data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A very bizarre weighted coin comes up heads with probability $\\frac12$, tails with probability $\\frac13$, and rests on its edge with probability $\\frac16$. If it comes up heads, I win 1 dollar. If it comes up tails, I win 3 dollars. But if it lands on its edge, I lose 5 dollars. What is the... | {
"ground_truth": "\\$\\dfrac23 \\approx \\$0.67",
"style": "rule"
} | {
"index": "4900",
"split": "train"
} |
math | [
{
"content": "A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swi... | {
"ground_truth": "550",
"style": "rule"
} | {
"index": "4901",
"split": "train"
} |
math | [
{
"content": "Suppose that all four of the numbers \\[2 - \\sqrt{5}, \\;4+\\sqrt{10}, \\;14 - 2\\sqrt{7}, \\;-\\sqrt{2}\\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "4902",
"split": "train"
} |
math | [
{
"content": "Compute $35^{1723} - 16^{1723}$ modulo 6. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "4903",
"split": "train"
} |
math | [
{
"content": "Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\\text{cm}$ . Another line intersects the circles at $E,F$ , res... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4904",
"split": "train"
} |
math | [
{
"content": "Suppose $A B C D$ is a convex quadrilateral with $\\angle A B D=105^{\\circ}, \\angle A D B=15^{\\circ}, A C=7$, and $B C=C D=5$. Compute the sum of all possible values of $B D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{291}",
"style": "rule"
} | {
"index": "4905",
"split": "train"
} |
math | [
{
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\cdot \\sin ^{4} x \\cos ^{4} x \\, dx\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{8}",
"style": "rule"
} | {
"index": "4906",
"split": "train"
} |
math | [
{
"content": "**The first term of a sequence is $2089$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $2089^{\\text{th}}$ term of the sequence?** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "4907",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number that cannot be written in the form \\(\\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\\), where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are natural numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4908",
"split": "train"
} |
math | [
{
"content": "Given that the radius of the base of a cone is $2$ and the area of the unfolded side of the cone is $8\\pi$, find the volume of the inscribed sphere in the cone. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{32\\sqrt{3}}{27}\\pi",
"style": "rule"
} | {
"index": "4909",
"split": "train"
} |
math | [
{
"content": "Cat and Claire are having a conversation about Cat's favorite number. Cat says, \"My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!\" \n\nClaire asks, \"If you picked a digit of your favor... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4910",
"split": "train"
} |
math | [
{
"content": "Given \\( x_{i} \\in \\{-1,1\\}, i=1,2,\\cdots,2021 \\), and \\( x_{1}+x_{2}+\\cdots+x_{k} \\geq 0 \\) for \\( k=1,2,\\cdots,2020 \\), with \\( x_{1}+x_{2}+\\cdots+x_{2021}=-1 \\). How many ordered arrays \\( (x_{1}, x_{2}, \\cdots, x_{2021}) \\) are there? Please output the final answer within \\... | {
"ground_truth": "\\frac{1}{1011} \\binom{2020}{1010}",
"style": "rule"
} | {
"index": "4911",
"split": "train"
} |
math | [
{
"content": "**A person rolls seven standard, six-sided dice. What is the probability that there is at least one pair but no three dice show the same value?** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{315}{972}",
"style": "rule"
} | {
"index": "4912",
"split": "train"
} |
math | [
{
"content": "The circles \\(O_{1}\\) and \\(O_{2}\\) touch the circle \\(O_{3}\\) with radius 13 at points \\(A\\) and \\(B\\) respectively and pass through its center \\(O\\). These circles intersect again at point \\(C\\). It is known that \\(OC = 12\\). Find \\(AB\\). Please output the final answer within \... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4913",
"split": "train"
} |
math | [
{
"content": "The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "4914",
"split": "train"
} |
math | [
{
"content": "There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts and snowy areas. In a ... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4915",
"split": "train"
} |
math | [
{
"content": "Ponchik was having a snack at a roadside café when a bus passed by. Three pastries after the bus, a motorcycle passed by Ponchik, and three pastries after that, a car passed by. Syrupchik, who was snacking at another café on the same road, saw them in a different order: first the bus, after three ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "4916",
"split": "train"
} |
math | [
{
"content": "A line that passes through the two foci of the hyperbola $$\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola. Please output the final answer within... | {
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}",
"style": "rule"
} | {
"index": "4917",
"split": "train"
} |
math | [
{
"content": "For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\\{m, m+4, m+10, n+1, n+2, 2n\\}$ are equal to $n$. What is $m+n$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "4918",
"split": "train"
} |
math | [
{
"content": "What is the greatest common divisor of $121^2 + 233^2 + 345^2$ and $120^2 + 232^2 + 346^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "4919",
"split": "train"
} |
math | [
{
"content": "Determine the value of the expression\n\\[\\log_3 (64 + \\log_3 (64 + \\log_3 (64 + \\cdots))),\\]\nassuming it is positive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.8",
"style": "rule"
} | {
"index": "4920",
"split": "train"
} |
math | [
{
"content": "In a speech competition, judges will score participants based on the content, delivery, and effectiveness of the speech, with weights of $4:4:2$ respectively. If a student receives scores of $91$, $94$, and $90$ in these three aspects, then the student's total score is ______ points. Please output... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "4921",
"split": "train"
} |
math | [
{
"content": "Given that line $MN$ passes through the left focus $F$ of the ellipse $\\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\\frac{|PQ|^{2}}{|M... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "4922",
"split": "train"
} |
math | [
{
"content": "What is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "4923",
"split": "train"
} |
math | [
{
"content": "A man has part of $4500 invested at 4% and the rest at 6%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.8\\%",
"style": "rule"
} | {
"index": "4924",
"split": "train"
} |
math | [
{
"content": "A target is a triangle divided by three sets of parallel lines into 100 equal equilateral triangles with unit sides. A sniper shoots at the target. He aims at a triangle and hits either it or one of the adjacent triangles sharing a side. He can see the results of his shots and can choose when to s... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "4925",
"split": "train"
} |
math | [
{
"content": "A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11.0",
"style": "rule"
} | {
"index": "4926",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of\n\\[\\frac{2}{a} + \\frac{4}{b}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "4927",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{2}-2x$ where $x \\in [-2,a]$. Find the minimum value of $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "4928",
"split": "train"
} |
math | [
{
"content": "The value of \\( \\frac{1}{2} + \\frac{2}{4} + \\frac{4}{8} + \\frac{8}{16} \\) is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "4929",
"split": "train"
} |
math | [
{
"content": "The sum of the first $2011$ terms of a geometric sequence is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "542",
"style": "rule"
} | {
"index": "4930",
"split": "train"
} |
math | [
{
"content": "Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{32768}",
"style": "rule"
} | {
"index": "4931",
"split": "train"
} |
math | [
{
"content": "The line $y = 2x + 7$ is to be parameterized using vectors. Which of the following options are valid parameterizations?\n\n(A) $\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}$\n\n(B) $\\begin{pmatrix} x \\\\ y \\end... | {
"ground_truth": "\\text{B,E}",
"style": "rule"
} | {
"index": "4932",
"split": "train"
} |
math | [
{
"content": "Find the largest value of $t$ such that \\[\\frac{13t^2 - 34t + 12}{3t - 2 } + 5t = 6t - 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "4933",
"split": "train"
} |
math | [
{
"content": "The regular octagon \\( A B C D E F G H \\) is inscribed in a circle. Points \\( P \\) and \\( Q \\) are on the circle, with \\( P \\) between \\( C \\) and \\( D \\), such that \\( A P Q \\) is an equilateral triangle. It is possible to inscribe a regular \\( n \\)-sided polygon, one of whose sid... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "4934",
"split": "train"
} |
math | [
{
"content": "The volume of a sphere is $72\\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bottom) of the cylinder.... | {
"ground_truth": "90\\pi \\sqrt[3]{4}",
"style": "rule"
} | {
"index": "4935",
"split": "train"
} |
math | [
{
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 \\angle D$ and $\\angle BAC = k \\pi$ in radians, then find $k$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{5}{11}",
"style": "rule"
} | {
"index": "4936",
"split": "train"
} |
math | [
{
"content": "A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-3k-10\\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "4937",
"split": "train"
} |
math | [
{
"content": "Given that $\\theta$ is an angle in the third quadrant, if $\\sin^{4}{\\theta} + \\cos^{4}{\\theta} = \\frac{5}{9}$, find the value of $\\sin{2\\theta}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "4938",
"split": "train"
} |
math | [
{
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\\le x\\le 4$ and $1\\le y\\le 4$?\n$\\text{(A) } 496\\quad \\text{(B) } 500\\quad \\text{(C) } 512\\quad \\text{(D) } 516\\quad \\text{(E) } 560$ Please ... | {
"ground_truth": "516",
"style": "rule"
} | {
"index": "4939",
"split": "train"
} |
math | [
{
"content": "A function $f$ is defined by $f(z) = i\\overline{z}$, where $i^2 = -1$ and $\\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "4940",
"split": "train"
} |
math | [
{
"content": "A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "4941",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\begin{cases}f(x+1), & x < 4 \\\\ 2^{x}, & x\\geq 4\\end{cases}$, evaluate $f(2+\\log_{2}3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "4942",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, two side lengths are $2$ and $3$, and the cosine value of the included angle is $\\frac{1}{3}$. Find the radius of the circumscribed circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9\\sqrt{2}}{8}",
"style": "rule"
} | {
"index": "4943",
"split": "train"
} |
math | [
{
"content": "Given an isosceles triangle \\( \\triangle ABC \\) with base angles \\( \\angle ABC = \\angle ACB = 50^\\circ \\), points \\( D \\) and \\( E \\) lie on \\( BC \\) and \\( AC \\) respectively. Lines \\( AD \\) and \\( BE \\) intersect at point \\( P \\). Given \\( \\angle ABE = 30^\\circ \\) and \... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "4944",
"split": "train"
} |
math | [
{
"content": "The three different points \\(A(x_1, y_1)\\), \\(B\\left(4, \\frac{9}{5}\\right)\\), and \\(C(x_2, y_2)\\) on the ellipse \\(\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1\\), along with the focus \\(F(4,0)\\) have distances that form an arithmetic sequence. If the perpendicular bisector of line segment... | {
"ground_truth": "5/4",
"style": "rule"
} | {
"index": "4945",
"split": "train"
} |
math | [
{
"content": "Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{223}{286}",
"style": "rule"
} | {
"index": "4946",
"split": "train"
} |
math | [
{
"content": "Among the parts processed by a worker, on average, $4 \\%$ are non-standard. Find the probability that among the 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the considered sample of 30 parts, and what is its probability? Ple... | {
"ground_truth": "0.305",
"style": "rule"
} | {
"index": "4947",
"split": "train"
} |
math | [
{
"content": "In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\\overline{BC}$, $\\overline{CD}$, and $\\overline{AD}$, respectively. Point $H$ is the midpoint of $\\overline{GE}$. What is the area of the shaded region? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "4948",
"split": "train"
} |
math | [
{
"content": "Given the graph of $y=\\sqrt{2}\\sin 3x$, calculate the horizontal shift required to obtain the graph of $y=\\sin 3x+\\cos 3x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\pi}{12}",
"style": "rule"
} | {
"index": "4949",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ in the following expressions: \n(1) $8x^3 = 27$; (2) $(x-2)^2 = 3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{3} + 2",
"style": "rule"
} | {
"index": "4950",
"split": "train"
} |
math | [
{
"content": "Vasya has:\na) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;\nb) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;\nc) a book of lyrical poems with a height of 40 cm, published by Vasya himself.\n\nVasya wants to arrange these books on... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "4951",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[x^2 + y^2 + \\frac{4}{(x + y)^2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "4952",
"split": "train"
} |
math | [
{
"content": "On Monday, I worked $t-6$ hours and earned $2t-5$ dollars per hour. My friend Andrew worked $2t-8$ hours but only earned $t-5$ dollars an hour. As it turns out, we earned the same amount. What is the value of $t$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4953",
"split": "train"
} |
math | [
{
"content": "Given that in triangle PQR, side PR = 6 cm and side PQ = 10 cm, point S is the midpoint of QR, and the length of the altitude from P to QR is 4 cm, calculate the length of QR. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "4954",
"split": "train"
} |
math | [
{
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$? Please output the... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4955",
"split": "train"
} |
math | [
{
"content": "Consider the equation $p = 15q^2 - 5$. Determine the value of $q$ when $p = 40$.\nA) $q = 1$\nB) $q = 2$\nC) $q = \\sqrt{3}$\nD) $q = \\sqrt{6}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "q = \\sqrt{3}",
"style": "rule"
} | {
"index": "4956",
"split": "train"
} |
math | [
{
"content": "Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & & B& B & A_6\\\\ & & & \\mathbf{4} & \\mathbf{1} & B_6\\\\& & + & A & \\mathbf{1} & \\mathbf{5_6}\\\\ \\cline{2-6} & & A & \\mathbf{1} & \\mathbf{5} & \\mathbf{... | {
"ground_truth": "1_6",
"style": "rule"
} | {
"index": "4957",
"split": "train"
} |
math | [
{
"content": "Toss a fair coin, with the probability of landing heads or tails being $\\frac {1}{2}$ each. Construct the sequence $\\{a_n\\}$ such that\n$$\na_n= \n\\begin{cases} \n1 & \\text{if the } n\\text{-th toss is heads,} \\\\ \n-1 & \\text{if the } n\\text{-th toss is tails.}\n\\end{cases}\n$$\nDefine $... | {
"ground_truth": "\\frac {13}{128}",
"style": "rule"
} | {
"index": "4958",
"split": "train"
} |
math | [
{
"content": "A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{16}",
"style": "rule"
} | {
"index": "4959",
"split": "train"
} |
math | [
{
"content": "Given that \\( O \\) is the circumcenter of \\( \\triangle ABC \\), and the equation\n\\[\n\\overrightarrow{A O} \\cdot \\overrightarrow{B C} + 2 \\overrightarrow{B O} \\cdot \\overrightarrow{C A} + 3 \\overrightarrow{C O} \\cdot \\overrightarrow{A B} = 0,\n\\]\nfind the minimum value of \\( \\fra... | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "4960",
"split": "train"
} |
math | [
{
"content": "A natural number greater than 1 is called \"good\" if it is exactly equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten \"good\" natural numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "182",
"style": "rule"
} | {
"index": "4961",
"split": "train"
} |
math | [
{
"content": "Given the number 105,000, express it in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.05\\times 10^{5}",
"style": "rule"
} | {
"index": "4962",
"split": "train"
} |
math | [
{
"content": "Let $O$ be the origin. A variable plane has a distance of 1 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the centroid of triangle $ABC.$ Find\n\\[\\frac{1}{p^2} + \\frac{1}{q^2} + \\frac{1}{r^2}.... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "4963",
"split": "train"
} |
math | [
{
"content": "Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the ... | {
"ground_truth": "408",
"style": "rule"
} | {
"index": "4964",
"split": "train"
} |
math | [
{
"content": "200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: \"I am a liar,\" t... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "4965",
"split": "train"
} |
math | [
{
"content": "We traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 hours, and the driver ... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "4966",
"split": "train"
} |
math | [
{
"content": "Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{2}{3}",
"style": "rule"
} | {
"index": "4967",
"split": "train"
} |
math | [
{
"content": "The sequence 1,000,000; 500,000; 250,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15625",
"style": "rule"
} | {
"index": "4968",
"split": "train"
} |
math | [
{
"content": "On a $3 \\times 3$ chessboard, each square contains a Chinese knight with $\\frac{1}{2}$ probability. What is the probability that there are two Chinese knights that can attack each other? (In Chinese chess, a Chinese knight can attack any piece which is two squares away from it in a particular di... | {
"ground_truth": "\\frac{79}{256}",
"style": "rule"
} | {
"index": "4969",
"split": "train"
} |
math | [
{
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "114",
"style": "rule"
} | {
"index": "4970",
"split": "train"
} |
math | [
{
"content": "Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "4971",
"split": "train"
} |
math | [
{
"content": "Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "7/27",
"style": "rule"
} | {
"index": "4972",
"split": "train"
} |
math | [
{
"content": "What is the largest value of $x$ such that the expression \\[\\dfrac{x+1}{8x^2-65x+8}\\] is not defined? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "4973",
"split": "train"
} |
math | [
{
"content": "Determine the area of the Crescent Gemini. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17\\pi}{4}",
"style": "rule"
} | {
"index": "4974",
"split": "train"
} |
math | [
{
"content": "Given the function $y=2\\sin \\left(x+ \\frac {\\pi}{6}\\right)\\cos \\left(x+ \\frac {\\pi}{6}\\right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\\sin 2x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "4975",
"split": "train"
} |
math | [
{
"content": "For how many real numbers $a$ does the quadratic equation $x^2 + ax + 12a = 0$ have only integer roots for $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4976",
"split": "train"
} |
math | [
{
"content": "In how many ways can 4 numbers be chosen from the first 1000 natural numbers \\( 1,2, \\ldots, 1000 \\) to form an increasing arithmetic progression? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "166167",
"style": "rule"
} | {
"index": "4977",
"split": "train"
} |
math | [
{
"content": "In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:\n(i) Less than $12$ employees have a more difficult work;\n(ii) At least $30$ employees take a higher salary.\nAssuming that an employee ei... | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "4978",
"split": "train"
} |
math | [
{
"content": "The expression $\\circ \\ 1\\ \\circ \\ 2 \\ \\circ 3 \\ \\circ \\dots \\circ \\ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are t... | {
"ground_truth": "1006",
"style": "rule"
} | {
"index": "4979",
"split": "train"
} |
math | [
{
"content": "Seven students count from 1 to 1000 as follows:\nAlice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.\nBarbara says all of the numbers that Alice doesn't say, except she also skips th... | {
"ground_truth": "365",
"style": "rule"
} | {
"index": "4980",
"split": "train"
} |
math | [
{
"content": "Let $Q$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $Q$ and the point $\\left(\\frac{1}{2}, \\frac{1}{4} \\right)$ is greater than or equal to $1$ can be writt... | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "4981",
"split": "train"
} |
math | [
{
"content": "Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4982",
"split": "train"
} |
math | [
{
"content": "A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "4983",
"split": "train"
} |
math | [
{
"content": "Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\\angle AXB$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5 \\sqrt{2}-3",
"style": "rule"
} | {
"index": "4984",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the first $n$ terms of an arithmetic sequence $\\{a\\_n\\}$ is $S\\_n$, with $a\\_2 = 4$ and $S\\_{10} = 110$, find the minimum value of $\\frac{S\\_n + 64}{a\\_n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{2}",
"style": "rule"
} | {
"index": "4985",
"split": "train"
} |
math | [
{
"content": "Let $m,n$ be natural numbers such that $\\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "4986",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{\\alpha}$, $\\overrightarrow{\\beta}$, $\\overrightarrow{\\gamma}$ satisfy $|\\overrightarrow{\\alpha}|=1$, $|\\overrightarrow{\\alpha}-\\overrightarrow{\\beta}|=|\\overrightarrow{\\beta}|$, $(\\overrightarrow{\\alpha}-\\overrightarrow{\\gamma}) \\cdot (\\overrighta... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "4987",
"split": "train"
} |
math | [
{
"content": "Given the function\n\\[ x(t)=5(t+1)^{2}+\\frac{a}{(t+1)^{5}}, \\]\nwhere \\( a \\) is a constant. Find the minimum value of \\( a \\) such that \\( x(t) \\geqslant 24 \\) for all \\( t \\geqslant 0 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{\\left(\\frac{24}{7}\\right)^7}",
"style": "rule"
} | {
"index": "4988",
"split": "train"
} |
math | [
{
"content": "Determine how many integers are in the list containing the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "871",
"style": "rule"
} | {
"index": "4989",
"split": "train"
} |
math | [
{
"content": "Eleven percent of what number is seventy-seven? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "700",
"style": "rule"
} | {
"index": "4990",
"split": "train"
} |
math | [
{
"content": "Simplify $(2-3z) - (3+4z)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1-7z",
"style": "rule"
} | {
"index": "4991",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $2\\sin 2C\\cdot\\cos C-\\sin 3C= \\sqrt {3}(1-\\cos C)$.\n(1) Find the measure of angle $C$;\n(2) If $AB=2$, and $\\sin C+\\sin (B-A)=2\\sin 2A$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {2 \\sqrt {3}}{3}",
"style": "rule"
} | {
"index": "4992",
"split": "train"
} |
math | [
{
"content": "Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "4993",
"split": "train"
} |
math | [
{
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "4994",
"split": "train"
} |
math | [
{
"content": "The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "4995",
"split": "train"
} |
math | [
{
"content": "A line that always passes through a fixed point is given by the equation $mx - ny - m = 0$, and it intersects with the parabola $y^2 = 4x$ at points $A$ and $B$. Find the number of different selections of distinct elements $m$ and $n$ from the set ${-3, -2, -1, 0, 1, 2, 3}$ such that $|AB| < 8$. P... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "4996",
"split": "train"
} |
math | [
{
"content": "What is the ratio of the volume of cone $A$ to the volume of cone $B$? Express your answer as a common fraction. [asy]\nsize(260);\ndraw(ellipse((0,0),14.8,6),S);\nlabel(\"Cone $A$\", (0,-6), S);\ndraw((0,28.3)--(0,0),dashed);\nlabel(\"$28.3$\",(0,14),SW);\ndraw((-14.8,0)--(0,28.3)--(14.8,0));\ndr... | {
"ground_truth": "\\frac{148}{283}",
"style": "rule"
} | {
"index": "4997",
"split": "train"
} |
math | [
{
"content": "At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances\nFin... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "4998",
"split": "train"
} |
math | [
{
"content": "Let $F_n$ be the Fibonacci sequence, that is, $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute\n\\[\\sum_{n=0}^{\\infty} \\frac{F_n}{10^n}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{10}{89}",
"style": "rule"
} | {
"index": "4999",
"split": "train"
} |
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