messages listlengths 1 1 | dataset listlengths 1 1 | ground_truth listlengths 1 1 | quality int64 0 10 ⌀ | id stringlengths 12 40 |
|---|---|---|---|---|
[
{
"content": "Find the largest prime factor of $11236$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"53"
] | 10 | deepscaler-gt_200_0 |
[
{
"content": "In triangle \\( A B C \\) with side \\( A C = 8 \\), a bisector \\( B L \\) is drawn. It is known that the areas of triangles \\( A B L \\) and \\( B L C \\) are in the ratio \\( 3: 1 \\). Find the bisector \\( B L \\), for which the height dropped from vertex \\( B \\) to the base \\( A C \\) wil... | [
"math"
] | [
"3\\sqrt{2}"
] | 10 | deepscaler-gt_201_0 |
[
{
"content": "Find the ordered pair $(a,b)$ of integers such that\n\\[\\sqrt{9 - 8 \\sin 50^\\circ} = a + b \\csc 50^\\circ.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"(3,-1)"
] | 10 | deepscaler-gt_202_0 |
[
{
"content": "A jar contains $29\\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user... | [
"math"
] | [
"14\\frac{6}{7}"
] | 10 | deepscaler-gt_203_0 |
[
{
"content": "Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\\angle A P B=60^{\\circ}, \\angle B Q C=36^{\\circ}$, and $\\angle C O A=72^{\\circ}$. ... | [
"math"
] | [
"90"
] | 10 | deepscaler-gt_204_0 |
[
{
"content": "A merchant placed on display some dresses, each with a marked price. He then posted a sign \"$1/3$ off on these dresses.\" The cost of the dresses was $3/4$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was: Let's think step by step and output the fina... | [
"math"
] | [
"\\frac{1}{2}"
] | 10 | deepscaler-gt_205_0 |
[
{
"content": "Find the number of subsets of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ that are subsets of neither $\\{1, 2, 3, 4, 5\\}$ nor $\\{4, 5, 6, 7, 8\\}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"196"
] | 10 | deepscaler-gt_206_0 |
[
{
"content": "Eight points are spaced around at intervals of one unit around a $2 \\times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?\n[asy]\nsize((50));\ndot((5,0));\ndot((5,5));\ndot((0,5));\ndot((-5,5));\ndot((-5,0));\ndot((... | [
"math"
] | [
"\\frac{2}{7}"
] | 10 | deepscaler-gt_207_0 |
[
{
"content": "Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\\triangle PQR$. Let's think step by step and output the final answer within \\boxed{}.",
"role"... | [
"math"
] | [
"198"
] | 10 | deepscaler-gt_208_0 |
[
{
"content": "The graph of an equation \\[\\sqrt{(x-3)^2 + (y+4)^2} + \\sqrt{(x+5)^2 + (y-8)^2} = 20.\\]is an ellipse. What is the distance between its foci? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"4\\sqrt{13}"
] | 10 | deepscaler-gt_209_0 |
[
{
"content": "Three friends are driving cars on a road in the same direction. At a certain moment, they are positioned relative to each other as follows: Andrews is at a certain distance behind Brooks, and Carter is at a distance twice the distance from Andrews to Brooks, ahead of Brooks. Each driver is traveli... | [
"math"
] | [
"6.666666666666667"
] | 10 | deepscaler-gt_210_0 |
[
{
"content": "Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\\overline{QR}$ and $\\overline{RS},$ respectively, so that $\\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\\overline{PM}.$ The length of a side of this smalle... | [
"math"
] | [
"12"
] | 10 | deepscaler-gt_211_0 |
[
{
"content": "Use the method of random simulation experiments to estimate the probability of having exactly two days of rain in three days: First, use a calculator to generate random integers between \\\\(0\\\\) and \\\\(9\\\\), with \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(4\\\\) representing rain, and \\\\... | [
"math"
] | [
"0.25"
] | 10 | deepscaler-gt_212_0 |
[
{
"content": "An ellipse has a focus at coordinates $\\left(0,-\\sqrt {2}\\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"-\\dfrac{2}{5}"
] | 10 | deepscaler-gt_213_0 |
[
{
"content": "How many solutions does the system have: $ \\{\\begin{matrix}&(3x+2y) *(\\frac{3}{x}+\\frac{1}{y})=2 & x^2+y^2\\leq 2012 \\end{matrix} $ where $ x,y $ are non-zero integers Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"102"
] | 10 | deepscaler-gt_214_0 |
[
{
"content": "If two lines \\( l \\) and \\( m \\) have equations \\( y = -2x + 8 \\), and \\( y = -3x + 9 \\), what is the probability that a point randomly selected in the 1st quadrant and below \\( l \\) will fall between \\( l \\) and \\( m \\)? Express your answer as a decimal to the nearest hundredth. Let... | [
"math"
] | [
"0.16"
] | 10 | deepscaler-gt_215_0 |
[
{
"content": "The Antarctican language has an alphabet of just 16 letters. Interestingly, every word in the language has exactly 3 letters, and it is known that no word's first letter equals any word's last letter (for instance, if the alphabet were $\\{a, b\\}$ then $a a b$ and aaa could not both be words in t... | [
"math"
] | [
"1024"
] | 10 | deepscaler-gt_216_0 |
[
{
"content": "Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"A=B"
] | 10 | deepscaler-gt_217_0 |
[
{
"content": "At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current ... | [
"math"
] | [
"2.5"
] | 10 | deepscaler-gt_218_0 |
[
{
"content": "Given the ellipse $$C: \\frac {x^{2}}{4}+ \\frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\\overrightarrow {OM}$$ and... | [
"math"
] | [
"\\sqrt{2}"
] | 10 | deepscaler-gt_219_0 |
[
{
"content": "Given the line $x+2y=a$ intersects the circle $x^2+y^2=4$ at points $A$ and $B$, and $|\\vec{OA}+ \\vec{OB}|=|\\vec{OA}- \\vec{OB}|$, where $O$ is the origin, determine the value of the real number $a$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | [
"math"
] | [
"-\\sqrt{10}"
] | 10 | deepscaler-gt_220_0 |
[
{
"content": "Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"45^\\circ"
] | 10 | deepscaler-gt_221_0 |
[
{
"content": "Let \\(\\triangle A B C\\) be a right triangle with right angle \\(C\\). Let \\(I\\) be the incenter of \\(A B C\\), and let \\(M\\) lie on \\(A C\\) and \\(N\\) on \\(B C\\), respectively, such that \\(M, I, N\\) are collinear and \\(\\overline{M N}\\) is parallel to \\(A B\\). If \\(A B=36\\) an... | [
"math"
] | [
"252"
] | 10 | deepscaler-gt_222_0 |
[
{
"content": "Given \\( 0 \\leq m-n \\leq 1 \\) and \\( 2 \\leq m+n \\leq 4 \\), when \\( m - 2n \\) reaches its maximum value, what is the value of \\( 2019m + 2020n \\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"2019"
] | 10 | deepscaler-gt_223_0 |
[
{
"content": "Given the function $f(x)= \\sqrt{2}\\sin \\left( 2x- \\frac{\\pi}{4} \\right)$, where $x\\in\\mathbb{R}$, if the maximum and minimum values of $f(x)$ in the interval $\\left[ \\frac{\\pi}{8}, \\frac{3\\pi}{4} \\right]$ are $a$ and $b$ respectively, then the value of $a+b$ is ______. Let's think st... | [
"math"
] | [
"\\sqrt{2}-1"
] | 10 | deepscaler-gt_224_0 |
[
{
"content": "Given a fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \\perp l$, with the foot of the perpendicular being $Q$, and $\\left( \\overrightarrow{PC}+\\frac{1}{2}\\overrightarrow{PQ} \\right)\\cdot \\left( \\overrightarrow{PC}-\\frac{1}{2}\\overrightarrow{... | [
"math"
] | [
"12-4\\sqrt{3}"
] | 10 | deepscaler-gt_225_0 |
[
{
"content": "The Hangzhou Asian Games are underway, and table tennis, known as China's \"national sport,\" is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other ... | [
"math"
] | [
"\\frac{3}{4}"
] | 10 | deepscaler-gt_226_0 |
[
{
"content": "Given the function $f(x)=-\\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"-4"
] | 10 | deepscaler-gt_227_0 |
[
{
"content": "The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{8k-168}{k-12}"
] | 10 | deepscaler-gt_228_0 |
[
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, which form a geometric progression. Also, $(2a-c)\\cos B = b\\cos C$.\n\n(Ⅰ) Find the magnitude of angle $B$; \n(Ⅱ) Calculate $\\frac{1}{\\tan A} + \\frac{1}{\\tan C}$. Let's think step by step... | [
"math"
] | [
"\\frac{2 \\sqrt{3}}{3}"
] | 10 | deepscaler-gt_229_0 |
[
{
"content": "Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second ... | [
"math"
] | [
"28"
] | 10 | deepscaler-gt_230_0 |
[
{
"content": "A bag contains only blue balls and green balls. There are $6$ blue balls. If the probability of drawing a blue ball at random from this bag is $\\frac{1}{4}$, then the number of green balls in the bag is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | [
"math"
] | [
"18"
] | 10 | deepscaler-gt_231_0 |
[
{
"content": "Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.\n\nMasha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.\n\nExactly one of the six numbers... | [
"math"
] | [
"17"
] | 10 | deepscaler-gt_232_0 |
[
{
"content": "A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{1}{4}"
] | 10 | deepscaler-gt_233_0 |
[
{
"content": "Let X be a set containing 10 elements, and A, B be two disjoint subsets of X, containing 3 and 4 elements respectively. Calculate the number of subsets of X that contain neither A nor B. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"840"
] | 10 | deepscaler-gt_234_0 |
[
{
"content": "Given the word 'ARROW', find the probability that a random arrangement of its letters will have both R's next to each other. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{2}{5}"
] | 10 | deepscaler-gt_235_0 |
[
{
"content": "Point $P$ lies inside equilateral triangle $A B C$ so that $\\angle B P C=120^{\\circ}$ and $A P \\sqrt{2}=B P+C P$. $\\frac{A P}{A B}$ can be written as $\\frac{a \\sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\\operatorname{gcd}(a, c)=1$. Find $100 a+10 b... | [
"math"
] | [
"255"
] | 10 | deepscaler-gt_236_0 |
[
{
"content": "Let \\\\(x,y\\\\) satisfy the constraint conditions \\\\(\\begin{cases} & x+y-2\\geqslant 0 \\\\\\\\ & x-y+1\\geqslant 0 \\\\\\\\ & x\\leqslant 3 \\end{cases}\\\\). If the minimum value of \\\\(z=mx+y\\\\) is \\\\(-3\\\\), then the value of \\\\(m\\\\) is . Let's think step by step and output... | [
"math"
] | [
"- \\dfrac{2}{3}"
] | 10 | deepscaler-gt_237_0 |
[
{
"content": "Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{27}{5}"
] | 10 | deepscaler-gt_238_0 |
[
{
"content": "Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user... | [
"math"
] | [
"\\frac{5}{16}"
] | 10 | deepscaler-gt_239_0 |
[
{
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{5} + \\frac{y^2}{10} = 1,\\]\nso that its sides are parallel to the coordinate axes. Find the area of the square. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{40}{3}"
] | 10 | deepscaler-gt_240_0 |
[
{
"content": "What is the largest integer less than or equal to \\(\\sqrt[3]{(2010)^{3}+3 \\times(2010)^{2}+4 \\times 2010+1}\\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"2011"
] | 10 | deepscaler-gt_241_0 |
[
{
"content": "Given the cyclist encounters red lights at each of 4 intersections with probability $\\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters the first red light a... | [
"math"
] | [
"\\frac{4}{27}"
] | 10 | deepscaler-gt_242_0 |
[
{
"content": "Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number. Let's think step by step and output the final answer within \\boxed... | [
"math"
] | [
"\\frac{5}{9}"
] | 10 | deepscaler-gt_243_0 |
[
{
"content": "Let $\\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\\alpha)\\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"P(x)\\equiv 0"
] | 10 | deepscaler-gt_244_0 |
[
{
"content": "The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the remaining cube when the freshly-cut face is placed on a table? Let's think step by step and output the final answer within \\boxed{}.",... | [
"math"
] | [
"\\frac{2\\sqrt{3}}{3}"
] | 10 | deepscaler-gt_245_0 |
[
{
"content": "Given 5 points \\( A, B, C, D, E \\) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment? Let's think step by step and output the final answer within \\boxed{}.",
"rol... | [
"math"
] | [
"135"
] | 10 | deepscaler-gt_246_0 |
[
{
"content": "Find all real numbers \\( p \\) such that the cubic equation \\( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \\) has two roots that are natural numbers. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"76"
] | 10 | deepscaler-gt_247_0 |
[
{
"content": "For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"16"
] | 10 | deepscaler-gt_248_0 |
[
{
"content": "Calculate $1010101_2 + 110011_2$ and express your answer in base $10$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"136"
] | 10 | deepscaler-gt_249_0 |
[
{
"content": "A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\\begin{cases} x=2-\\frac{1}{2}t \\\\ y=-1+\\frac{1}{2}t \\end{cases} (t\\text{ is the parameter})$ has a length of $\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | [
"math"
] | [
"\\sqrt{14}"
] | 10 | deepscaler-gt_250_0 |
[
{
"content": "What is the sum of the greatest common divisor of $45$ and $4410$ and the least common multiple of $45$ and $4410$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"4455"
] | 10 | deepscaler-gt_251_0 |
[
{
"content": "Let $b_1, b_2, \\dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \\geq 1$. Find \n\\[\n\\sum_{n=1}^\\infty \\frac{b_n}{3^{n+1}}.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{2}{5}"
] | 10 | deepscaler-gt_252_0 |
[
{
"content": "Given $6$ cards labeled $1$, $2$, $3$, $4$, $5$, and $6$ are drawn without replacement, calculate the probability that the product of the numbers of the $2$ cards is a multiple of $4$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{2}{5}"
] | 10 | deepscaler-gt_253_0 |
[
{
"content": "Equilateral $\\triangle ABC$ has side length $\\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\\triangle ABC$, with $BD_1 = BD_2 = \\sqrt{11}$. Find $\\sum_{k=1}^4(CE_k)^2$. Let's think step by step and output the final answer with... | [
"math"
] | [
"677"
] | 10 | deepscaler-gt_254_0 |
[
{
"content": "Let \\( a, b, c \\) be real numbers satisfying \\( 9a^2 + 4b^2 + 36c^2 = 4 \\). Find the minimum value of \\( 3a + 6b + 12c \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"-2\\sqrt{14}"
] | 10 | deepscaler-gt_255_0 |
[
{
"content": "Given that the discrete random variable \\\\(\\xi\\\\) follows a normal distribution \\\\(N \\sim (2,1)\\\\), and \\\\(P(\\xi < 3) = 0.968\\\\), then \\\\(P(1 < \\xi < 3) =\\\\) \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"0.936"
] | 10 | deepscaler-gt_256_0 |
[
{
"content": "Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"671"
] | 10 | deepscaler-gt_257_0 |
[
{
"content": "Let $x_1,$ $x_2,$ $x_3,$ $x_4$ be the roots of the polynomial $f(x) = x^4 - x^3 + x^2 + 1$. Define $g(x) = x^2 - 3$. Find the product:\n\\[ g(x_1) g(x_2) g(x_3) g(x_4). \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"142"
] | 10 | deepscaler-gt_258_0 |
[
{
"content": "Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"50"
] | 10 | deepscaler-gt_259_0 |
[
{
"content": "Given the curve E with the polar coordinate equation 4(ρ^2^-4)sin^2^θ=(16-ρ^2)cos^2^θ, establish a rectangular coordinate system with the non-negative semi-axis of the polar axis as the x-axis and the pole O as the coordinate origin.\n(1) Write the rectangular coordinate equation of the curve E;\n... | [
"math"
] | [
"\\sqrt{10}"
] | 10 | deepscaler-gt_260_0 |
[
{
"content": "Given the function \\( f(x) = 5(x+1)^{2} + \\frac{a}{(x+1)^{5}} \\) for \\( a > 0 \\), find the minimum value of \\( a \\) such that \\( f(x) \\geqslant 24 \\) when \\( x \\geqslant 0 \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"2 \\sqrt{\\left(\\frac{24}{7}\\right)^7}"
] | 10 | deepscaler-gt_261_0 |
[
{
"content": "In the $x-y$ plane, draw a circle of radius 2 centered at $(0,0)$. Color the circle red above the line $y=1$, color the circle blue below the line $y=-1$, and color the rest of the circle white. Now consider an arbitrary straight line at distance 1 from the circle. We color each point $P$ of the l... | [
"math"
] | [
"\\frac{2}{3}"
] | 10 | deepscaler-gt_262_0 |
[
{
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"1286"
] | 10 | deepscaler-gt_263_0 |
[
{
"content": "Suppose three hoses, X, Y, and Z, are used to fill a pool. Hoses X and Y together take 3 hours, while hose Y working alone takes 9 hours. Hoses X and Z together take 4 hours to fill the same pool. All three hoses working together can fill the pool in 2.5 hours. How long does it take for hose Z wor... | [
"math"
] | [
"15"
] | 10 | deepscaler-gt_264_0 |
[
{
"content": "Let \\( w \\) be a complex number such that \\( |w - 3 + 2i| = 4 \\). Find the minimum value of\n\\[\n|w + 1 + 2i|^2 + |w - 7 - 2i|^2.\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"48"
] | 10 | deepscaler-gt_265_0 |
[
{
"content": "Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$ directly above $O$.... | [
"math"
] | [
"30\\sqrt{11}"
] | 10 | deepscaler-gt_266_0 |
[
{
"content": "The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find\n\\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"8"
] | 10 | deepscaler-gt_267_0 |
[
{
"content": "The numbers \\( x \\) and \\( y \\) are such that the equalities \\( \\operatorname{ctg} x - \\operatorname{ctg} y = 2 \\) and \\( 5 \\sin (2x - 2y) = \\sin 2x \\sin 2y \\) hold. Find \\( \\operatorname{tg} x \\operatorname{tg} y \\). Let's think step by step and output the final answer within \\b... | [
"math"
] | [
"-\\frac{6}{5}"
] | 10 | deepscaler-gt_268_0 |
[
{
"content": "Let $ABC$ be an acute triangle with incenter $I$ ; ray $AI$ meets the circumcircle $\\Omega$ of $ABC$ at $M \\neq A$ . Suppose $T$ lies on line $BC$ such that $\\angle MIT=90^{\\circ}$ .\nLet $K$ be the foot of the altitude from $I$ to $\\overline{TM}$ . Given that $\\sin B = ... | [
"math"
] | [
"128"
] | 10 | deepscaler-gt_269_0 |
[
{
"content": "Let $A B C D$ be a rectangle, and let $E$ and $F$ be points on segment $A B$ such that $A E=E F=F B$. If $C E$ intersects the line $A D$ at $P$, and $P F$ intersects $B C$ at $Q$, determine the ratio of $B Q$ to $C Q$. Let's think step by step and output the final answer within \\boxed{}.",
"r... | [
"math"
] | [
"\\frac{1}{3}"
] | 10 | deepscaler-gt_270_0 |
[
{
"content": "The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}... | [
"math"
] | [
"66"
] | 10 | deepscaler-gt_271_0 |
[
{
"content": "The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,\n\\[(x+y)^r=x^r+rx^{r-1}y^1+\\frac{r(r-1)}2x^{r-2}y^2+\\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\\cdots\\]What are the first three digits to the right of the decimal point i... | [
"math"
] | [
"428"
] | 10 | deepscaler-gt_272_0 |
[
{
"content": "In the Cartesian coordinate system $xoy$, point $P(0, \\sqrt{3})$ is given. The parametric equation of curve $C$ is $\\begin{cases} x = \\sqrt{2} \\cos \\varphi \\\\ y = 2 \\sin \\varphi \\end{cases}$ (where $\\varphi$ is the parameter). A polar coordinate system is established with the origin as ... | [
"math"
] | [
"\\sqrt{14}"
] | 10 | deepscaler-gt_273_0 |
[
{
"content": "A dice is repeatedly rolled, and the upward-facing number is recorded for each roll. The rolling stops once three different numbers are recorded. If the sequence stops exactly after five rolls, calculate the total number of distinct recording sequences for these five numbers. Let's think step by s... | [
"math"
] | [
"840"
] | 10 | deepscaler-gt_274_0 |
[
{
"content": "Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more th... | [
"math"
] | [
"351"
] | 10 | deepscaler-gt_275_0 |
[
{
"content": "Given that $\\angle A$ and $\\angle B$ are within the interval $\\left(0, \\frac{\\pi}{2}\\right)$, and that $\\frac{\\sin A}{\\sin B} = \\sin (A+B)$, find the maximum value of $\\tan A$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"4/3"
] | 10 | deepscaler-gt_276_0 |
[
{
"content": "For how many positive integers $x$ is $\\log_{10}(x-40) + \\log_{10}(60-x) < 2$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"18"
] | 10 | deepscaler-gt_277_0 |
[
{
"content": "In the diagram, \\( AB \\) and \\( CD \\) intersect at \\( E \\). If \\(\\triangle BCE\\) is equilateral and \\(\\triangle ADE\\) is a right-angled triangle, what is the value of \\( x \\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"30"
] | 10 | deepscaler-gt_278_0 |
[
{
"content": "Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"2\\sqrt{2}"
] | 10 | deepscaler-gt_279_0 |
[
{
"content": "$ABCDEFGH$ is a cube. Find $\\sin \\angle BAE$, where $E$ is the top vertex directly above $A$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{1}{\\sqrt{2}}"
] | 10 | deepscaler-gt_280_0 |
[
{
"content": "In quadrilateral \\(ABCD\\), we have \\(AB=5\\), \\(BC=6\\), \\(CD=5\\), \\(DA=4\\), and \\(\\angle ABC=90^\\circ\\). Let \\(AC\\) and \\(BD\\) meet at \\(E\\). Compute \\(\\frac{BE}{ED}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\sqrt{3}"
] | 10 | deepscaler-gt_281_0 |
[
{
"content": "Given an ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \\dfrac{\\sqrt{2}}{2}$, the length of the minor axis is $2$. \n$(1)$ Find the equation of the ellipse; \n\n$(2)$ Point $A$ is a movi... | [
"math"
] | [
"\\sqrt{2}"
] | 10 | deepscaler-gt_282_0 |
[
{
"content": "Given that when 81849, 106392, and 124374 are divided by an integer \\( n \\), the remainders are equal. If \\( a \\) is the maximum value of \\( n \\), find \\( a \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"243"
] | 10 | deepscaler-gt_283_0 |
[
{
"content": "The equations of the asymptotes of the hyperbola \\\\( \\frac {x^{2}}{3}- \\frac {y^{2}}{6}=1 \\\\) are \\_\\_\\_\\_\\_\\_, and its eccentricity is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\sqrt {3}"
] | 10 | deepscaler-gt_284_0 |
[
{
"content": "What is the value of $\\frac{11! - 10!}{9!}$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"100"
] | 10 | deepscaler-gt_285_0 |
[
{
"content": "Compute the number of real solutions $(x, y, z, w)$ to the system of equations: $$\\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\\\ y & =w+x+w x y & w & =y+z+y z w \\end{array}$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"5"
] | 10 | deepscaler-gt_286_0 |
[
{
"content": "When $\\frac{1}{909}$ is expressed as a decimal, what is the sum of the first 30 digits after the decimal point? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"14"
] | 10 | deepscaler-gt_287_0 |
[
{
"content": "$\\triangle ABC$ and $\\triangle DBC$ share $BC$. $AB = 5\\ \\text{cm}$, $AC = 12\\ \\text{cm}$, $DC = 8\\ \\text{cm}$, and $BD = 20\\ \\text{cm}$. What is the least possible integral number of centimeters in $BC$?\n\n[asy]\nsize(100); import graph; currentpen = fontsize(10pt);\npair B = (0,0), C ... | [
"math"
] | [
"13"
] | 10 | deepscaler-gt_288_0 |
[
{
"content": "For each pair of real numbers \\((x, y)\\) with \\(0 \\leq x \\leq y \\leq 1\\), consider the set\n\\[ A = \\{ x y, x y - x - y + 1, x + y - 2 x y \\}. \\]\n\nLet the maximum value of the elements in set \\(A\\) be \\(M(x, y)\\). Find the minimum value of \\(M(x, y)\\). Let's think step by step an... | [
"math"
] | [
"4/9"
] | 10 | deepscaler-gt_289_0 |
[
{
"content": "For the largest natural \\( m \\), when will the product \\( m! \\cdot 2022! \\) be a factorial of a natural number? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"2022! - 1"
] | 10 | deepscaler-gt_290_0 |
[
{
"content": "Given the function $f(x)=\\frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))$ where $n \\in \\mathbb{N}^{*}$, vector $\\overrightarrow{a}=(0,1)$, and $\\theta_{n}$ is the angle between vector $\\overrightarrow{OA}_{n}$ and $\\overrightarrow{a}$. Compute the value of $\\frac{\... | [
"math"
] | [
"\\frac{2016}{2017}"
] | 10 | deepscaler-gt_291_0 |
[
{
"content": "Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"21"
] | 10 | deepscaler-gt_292_0 |
[
{
"content": "Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ whe... | [
"math"
] | [
"489"
] | 10 | deepscaler-gt_293_0 |
[
{
"content": "What is the least positive integer that satisfies the following conditions?\n\na) When divided by 2, the remainder is 1.\n\nb) When divided by 3, the remainder is 2.\n\nc) When divided by 4, the remainder is 3.\n\nd) When divided by 5, the remainder is 4. Let's think step by step and output the fi... | [
"math"
] | [
"59"
] | 10 | deepscaler-gt_294_0 |
[
{
"content": "The number halfway between $\\frac{1}{6}$ and $\\frac{1}{4}$ is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"\\frac{5}{24}"
] | 10 | deepscaler-gt_295_0 |
[
{
"content": "How many positive integers less than $555$ are either a perfect cube or a perfect square? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"29"
] | 10 | deepscaler-gt_296_0 |
[
{
"content": "180 grams of 920 purity gold was alloyed with 100 grams of 752 purity gold. What is the purity of the resulting alloy? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | [
"math"
] | [
"860"
] | 10 | deepscaler-gt_297_0 |
[
{
"content": "A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\\{A, B, C\\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? Let's think step ... | [
"math"
] | [
"385"
] | 10 | deepscaler-gt_298_0 |
[
{
"content": "In the diagram, if $\\angle PQR = 48^\\circ$, what is the measure of $\\angle PMN$? [asy]\n\nsize(6cm);\n\npair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * dir(-24);\n\nlabel(\"$M$\", m, N); label(\"$R$\", r, N); label(\"$P$\", p... | [
"math"
] | [
"66^\\circ"
] | 10 | deepscaler-gt_299_0 |
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