data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let $x_1,$ $x_2,$ $\\dots,$ $x_{100}$ be real numbers such that $x_1 + x_2 + \\dots + x_{100} = 1$ and\n\\[\\frac{x_1}{1 - x_1} + \\frac{x_2}{1 - x_2} + \\dots + \\frac{x_{100}}{1 - x_{100}} = 1.\\]Find\n\\[\\frac{x_1^2}{1 - x_1} + \\frac{x_2^2}{1 - x_2} + \\dots + \\frac{x_{100}^2}{1 - x_{100}}.\... | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "33800",
"split": "train"
} |
math | [
{
"content": "Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F. Please output the final answer within \\boxed{}."... | {
"ground_truth": "36\\pi",
"style": "rule"
} | {
"index": "33801",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\\frac{a + c}{a - c}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "33802",
"split": "train"
} |
math | [
{
"content": "Given a positive integer \\( n \\geqslant 2 \\), positive real numbers \\( a_1, a_2, \\ldots, a_n \\), and non-negative real numbers \\( b_1, b_2, \\ldots, b_n \\), which satisfy the following conditions:\n(a) \\( a_1 + a_2 + \\cdots + a_n + b_1 + b_2 + \\cdots + b_n = n \\);\n(b) \\( a_1 a_2 \\cd... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33803",
"split": "train"
} |
math | [
{
"content": "Find the distance between the vertices of the hyperbola $9x^2 + 54x - y^2 + 10y + 55 = 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "33804",
"split": "train"
} |
math | [
{
"content": "Convert $12012_3$ to a base 10 integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "33805",
"split": "train"
} |
math | [
{
"content": "Determine the number of integers $2 \\leq n \\leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "33806",
"split": "train"
} |
math | [
{
"content": "How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "33807",
"split": "train"
} |
math | [
{
"content": "Given a set of data pairs (3,y_{1}), (5,y_{2}), (7,y_{3}), (12,y_{4}), (13,y_{5}) corresponding to variables x and y, the linear regression equation obtained is \\hat{y} = \\frac{1}{2}x + 20. Calculate the value of \\sum\\limits_{i=1}^{5}y_{i}. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "33808",
"split": "train"
} |
math | [
{
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{35}",
"style": "rule"
} | {
"index": "33809",
"split": "train"
} |
math | [
{
"content": "In 500 kg of ore, there is a certain amount of iron. After removing 200 kg of impurities, which contain on average 12.5% iron, the iron content in the remaining ore increased by 20%. What amount of iron remains in the ore? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "187.5",
"style": "rule"
} | {
"index": "33810",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33811",
"split": "train"
} |
math | [
{
"content": "Suppose that \\(a, b, c, d\\) are real numbers satisfying \\(a \\geq b \\geq c \\geq d \\geq 0\\), \\(a^2 + d^2 = 1\\), \\(b^2 + c^2 = 1\\), and \\(ac + bd = \\frac{1}{3}\\). Find the value of \\(ab - cd\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "33812",
"split": "train"
} |
math | [
{
"content": "Calculate the product of $1101_2 \\cdot 111_2$. Express your answer in base 2. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10010111_2",
"style": "rule"
} | {
"index": "33813",
"split": "train"
} |
math | [
{
"content": "Let \\mathbb{N} denote the natural numbers. Compute the number of functions $f: \\mathbb{N} \\rightarrow\\{0,1, \\ldots, 16\\}$ such that $$f(x+17)=f(x) \\quad \\text { and } \\quad f\\left(x^{2}\\right) \\equiv f(x)^{2}+15 \\quad(\\bmod 17)$$ for all integers $x \\geq 1$ Please output the final a... | {
"ground_truth": "12066",
"style": "rule"
} | {
"index": "33814",
"split": "train"
} |
math | [
{
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "33815",
"split": "train"
} |
math | [
{
"content": "Two cars, $A$ and $B$, depart from one city to another. In the first 5 minutes, they traveled the same distance. Then, due to an engine failure, $B$ had to reduce its speed to 2/5 of its original speed, and thus arrived at the destination 15 minutes after car $A$, which continued at a constant spe... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "33816",
"split": "train"
} |
math | [
{
"content": "Solve the inequality\n\\[|x - 1| + |x + 2| < 5.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-3,2)",
"style": "rule"
} | {
"index": "33817",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\\frac{n}{2520}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "33818",
"split": "train"
} |
math | [
{
"content": "The increasing sequence consists of all those positive integers which are either powers of 2, powers of 3, or sums of distinct powers of 2 and 3. Find the $50^{\\rm th}$ term of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "33819",
"split": "train"
} |
math | [
{
"content": "Two circles $\\Gamma_{1}$ and $\\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\\Gamma_{1}$. When it reaches $\\Gamma_{1}$, it bounces off the circumference and heads back towards $\\Gamma_{2}$. The particle continues b... | {
"ground_truth": "403",
"style": "rule"
} | {
"index": "33820",
"split": "train"
} |
math | [
{
"content": "In a $16 \\times 16$ table of integers, each row and column contains at most 4 distinct integers. What is the maximum number of distinct integers that there can be in the whole table? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "33821",
"split": "train"
} |
math | [
{
"content": "Determine all positive integers $ n\\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \\[a \\equiv b \\pmod n\\qquad\\text{if and only if}\\qquad ab\\equiv 1 \\pmod n.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2, 3, 4, 6, 8, 12, 24",
"style": "rule"
} | {
"index": "33822",
"split": "train"
} |
math | [
{
"content": "Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After drivin... | {
"ground_truth": "82.2",
"style": "rule"
} | {
"index": "33823",
"split": "train"
} |
math | [
{
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest co... | {
"ground_truth": "\\$187.12",
"style": "rule"
} | {
"index": "33824",
"split": "train"
} |
math | [
{
"content": "What is the least five-digit positive integer which is congruent to 7 (mod 21)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,003",
"style": "rule"
} | {
"index": "33825",
"split": "train"
} |
math | [
{
"content": "If $N$ is represented as $11000_2$ in binary, what is the binary representation of the integer that comes immediately before $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$10111_2$",
"style": "rule"
} | {
"index": "33826",
"split": "train"
} |
math | [
{
"content": "In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "33827",
"split": "train"
} |
math | [
{
"content": "Given the inequality (e-a)e^x + x + b + 1 ≤ 0, where e is the natural constant, find the maximum value of $\\frac{b+1}{a}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{e}",
"style": "rule"
} | {
"index": "33828",
"split": "train"
} |
math | [
{
"content": "Compute the square of 1085 without using a calculator. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1177225",
"style": "rule"
} | {
"index": "33829",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer that is divisible by the first three prime numbers greater than 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "33830",
"split": "train"
} |
math | [
{
"content": "8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.\n\na) How many four-digit numbers are liked by both Andrey and Tanya?\n\nb) Find the total sum of the digits of all such four-digit numbers. Please output the fi... | {
"ground_truth": "14580",
"style": "rule"
} | {
"index": "33831",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the number of ordered pairs of nonempty sets $\\mathcal{A}$ and $\\mathcal{B}$ that have the following properties:\n$\\mathcal{A} \\cup \\mathcal{B} = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$,\n$\\mathcal{A} \\cap \\mathcal{B} = \\emptyset$,\nThe number of elements of $\\mathcal{A}$ is not an ... | {
"ground_truth": "772",
"style": "rule"
} | {
"index": "33832",
"split": "train"
} |
math | [
{
"content": "How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "33833",
"split": "train"
} |
math | [
{
"content": "What time is it 2017 minutes after 20:17? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "05:54",
"style": "rule"
} | {
"index": "33834",
"split": "train"
} |
math | [
{
"content": "Given the following definition: We call a pair of rational numbers $a$ and $b$ that satisfy the equation $a-b=ab+1$ as \"companion rational number pairs,\" denoted as $\\left(a,b\\right)$. For example, $3-\\frac{1}{2}=3\\times \\frac{1}{2}+1$, $5-\\frac{2}{3}=5\\times \\frac{2}{3}+1$, so the pairs... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "33835",
"split": "train"
} |
math | [
{
"content": "Given points $A(-2,-2)$, $B(-2,6)$, $C(4,-2)$, and point $P$ moving on the circle $x^{2}+y^{2}=4$, find the maximum value of $|PA|^{2}+|PB|^{2}+|PC|^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "33836",
"split": "train"
} |
math | [
{
"content": "In the diagram, $COB$ is a sector of a circle with $\\angle COB=45^\\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10 - 5 \\sqrt{2+\\sqrt{2}}",
"style": "rule"
} | {
"index": "33837",
"split": "train"
} |
math | [
{
"content": "In the geometric sequence $\\{a_n\\}$, if $a_n > a_{n+1}$, and $a_7 \\cdot a_{14} = 6, a_4 + a_{17} = 5$, calculate $\\frac{a_5}{a_{18}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "33838",
"split": "train"
} |
math | [
{
"content": "Given that the radius of a hemisphere is 2, calculate the maximum lateral area of the inscribed cylinder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "33839",
"split": "train"
} |
math | [
{
"content": "A function $g$ from the integers to the integers is defined as follows:\n\\[g(n) = \\left\\{\n\\begin{array}{cl}\nn + 5 & \\text{if $n$ is odd}, \\\\\nn/2 & \\text{if $n$ is even}.\n\\end{array}\n\\right.\\]\nSuppose $m$ is odd and $g(g(g(m))) = 39.$ Find $m.$ Please output the final answer withi... | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "33840",
"split": "train"
} |
math | [
{
"content": "If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "33841",
"split": "train"
} |
math | [
{
"content": "Determine the set of all real numbers $p$ for which the polynomial $Q(x)=x^{3}+p x^{2}-p x-1$ has three distinct real roots. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "p>1 \\text{ and } p<-3",
"style": "rule"
} | {
"index": "33842",
"split": "train"
} |
math | [
{
"content": "Given that \\( x \\) and \\( y \\) are positive numbers, determine the minimum value of \\(\\left(x+\\frac{1}{y}\\right)^{2}+\\left(y+\\frac{1}{2x}\\right)^{2}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 + 2 \\sqrt{2}",
"style": "rule"
} | {
"index": "33843",
"split": "train"
} |
math | [
{
"content": "Let $PQRST$ be a convex pentagon with $PQ \\parallel RT, QR \\parallel PS, QS \\parallel PT, \\angle PQR=100^\\circ, PQ=4, QR=7,$ and $PT = 21.$ Given that the ratio between the area of triangle $PQR$ and the area of triangle $RST$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers... | {
"ground_truth": "232",
"style": "rule"
} | {
"index": "33844",
"split": "train"
} |
math | [
{
"content": "A class has $50$ students. The math scores $\\xi$ of an exam follow a normal distribution $N(100, 10^{2})$. Given that $P(90 \\leqslant \\xi \\leqslant 100)=0.3$, estimate the number of students with scores of $110$ or higher. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "33845",
"split": "train"
} |
math | [
{
"content": "The writer Arthur has $n \\ge1$ co-authors who write books with him. Each book has a list of authors including Arthur himself. No two books have the same set of authors. At a party with all his co-author, each co-author writes on a note how many books they remember having written with Arthur. Insp... | {
"ground_truth": "$n\\le6$",
"style": "rule"
} | {
"index": "33846",
"split": "train"
} |
math | [
{
"content": "Two positive real numbers have geometric mean $\\sqrt{3}$ and harmonic mean $\\frac{3}{2}.$ Enter the two numbers, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1,3",
"style": "rule"
} | {
"index": "33847",
"split": "train"
} |
math | [
{
"content": "What is $\\frac{2+4+6}{1+3+5} - \\frac{1+3+5}{2+4+6}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} | {
"index": "33848",
"split": "train"
} |
math | [
{
"content": "Given points $A(-1, 2, 0)$, $B(5, 2, -1)$, $C(2, -1, 4)$, and $D(-2, 2, -1)$ in space, find:\n\na) the distance from vertex $D$ of tetrahedron $ABCD$ to the intersection point of the medians of the base $ABC$;\n\nb) the equation of the plane $ABC$;\n\nc) the height of the tetrahedron from vertex $... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "33849",
"split": "train"
} |
math | [
{
"content": "A furniture store received a batch of office chairs that were identical except for their colors: 15 chairs were black and 18 were brown. The chairs were in demand and were being bought in a random order. At some point, a customer on the store's website discovered that only two chairs were left for... | {
"ground_truth": "0.489",
"style": "rule"
} | {
"index": "33850",
"split": "train"
} |
math | [
{
"content": "A permutation of a finite set is a one-to-one function from the set to itself; for instance, one permutation of $\\{1,2,3,4\\}$ is the function $\\pi$ defined such that $\\pi(1)=1, \\pi(2)=3$, $\\pi(3)=4$, and $\\pi(4)=2$. How many permutations $\\pi$ of the set $\\{1,2, \\ldots, 10\\}$ have the p... | {
"ground_truth": "945",
"style": "rule"
} | {
"index": "33851",
"split": "train"
} |
math | [
{
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"on the cells.\" Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detec... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "33852",
"split": "train"
} |
math | [
{
"content": "Compute $\\arccos(\\cos 9).$ All functions are in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9 - 2\\pi",
"style": "rule"
} | {
"index": "33853",
"split": "train"
} |
math | [
{
"content": "Let the focus of the parabola $y^{2}=8x$ be $F$, and its directrix be $l$. Let $P$ be a point on the parabola, and $PA\\perpendicular l$ with $A$ being the foot of the perpendicular. If the angle of inclination of the line $PF$ is $120^{\\circ}$, then $|PF|=$ ______. Please output the final answer... | {
"ground_truth": "\\dfrac{8}{3}",
"style": "rule"
} | {
"index": "33854",
"split": "train"
} |
math | [
{
"content": "In parallelogram \\(ABCD\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\), \\(AB = 73\\) and \\(BC = 88\\). The angle bisector of \\(\\angle ABC\\) intersects segment \\(AD\\) at point \\(E\\) and ray \\(CD\\) at point \\(F\\). Find the length of segment \\(EF\\).\n\n1. 9\n2. 13\n3. 12\n4. 15 ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "33855",
"split": "train"
} |
math | [
{
"content": "A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola. Please output the final ... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "33856",
"split": "train"
} |
math | [
{
"content": "If real numbers \\(a\\), \\(b\\), and \\(c\\) satisfy \\(a^{2} + b^{2} + c^{2} = 9\\), then what is the maximum value of the algebraic expression \\((a - b)^{2} + (b - c)^{2} + (c - a)^{2}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "33857",
"split": "train"
} |
math | [
{
"content": "Given $f(x)=\\cos x+\\cos (x+ \\frac {π}{2}).$\n(1) Find $f( \\frac {π}{12})$;\n(2) Suppose $α$ and $β∈(- \\frac {π}{2},0)$, $f(α+ \\frac {3π}{4})=- \\frac {3 \\sqrt {2}}{5}$, $f( \\frac {π}{4}-β)=- \\frac {5 \\sqrt {2}}{13}$, find $\\cos (α+β)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac {16}{65}",
"style": "rule"
} | {
"index": "33858",
"split": "train"
} |
math | [
{
"content": "Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "33859",
"split": "train"
} |
math | [
{
"content": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "33860",
"split": "train"
} |
math | [
{
"content": "Find the area of a triangle with side lengths 14, 48, and 50. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "33861",
"split": "train"
} |
math | [
{
"content": "A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "33862",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number with 6 as the last digit, such that if the final 6 is moved to the front of the number it is multiplied by 4. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "153846",
"style": "rule"
} | {
"index": "33863",
"split": "train"
} |
math | [
{
"content": "Given that bag A contains 3 white balls and 5 black balls, and bag B contains 4 white balls and 6 black balls, calculate the probability that the number of white balls in bag A does not decrease after a ball is randomly taken from bag A and put into bag B, and a ball is then randomly taken from ba... | {
"ground_truth": "\\frac{35}{44}",
"style": "rule"
} | {
"index": "33864",
"split": "train"
} |
math | [
{
"content": "A regular $n$-gon is inscribed in a circle with radius $R$, and its area is equal to $3 R^{2}$. Find $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33865",
"split": "train"
} |
math | [
{
"content": "Given a real number \\(a\\), and for any \\(k \\in [-1, 1]\\), when \\(x \\in (0, 6]\\), the inequality \\(6 \\ln x + x^2 - 8x + a \\leq kx\\) always holds. Determine the maximum value of \\(a\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6 - 6 \\ln 6",
"style": "rule"
} | {
"index": "33866",
"split": "train"
} |
math | [
{
"content": "Andrey drove a car to the airport in a neighboring city. After one hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. He then sharply increased his speed, and as a result, covered the remaining distance at an average speed of 90 km/... | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "33867",
"split": "train"
} |
math | [
{
"content": "Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6 + 3\\sqrt{3}",
"style": "rule"
} | {
"index": "33868",
"split": "train"
} |
math | [
{
"content": "What is the positive difference of the solutions of $\\dfrac{s^2 - 4s - 22}{s + 3} = 3s + 8$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{2}",
"style": "rule"
} | {
"index": "33869",
"split": "train"
} |
math | [
{
"content": "How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "33870",
"split": "train"
} |
math | [
{
"content": "Points \\( A \\), \\( B \\), \\( C \\), \\( D \\), and \\( E \\) are located in 3-dimensional space with \\( AB = BC = CD = DE = 3 \\) and \\( \\angle ABC = \\angle CDE = 60^\\circ \\). Additionally, the plane of triangle \\( ABC \\) is perpendicular to line \\( \\overline{DE} \\). Determine the a... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "33871",
"split": "train"
} |
math | [
{
"content": "Given that $m$ is an integer and $0 < 3m < 27$, what is the sum of all possible integer values of $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "33872",
"split": "train"
} |
math | [
{
"content": "What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33873",
"split": "train"
} |
math | [
{
"content": "The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "33874",
"split": "train"
} |
math | [
{
"content": "Tom's graduating class has 288 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 10 rows and at least 15 students in each row, then there can be $x$ students in each row. What is the sum of all possible value... | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "33875",
"split": "train"
} |
math | [
{
"content": "Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \\( 20 \\text{ cm} \\), what is the outside perimeter of the mosaic? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "33876",
"split": "train"
} |
math | [
{
"content": "In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three sides each of leng... | {
"ground_truth": "409",
"style": "rule"
} | {
"index": "33877",
"split": "train"
} |
math | [
{
"content": "If $a$ and $b$ are nonzero real numbers such that $\\left| a \\right| \\ne \\left| b \\right|$ , compute the value of the expression\n\\[\n \\left( \\frac{b^2}{a^2} + \\frac{a^2}{b^2} - 2 \\right) \\times\n \\left( \\frac{a + b}{b - a} + \\frac{b - a}{a + b} \\right) \\times\n \\left( \n ... | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "33878",
"split": "train"
} |
math | [
{
"content": "Express the number $15.7$ billion in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.57\\times 10^{9}",
"style": "rule"
} | {
"index": "33879",
"split": "train"
} |
math | [
{
"content": "A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "33880",
"split": "train"
} |
math | [
{
"content": "Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{4q+1}",
"style": "rule"
} | {
"index": "33881",
"split": "train"
} |
math | [
{
"content": "The average (mean) of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "33882",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{5}{4\\sqrt{7} + 3\\sqrt{13}}$ and express your answer in the form $\\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$ where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$? Please output the final answer within \\... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "33883",
"split": "train"
} |
math | [
{
"content": "Yvan and Zoé play the following game. Let \\( n \\in \\mathbb{N} \\). The integers from 1 to \\( n \\) are written on \\( n \\) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.\n... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "33884",
"split": "train"
} |
math | [
{
"content": "Regular decagon $P_1 P_2 \\dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product\n\\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dotsm (x_{10} + y_{10} i).\\] Please output the final a... | {
"ground_truth": "1023",
"style": "rule"
} | {
"index": "33885",
"split": "train"
} |
math | [
{
"content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bas... | {
"ground_truth": "\\sqrt{2}+\\frac{1}{2}",
"style": "rule"
} | {
"index": "33886",
"split": "train"
} |
math | [
{
"content": "Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in 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": "33887",
"split": "train"
} |
math | [
{
"content": "Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "33888",
"split": "train"
} |
math | [
{
"content": "On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. Whe... | {
"ground_truth": "\\frac{35}{72}",
"style": "rule"
} | {
"index": "33889",
"split": "train"
} |
math | [
{
"content": "Let $[ABCD]$ be a convex quadrilateral with area $2014$ , and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$ . Let $R$ be the intersection of $[AC]$ and $[PQ]$ . Determine $\\frac{\\overline{RC}}{\\overline{RA}}$ .... | {
"ground_truth": "2013",
"style": "rule"
} | {
"index": "33890",
"split": "train"
} |
math | [
{
"content": "Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the ... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "33891",
"split": "train"
} |
math | [
{
"content": "A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1223456",
"style": "rule"
} | {
"index": "33892",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\ln x + \\ln (ax+1) - \\frac {3a}{2}x + 1$ ($a \\in \\mathbb{R}$).\n\n(1) Discuss the intervals of monotonicity for the function $f(x)$.\n\n(2) When $a = \\frac {2}{3}$, if the inequality $xe^{x-\\frac {1}{2}} + m \\geqslant f(x)$ holds true for all $x$, find the minimu... | {
"ground_truth": "\\ln \\frac {2}{3}",
"style": "rule"
} | {
"index": "33893",
"split": "train"
} |
math | [
{
"content": "Yan is at a point between his house and a park. He has two options to reach the park: He can either walk directly to the park or he can walk back to his house and use his scooter to reach the park. He scoots 10 times faster than he walks, and both routes take the same amount of time. Determine the... | {
"ground_truth": "\\frac{9}{11}",
"style": "rule"
} | {
"index": "33894",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be angles such that $\\sin (a + b) = \\frac{3}{4}$ and $\\sin (a - b) = \\frac{1}{2}.$ Find $\\frac{\\tan a}{\\tan b}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "33895",
"split": "train"
} |
math | [
{
"content": "Given a list of the first 12 positive integers such that for each $2\\le i\\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2048",
"style": "rule"
} | {
"index": "33896",
"split": "train"
} |
math | [
{
"content": "Given the numbers \\(-2, -1, 0, 1, 2\\), arrange them in some order. Compute the difference between the largest and smallest possible values that can be obtained using the iterative average procedure. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.125",
"style": "rule"
} | {
"index": "33897",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of\n\\[\\frac{4}{x} + \\frac{9}{y} + \\frac{16}{z}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "33898",
"split": "train"
} |
math | [
{
"content": "Let's call a natural number \"remarkable\" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit \"remarkable\" numbers exist? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1800",
"style": "rule"
} | {
"index": "33899",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.