data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $W, S$ be as in problem 32. Let $A$ be the least positive integer such that an acute triangle with side lengths $S, A$, and $W$ exists. Find $A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 22549,
"solution": "There are two solutions to the alphametic in problem 32: $36 \\times 686=24696$ and $86 \\times 636=54696$. So $(W, S)$ may be $(3,2)$ or $(8,5)$. If $(W, S)=(3,2)$, then by problem (3) $A=3$, but then by problem $31 W=4$, a contradiction. So, $(W, S)$ must be $(8,... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $W X Y Z$ has $W X=4, W Z=3$, and $Z V=3$. The rectangle is curled without overlapping into a cylinder so that sides $W Z$ and $X Y$ touch each other. In other words, $W$ touches $X$ and $Z$ touches $Y$. The shortest distance from $W$ to $V$ through the inside of the ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 22550,
"solution": "When the cylinder is created, $W$ and $X$ touch and $Z$ and $Y$ touch. This means that $W Y$ is vertical and so is perpendicular to the plane of the circular base of the cylinder. This means that $\\triangle V Y W$ is right-angled at $Y$. By the Pythagorean Theorem... |
deepscaler | [
{
"role": "user",
"content": "The product of two consecutive integers is $20{,}412$. What is the sum of these two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "287"
} | {
"split": "train",
"index": 22551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the dividend to the divisor is 9:2, and the ratio of the divisor to the quotient is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 22552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(80 + 12\\sqrt{41})\\pi"
} | {
"split": "train",
"index": 22553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task c... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 22554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On May 19, 2022, Jinyan Airport in Dazhou was officially opened, covering a total area of 2940 acres, with an estimated investment of approximately 26.62 billion yuan. Express this amount in scientific notation. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "2.662 \\times 10^{9}"
} | {
"split": "train",
"index": 22555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{x} + \\frac{1}{y} = 3$ and $\\frac{1}{x} - \\frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{10}"
} | {
"split": "train",
"index": 22556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $D$ lies on $\\overline{BC}$ and $F$ lies on $\\overline{AB}.$ Let $\\overline{AD}$ and $\\overline{CF}$ intersect at $P.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, F, P;\n\nA = (1,4);\nB = (0,0);\nC = (6,0);\nD = interp(B,C,7/12);\nF = interp(A,B,5/14... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 22557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest multiple of $9$ which is smaller than $-70$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"split": "train",
"index": 22558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bending resistance of a beam with a rectangular cross-section is proportional to the product of its width and the square of its height. What should be the width of the beam section, cut from a round log with a diameter of \\( 15 \\sqrt{3} \\), to maximize its bending resist... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 22559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelogram with an acute angle of \\(60^{\\circ}\\). Find the ratio of the sides of the parallelogram if the ratio of the squares of the diagonals is \\(\\frac{1}{3}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 22560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\\overrightarrow {OA}\\cdot \\overrightarrow {OB}=m^{2}-2m$.\n\n1. Find the equation of the parabola $C$.\n2. Let $P$ be the point whe... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 22561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Circle $\\omega$ intersects $\\overline{AB}$ at $E$ and $B$, $\\overline{BC}$ at $B$ and $D$, and $\\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\\frac{DG}{EG} = \\frac{3}{4}$, length $DE=\\frac{a\\sqrt{b}}{c}$, w... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 22562,
"solution": "Call $DE=x$ and as a result $DF=EF=\\frac{x\\sqrt{2}}{2}, EG=\\frac{4x}{5}, GD=\\frac{3x}{5}$. Since $EFGD$ is cyclic we just need to get $DG$ and using LoS(for more detail see the $2$nd paragraph of Solution $2$) we get $AG=\\frac{5}{2}$ and using a similar argume... |
deepscaler | [
{
"role": "user",
"content": "Given Harry has 4 sisters and 6 brothers, and his sister Harriet has S sisters and B brothers, calculate the product of S and B. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 22563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually ma... | math | {
"style": "rule",
"ground_truth": "20.3"
} | {
"split": "train",
"index": 22564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum constant \\( k \\) such that for \\( x, y, z \\in \\mathbb{R}_+ \\), the following inequality holds:\n$$\n\\sum \\frac{x}{\\sqrt{y+z}} \\geqslant k \\sqrt{\\sum x},\n$$\n\nwhere \" \\( \\sum \\) \" denotes a cyclic sum. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}}"
} | {
"split": "train",
"index": 22565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:\n(1) If digits can be repeated, how many different five-digit even numbers can be formed?\n(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even integers are there between $\\frac{9}{2}$ and $\\frac{24}{1}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sector with a central angle of 120° and an area of $3\\pi$, which is used as the lateral surface area of a cone, find the surface area and volume of the cone. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}\\pi}{3}"
} | {
"split": "train",
"index": 22568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common factor of 68 and 92? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three different integers are randomly chosen from the set $$\\{ -6, -3, 0, 2, 5, 7 \\}$$. What is the probability that their sum is even? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{20}"
} | {
"split": "train",
"index": 22570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "X is the point (1994p, 7·1994p), where p is a prime, and O is the point (0, 0). How many triangles XYZ have Y and Z at lattice points, incenter at O, and YXZ as a right-angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 22571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the exact value of the expression\n\\[|\\pi - |\\pi - 7||.\\]Write your answer using only integers and $\\pi,$ without any absolute value signs. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 - 2 \\pi"
} | {
"split": "train",
"index": 22572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p>3$ be a prime and let $a_1,a_2,...,a_{\\frac{p-1}{2}}$ be a permutation of $1,2,...,\\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\\frac{p-1}{2}}$ if it for all $i,j\\in\\{1,2,...,\\frac{p-1}{2}\\}$ with $i\\not=j$ the res... | math | {
"style": "rule",
"ground_truth": "p \\geq 7"
} | {
"split": "train",
"index": 22573,
"solution": "\nGiven a prime number \\( p > 3 \\), we have a permutation \\( a_1, a_2, \\ldots, a_{\\frac{p-1}{2}} \\) of the set \\( \\{1, 2, \\ldots, \\frac{p-1}{2}\\} \\). The task is to determine for which primes \\( p \\) it is always possible to reconstruct the sequence \... |
deepscaler | [
{
"role": "user",
"content": "A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 22574,
"solution": "\nLet's analyze the problem step by step:\n\n1. **Define Variables:**\n - Let $t$ be the time it takes for the power boat to travel from dock $A$ to dock $B$.\n - Let $r$ be the speed of the river current (and also the speed of the raft, since it drifts with th... |
deepscaler | [
{
"role": "user",
"content": "A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\\vartriangle VAB$ .\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100\\sqrt{10}"
} | {
"split": "train",
"index": 22575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $\\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$ , as point $O$ moves along the triangle's sides. If the area of the triangle is $E$ , find the area of $\\Phi$ . Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2E"
} | {
"split": "train",
"index": 22576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line $x- \\sqrt {3}y+3=0$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{6}"
} | {
"split": "train",
"index": 22577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sophie has collected one of each of the first 25 U.S. state quarters based on the order in which the states joined the union. The following graph indicates the number of states that joined the union in each decade. Determine the fraction of Sophie's 25 coins that represent stat... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 22578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWe wrote the numbers from 1 to 2009 on a piece of paper. In the second step, we also wrote down twice each of these numbers on the paper, and then we erased the numbers that appeared twice. We repeat this process such that, in the $i$-th step, we write $i$ times each of the n... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 22579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained? Let's think step ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 22580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\) with bases \\(AB\\) and \\(CD\\), it holds that \\(|AD| = |CD|\\), \\(|AB| = 2|CD|\\), \\(|BC| = 24 \\text{ cm}\\), and \\(|AC| = 10 \\text{ cm}\\).\n\nCalculate the area of trapezoid \\(ABCD\\). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 22581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"split": "train",
"index": 22582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $C$ is the midpoint of $\\overline{AB}$, point $D$ is the midpoint of $\\overline{AC}$, point $E$ is the midpoint of $\\overline{AD}$, and point $F$ is the midpoint of $\\overline{AE}$. If $AF=3$, what is the number of units in the length of $\\overline{AB}$? Let's think ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 22584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. Determine the total number of different flags th... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 22585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"split": "train",
"index": 22586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two clos... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 22587,
"solution": "The volume of this pyramid can be found by the equation $V=\\frac{1}{3}bh$, where $b$ is the base and $h$ is the height. The base is easy, since it is a square and has area $15^2=225$.\n To find the height of the pyramid, the height of the four triangles is needed,... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\sqrt{2 -\\!\\sqrt{2 - \\!\\sqrt{2 - \\!\\sqrt{2 - \\cdots}}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 22588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $(2\\overrightarrow{a}+\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 22589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the system of equations \\begin{align*}\n3x+y&=a,\\\\\n2x+5y&=2a,\n\\end{align*} has a solution $(x,y)$ when $x=2$, compute $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"split": "train",
"index": 22590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\\frac{\\pi}{6}$, and the distance from origin to this line is $\\frac{\\sqrt{3}}{2}$.\n\n(1) Find the equation of the e... | math | {
"style": "rule",
"ground_truth": "k = \\frac{7}{6}"
} | {
"split": "train",
"index": 22591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 7! + 5 \\cdot 5! + 3 \\cdot 3! + 3!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35904"
} | {
"split": "train",
"index": 22592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{11}"
} | {
"split": "train",
"index": 22593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(BD\\) is perpendicular to \\(BC\\) and to \\(AD\\). If \\(AB = 52\\), \\(BC = 21\\), and \\(AD = 48\\), what is the length of \\(DC\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 22594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that... | math | {
"style": "rule",
"ground_truth": "5250\\pi"
} | {
"split": "train",
"index": 22595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six students are arranged into two rows, with 3 students per row, and the number of different arrangements must be calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 22596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a sequence $a_{i, j}$ of integers such that $a_{1, n}=n^{n}$ for $n \\geq 1$ and $a_{i, j}=a_{i-1, j}+a_{i-1, j+1}$ for all $i, j \\geq 1$. Find the last (decimal) digit of $a_{128,1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 22597,
"solution": "By applying the recursion multiple times, we find that $a_{1,1}=1, a_{2, n}=n^{n}+(n+1)^{n+1}$, and $a_{3, n}=n^{n}+2(n+1)^{n+1}+(n+2)^{n+2}$. At this point, we can conjecture and prove by induction that $a_{m, n}=\\sum_{k=0}^{m-1}\\binom{m-1}{k}(n+k)^{n+k}=\\sum_{... |
deepscaler | [
{
"role": "user",
"content": "Jasmine wishes to purchase some trading cards. She has $\\$7.50$ and the cards each cost $\\$0.85$, tax included. What is the most number of cards she can buy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 22598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest \\( n \\) so that the number of integers less than or equal to \\( n \\) and divisible by 3 equals the number divisible by 5 or 7 (or both). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 22599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( ABC \\) with \\( AB = 12 \\), \\( BC = 10 \\), and \\( \\angle ABC = 120^\\circ \\), find \\( R^2 \\), where \\( R \\) is the radius of the smallest circle that can contain this triangle. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 22600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the s... | math | {
"style": "rule",
"ground_truth": "5:4"
} | {
"split": "train",
"index": 22601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ ($0 < \\alpha < \\frac{\\pi}{2}$) passes through the point $(\\cos 2\\beta, 1+\\sin 3\\beta \\cos \\beta - \\cos 3\\beta \\sin \\beta)$, where $\\frac{\\pi}{2} < \\beta < \\pi$ and $\\beta \\neq \\frac{3\\pi}{4}$, calculate $\\alp... | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 22602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Alice $25$ minutes to clean her room. It takes Bob $\\frac{2}{5}$ of that amount of time to clean his room. How many minutes does it take Bob to clean his room? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 22603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a < b < c < d < e \\) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \\( e \\). Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "27.5"
} | {
"split": "train",
"index": 22604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 22605,
"solution": "1. **Understanding the Structure**: The sculpture is made up of 14 cubes, each with an edge of 1 meter. The cubes are arranged in a pyramid-like structure with three layers. The top layer has 1 cube, the middle layer has 4 cubes, and the bottom layer has 9 cubes.\n... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{AB} \\perp \\overrightarrow{AC}$, $|\\overrightarrow{AB}|= \\frac{1}{t}$, $|\\overrightarrow{AC}|=t$, and point $P$ is a point on the plane of $\\triangle ABC$ such that $\\overrightarrow{AP}= \\frac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|} + \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 22606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $5(\\cos a + \\cos b) + 4(\\cos a \\cos b + 1) = 0,$ then find all possible values of\n\\[\\tan \\frac{a}{2} \\tan \\frac{b}{2}.\\]Enter all the possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3,-3"
} | {
"split": "train",
"index": 22607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Appending three digits at the end of 2007, one obtains an integer \\(N\\) of seven digits. In order to get \\(N\\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "075"
} | {
"split": "train",
"index": 22608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $x(3x^2-2)-5(x^2-2x+7)$. Express your answer in the form $Ax^3+Bx^2+Cx+D.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3x^3-5x^2+8x-35"
} | {
"split": "train",
"index": 22609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^3 - 6x^2 - x + 3 = 0$ has three real roots $a$, $b$, $c$. Find $\\frac{1}{a^2} + \\frac{1}{b^2} + \\frac{1}{c^2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{9}"
} | {
"split": "train",
"index": 22610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \\cdot CD = AB \\cdot DA$. What is the largest possible value of $BD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\dfrac{425}{2}}"
} | {
"split": "train",
"index": 22611,
"solution": "1. **Assign Variables to Side Lengths**: Let $AB = a$, $BC = b$, $CD = c$, and $AD = d$.\n\n2. **Use the Law of Cosines**: Apply the Law of Cosines to triangles $\\triangle ABD$ and $\\triangle CBD$:\n \\[\n BD^2 = a^2 + d^2 - 2ad\\cos{\\angle BAD}\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "John drove continuously from 8:30 a.m. until 2:15 p.m. of the same day and covered a distance of 246 miles. What was his average speed in miles per hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42.78"
} | {
"split": "train",
"index": 22612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $(42 \\div (12 - 10 + 3))^{2} \\cdot 7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "493.92"
} | {
"split": "train",
"index": 22613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 22614,
"solution": "Let $E, F$, be the projections of $A, B$ on $C D$. A point whose projections lie on the sides must be contained in the square $A B F E$. Furthermore, the point must lie under the perpendicular to $A D$ at $A$ and the perpendicular to $B C$ at $B$, which have slopes... |
deepscaler | [
{
"role": "user",
"content": "If a computer executes the following program:\n1. Initial values: \\( x = 3 \\), \\( S = 0 \\).\n2. \\( x = x + 2 \\).\n3. \\( S = S + x \\).\n4. If \\( S \\geq 10000 \\), go to step 5; otherwise, go back to step 2.\n5. Print \\( x \\).\n6. Stop.\n\nThen the value printed at st... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 22615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\(a, b, c\\) are chosen such that \\(a < b < c\\). It is also known that the system of equations \\(2x + y = 2035\\) and \\(y = |x - a| + |x - b| + |x - c|\\) has exactly one solution. Find the minimum possible value of \\(c\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "1018"
} | {
"split": "train",
"index": 22616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all team... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 22617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number divisible by integers 1 through 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 22618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $E$-shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that the endpoints of the rays all lie on the segment, the segment is perpendicular to all three rays, both endpoints of the s... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22619,
"solution": "Define a $C$-shape to be an $E$-shape without the middle ray. Then, an $E$-shape consists of a ray and a $C$-shape. Two $C$-shapes can intersect at most 6 times, a $C$-shape and a ray can intersect at most 2 times, and two rays can intersect at most 1 time. Thus, t... |
deepscaler | [
{
"role": "user",
"content": "On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:\n\n| Number of Trees Planted | 5 | 6 | 7 |\n|--------------------------|-----|-----... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 22620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides $5$ times as fast as Bella walks. The distance between their houses is $2$ miles, whic... | math | {
"style": "rule",
"ground_truth": "704"
} | {
"split": "train",
"index": 22621,
"solution": "1. **Understanding the problem**: Bella and Ella start moving towards each other from their respective houses, which are 2 miles apart. Bella walks and Ella rides a bicycle, with Ella's speed being 5 times Bella's speed. We need to find how many steps Bella takes b... |
deepscaler | [
{
"role": "user",
"content": "A six-digit number 111aaa is the product of two consecutive positive integers \\( b \\) and \\( b+1 \\). Find the value of \\( b \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "333"
} | {
"split": "train",
"index": 22622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 22623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold. - Every little child gets at least one piece of candy. - No two little children have the same number of pieces of candy. - The number... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 22624,
"solution": "An optimal arrangement is 2-6-3-9-12-4-8. Note that at least two prime factors must appear. In addition, any prime factor that appears must appear in at least two non-prime powers unless it is not used as a common factor between any two adjacent little children. Th... |
deepscaler | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3, 5, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 5, -8),$ with radius $r.$ Find $r.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"split": "train",
"index": 22625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\\{ 1,2,\\dots, 1000\\}$ there exist two sets, whose intersection contains at least $m$ numbers. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 22626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular triangular pyramid \\( S A B C \\). Point \\( S \\) is the apex of the pyramid, \\( AB = 1 \\), \\( AS = 2 \\), \\( BM \\) is the median of triangle \\( ABC \\), and \\( AD \\) is the angle bisector of triangle \\( SAB \\). Find the length of segment \\( DM \\).... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{31}}{6}"
} | {
"split": "train",
"index": 22627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define \\(P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2)\\). How many integers \\(n\\) are there such that \\(P(n)\\leq 0\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5100"
} | {
"split": "train",
"index": 22628,
"solution": "We are given the polynomial \\( P(x) = (x-1^2)(x-2^2)\\cdots(x-100^2) \\) and need to determine how many integers \\( n \\) satisfy \\( P(n) \\leq 0 \\).\n\n1. **Evaluating \\( P(n) = 0 \\):**\n - \\( P(n) = 0 \\) when \\( n \\) is exactly one of the squares from... |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 22629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there exists a real number \\( a \\) such that the equation \\( x \\sqrt{a(x-a)} + y \\sqrt{a(y-a)} = \\sqrt{|\\log(x-a) - \\log(a-y)|} \\) holds in the real number domain, find the value of \\( \\frac{3x^2 + xy - y^2}{x^2 - xy + y^2} \\). Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 22630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the numbers $1$, $2$, $3$, $4$ to form a four-digit number without repeating digits, the number of four-digit numbers larger than $2134$ is _____. (Answer in digits) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 22631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\\frac{a+c}{2}\\geq\\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 22632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\\circ$ and $38^\\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$, in degrees.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair A, B, C, D, P, Q;... | math | {
"style": "rule",
"ground_truth": "40^\\circ"
} | {
"split": "train",
"index": 22633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bangladesh National Mathematical Olympiad 2016 Higher Secondary\n\n<u>**Problem 2:**</u>\n(a) How many positive integer factors does $6000$ have?\n\n(b) How many positive integer factors of $6000$ are not perfect squares? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 22634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the degree of the polynomial $(x^4+ax^7+bx+c)(x^3+dx^2+e)(x+f)$, where letters $a$ through $f$ are all nonzero constants? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 22635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two lines defined by the equations $y = mx + 4$ and $y = 3x + b$, where $m$ and $b$ are constants, intersect at the point $(6, 10)$. What is the value of $b + m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 22636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball thrown vertically upwards has a height above the ground that is a quadratic function of its travel time. Xiaohong throws two balls vertically upwards at intervals of 1 second. Assuming the height above the ground is the same at the moment of release for both balls, and b... | math | {
"style": "rule",
"ground_truth": "1.6"
} | {
"split": "train",
"index": 22637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y=ax^{2}+bx+c$ ($a\\neq 0$) with its axis of symmetry to the left of the $y$-axis, where $a$, $b$, $c \\in \\{-3,-2,-1,0,1,2,3\\}$, let the random variable $X$ be the value of \"$|a-b|$\". Then, the expected value $EX$ is \\_\\_\\_\\_\\_\\_. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\dfrac {8}{9}"
} | {
"split": "train",
"index": 22638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}. L... | math | {
"style": "rule",
"ground_truth": "13515"
} | {
"split": "train",
"index": 22639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point in the $xz$-plane that is equidistant from the points $(1,-1,0),$ $(2,1,2),$ and $(3,2,-1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{31}{10}, 0, \\frac{1}{5} \\right)"
} | {
"split": "train",
"index": 22640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A room measures 16 feet by 12 feet and includes a column with a square base of 2 feet on each side. Find the area in square inches of the floor that remains uncovered by the column. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27,072"
} | {
"split": "train",
"index": 22641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brown lives on a street with more than 20 but fewer than 500 houses (all houses are numbered sequentially: $1, 2, 3$, etc.). Brown discovered that the sum of all the numbers from the first house to his own house (inclusive) equals half the sum of all the numbers from the first ... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 22642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempte... | math | {
"style": "rule",
"ground_truth": "\\frac{349}{500}"
} | {
"split": "train",
"index": 22643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 22644,
"solution": "1. **Identify the common difference**: Given an arithmetic sequence, the common difference $d$ can be calculated using any two consecutive terms. From the terms $x$ and $2x$, we have:\n \\[\n d = 2x - x = x\n \\]\n However, since $2x$ is the fourth term and... |
deepscaler | [
{
"role": "user",
"content": "For positive integers $a$ and $b$, let $M(a, b)=\\frac{\\operatorname{lcm}(a, b)}{\\operatorname{gcd}(a, b)}$, and for each positive integer $n \\geq 2$, define $$x_{n}=M(1, M(2, M(3, \\ldots, M(n-2, M(n-1, n)) \\ldots)))$$ Compute the number of positive integers $n$ such that ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22645,
"solution": "The desired condition is that $x_{n}=5 x_{n+1}$ or $x_{n+1}=5 x_{n}$. Note that for any prime $p$, we have $\\nu_{p}(M(a, b))=\\left|\\nu_{p}(a)-\\nu_{p}(b)\\right|$. Furthermore, $\\nu_{p}(M(a, b)) \\equiv \\nu_{p}(a)+\\nu_{p}(b) \\bmod 2$. So, we have that $$\\nu... |
deepscaler | [
{
"role": "user",
"content": "Sides $AB, BC, CD$ and $DA$ of convex polygon $ABCD$ have lengths 3, 4, 12, and 13, respectively, and $\\angle CBA$ is a right angle. The area of the quadrilateral is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 22646,
"solution": "1. **Identify the triangles**: Given that $\\angle CBA$ is a right angle and the sides $AB = 3$ and $BC = 4$, triangle $ABC$ is a right triangle. By the Pythagorean theorem, the hypotenuse $AC$ can be calculated as follows:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\... |
deepscaler | [
{
"role": "user",
"content": "Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"split": "train",
"index": 22647,
"solution": "To solve this problem, we will use complementary counting to determine the probability that the product of the numbers rolled on the six dice is not divisible by $4$. We then subtract this probability from $1$ to find the probability that the product is divisible b... |
deepscaler | [
{
"role": "user",
"content": "The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation\n\\[f(a) + f(b) = n^2\\]holds. What is $f(2002)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 22648,
"solution": ""
} |
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