) Find the value of the real number $a$;\n(2) If the equation $\\frac{\\ln x}{f(x)}=x^{2}-2ex+m$ has exactly one real root, find the value of $m$. Please output the final answe... | {
"ground_truth": "e^{2}+\\frac{1}{e}",
"style": "rule"
} | {
"index": "31063",
"split": "train"
} |
math | [
{
"content": "Solve for $z$ in the following equation: $1-iz = -1 + iz$ (where $i^2 = -1$). Simplify your answer as much as possible. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-i",
"style": "rule"
} | {
"index": "31064",
"split": "train"
} |
math | [
{
"content": "In the geoboard shown, the points are evenly spaced vertically and horizontally. Segment $AB$ is drawn using two points, as shown. Point $C$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $ABC$ being isosceles? [asy]\ndraw((0,0)--(0,6)--(6,6)-... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "31065",
"split": "train"
} |
math | [
{
"content": "There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "31066",
"split": "train"
} |
math | [
{
"content": "King Middle School has $1200$ students. Each student takes $5$ classes a day. Each teacher teaches $4$ classes. Each class has $30$ students and $1$ teacher. How many teachers are there at King Middle School? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "31067",
"split": "train"
} |
math | [
{
"content": "There are 5 people standing in a row. The number of ways to arrange them so that there is exactly one person between A and B. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31068",
"split": "train"
} |
math | [
{
"content": "Suppose $\\cos S = 0.5$ in a right triangle where $SP = 10$. What is $SR$?\n\n[asy]\npair S,P,R;\nS = (0,0);\nP = (10,0);\nR = (0,10*tan(acos(0.5)));\ndraw(S--P--R--S);\ndraw(rightanglemark(S,P,R,18));\nlabel(\"$S$\",S,SW);\nlabel(\"$P$\",P,SE);\nlabel(\"$R$\",R,N);\nlabel(\"$10$\",P/2,S);\n[/asy]... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "31069",
"split": "train"
} |
math | [
{
"content": "Given that the domain of the function $f(x)$ is $\\mathbb{R}$, and for any $x \\in \\mathbb{Z}$, it holds that $f(x) = f(x-1) + f(x+1)$. If $f(-1) = 6$ and $f(1) = 7$, then $f(2012) + f(-2012) = \\boxed{?}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": "31070",
"split": "train"
} |
math | [
{
"content": "Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$ . Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1266",
"style": "rule"
} | {
"index": "31071",
"split": "train"
} |
math | [
{
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "749",
"style": "rule"
} | {
"index": "31072",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be unit vectors such that\n\\[\\mathbf{a} + \\mathbf{b} + \\sqrt{3} \\mathbf{c} = \\mathbf{0}.\\]Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.\n\nNote: A unit vector is a vector of magnitude 1. Please output the final answer... | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "31073",
"split": "train"
} |
math | [
{
"content": "When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.\nWhat is the remainder when $x+2uy$ is divided by $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "v",
"style": "rule"
} | {
"index": "31074",
"split": "train"
} |
math | [
{
"content": "In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \\(S, T, U,\\) and \\(V\\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \\(XY\\) is outside th... | {
"ground_truth": "29.3",
"style": "rule"
} | {
"index": "31075",
"split": "train"
} |
math | [
{
"content": "Suppose $X$ and $Y$ are digits in base $d > 8$ such that $\\overline{XY}_d + \\overline{XX}_d = 234_d$. Find $X_d - Y_d$ in base $d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "31076",
"split": "train"
} |
math | [
{
"content": "Find all real solutions to\n\\[\\frac{1}{(x - 1)(x - 2)} + \\frac{1}{(x - 2)(x - 3)} + \\frac{1}{(x - 3)(x - 4)} = \\frac{1}{6}.\\]Enter all solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7,-2",
"style": "rule"
} | {
"index": "31077",
"split": "train"
} |
math | [
{
"content": "A circle with center $A$ has radius $10$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $XY$ is the common external tangent to circle $A$ and circle $B$ at points $X$ and $Y$, respectively. What is the length of segment $AY$? Expre... | {
"ground_truth": "2\\sqrt{55}",
"style": "rule"
} | {
"index": "31078",
"split": "train"
} |
math | [
{
"content": "A stick 5 cm long, a stick 9 cm long, and a third stick $n$ cm long form a triangle. What is the sum of all possible whole number values of $n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "31079",
"split": "train"
} |
math | [
{
"content": "Let the line \\( p \\) be the perpendicular bisector of points \\( A = (20, 12) \\) and \\( B = (-4, 3) \\). Determine the point \\( C = (x, y) \\) where line \\( p \\) meets segment \\( AB \\), and calculate \\( 3x - 5y \\). Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-13.5",
"style": "rule"
} | {
"index": "31080",
"split": "train"
} |
math | [
{
"content": "Find the distance between the points $(-5,-2)$ and $(7,3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31081",
"split": "train"
} |
math | [
{
"content": "1. There are 5 different books, and we need to choose 3 books to give to 3 students, one book per student. There are a total of different ways to do this.\n2. There are 5 different books, and we want to buy 3 books to give to 3 students, one book per student. There are a total of different... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "31082",
"split": "train"
} |
math | [
{
"content": "A rectangular sheet of paper is 20 cm long and 12 cm wide. It is folded along its diagonal. What is the perimeter of the shaded region? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "31083",
"split": "train"
} |
math | [
{
"content": "Using the trapezoidal rule with an accuracy of 0.01, calculate $\\int_{2}^{3} \\frac{d x}{x-1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.6956",
"style": "rule"
} | {
"index": "31084",
"split": "train"
} |
math | [
{
"content": "Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "31085",
"split": "train"
} |
math | [
{
"content": "The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "31086",
"split": "train"
} |
math | [
{
"content": "Find the quadratic polynomial $p(x)$ such that $p(-2) = 13,$ $p(1) = -2,$ and $p(3) = 8.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2x^2 - 3x - 1",
"style": "rule"
} | {
"index": "31087",
"split": "train"
} |
math | [
{
"content": "Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one amon... | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "31088",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola with the equation $\\frac{x^{2}}{4} - \\frac{y^{2}}{9} = 1$, where $F\\_1$ and $F\\_2$ are its foci, and point $M$ lies on the hyperbola.\n(1) If $\\angle F\\_1 M F\\_2 = 90^{\\circ}$, find the area of $\\triangle F\\_1 M F\\_2$.\n(2) If $\\angle F\\_1 M F\\_2 = 60^{\\circ}$, w... | {
"ground_truth": "3 \\sqrt{3}",
"style": "rule"
} | {
"index": "31089",
"split": "train"
} |
math | [
{
"content": "A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "31090",
"split": "train"
} |
math | [
{
"content": "[asy]\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16);\ndraw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N);\nlabel(\"S\", (18,8));\nlabel(\"S\", (50,8));\... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "31091",
"split": "train"
} |
math | [
{
"content": "How many multiples of 10 are between 9 and 101? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31092",
"split": "train"
} |
math | [
{
"content": "Find a whole number, $M$, such that $\\frac{M}{5}$ is strictly between 9.5 and 10.5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "31093",
"split": "train"
} |
math | [
{
"content": "Given that $a > 0$, $b > 0$, and $\\frac{1}{a} + \\frac{2}{b} = 2$.\n(1) Find the minimum value of $ab$;\n(2) Find the minimum value of $a + 2b$, and find the corresponding values of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "31094",
"split": "train"
} |
math | [
{
"content": "A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the word MATH? Please out... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "31095",
"split": "train"
} |
math | [
{
"content": "Find $\\begin{pmatrix} -5 \\\\ 1 \\\\ -4 \\end{pmatrix} + \\begin{pmatrix} 0 \\\\ 8 \\\\ -4 \\end{pmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -5 \\\\ 9 \\\\ -8 \\end{pmatrix}",
"style": "rule"
} | {
"index": "31096",
"split": "train"
} |
math | [
{
"content": "Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\\angle ABC=90^{\\circ}$, as shown. What is the area of the quadrilateral? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "31097",
"split": "train"
} |
math | [
{
"content": "Let $g(x) = ax^7 + bx^3 + dx^2 + cx - 8$. If $g(-7) = 3$, then find $g(7)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-19",
"style": "rule"
} | {
"index": "31098",
"split": "train"
} |
math | [
{
"content": "A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to two decimal places. P... | {
"ground_truth": "5.97",
"style": "rule"
} | {
"index": "31099",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.