data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 21949,
"solution": "1. **Define the transformation**: Let $d_i$ be the number on the $i$-th die. We define a transformation where each die's number is replaced by $d_i' = 7 - d_i$. This transformation is an involution, meaning applying it twice returns the original value: $7 - (7 - d_... |
deepscaler | [
{
"role": "user",
"content": "Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs $(a,b)$ with $a,b\\in S$ and $a\\leq b$ is $\\frac{F_5(a,b)}{F_3(a,b)}$ an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 21950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$x = {1+\\frac{\\sqrt{2}}{1+\\frac{\\sqrt{2}}{1+...}}}$. Find $\\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\\frac{A+\\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A|+|B|+|C|$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\\widehat{AOD})=60^\\circ$ and the altitude of the trapezoid is $10$ , what is the area of the trapezoid? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "100\\sqrt{3}"
} | {
"split": "train",
"index": 21952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of minutes in a week is closest to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 21953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P(1, \\frac{\\sqrt{2}}{2})$ is on the ellipse, and $|PF_1| + |PF_2| = 2\\sqrt{2}$.\n\n$(1)$ Find the standard equation of ellipse $C$; \n... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 21954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John borrows $2000$ from Mary, who charges an interest rate of $6\\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than triple what he borrowed? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 21955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a rectangle, and let \\(E\\) and \\(F\\) be points on segment \\(AB\\) such that \\(AE = EF = FB\\). If \\(CE\\) intersects the line \\(AD\\) at \\(P\\), and \\(PF\\) intersects \\(BC\\) at \\(Q\\), determine the ratio of \\(BQ\\) to \\(CQ\\). Let's think step... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 21956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: y^2 = 8x$ with focus $F$ and directrix $l$. Let $P$ be a point on $l$, and $Q$ be a point where the line $PF$ intersects $C$. If $\\overrightarrow{FQ} = -4\\overrightarrow{FP}$, calculate $|QF|$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\begin{pmatrix} 1 & 1 & -2 \\\\ 0 & 4 & -3 \\\\ -1 & 4 & 3 \\end{pmatrix} \\begin{pmatrix} 2 & -2 & 0 \\\\ 1 & 0 & -3 \\\\ 4 & 0 & 0 \\end{pmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -5 & -2 & -3 \\\\ -8 & 0 & -12 \\\\ 14 & 2 & -12 \\end{pmatrix}"
} | {
"split": "train",
"index": 21958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the product \\( S = \\left(1+2^{-\\frac{1}{32}}\\right)\\left(1+2^{-\\frac{1}{16}}\\right)\\left(1+2^{-\\frac{1}{8}}\\right)\\left(1+2^{-\\frac{1}{4}}\\right)\\left(1+2^{-\\frac{1}{2}}\\right) \\), calculate the value of \\( S \\). Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}\\left(1 - 2^{-\\frac{1}{32}}\\right)^{-1}"
} | {
"split": "train",
"index": 21959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle PQR with right angle at P, suppose $\\sin Q = 0.6$. If the length of QP is 15, what is the length of QR? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 21960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 21961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[x^6 - x^3 - x^2 - x - 1 = q_1(x) q_2(x) \\dotsm q_m(x),\\]\nwhere each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \\dots + q_m(2).$ Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 21962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "First, a number \\( a \\) is randomly selected from the set \\(\\{1,2,3, \\cdots, 99,100\\}\\), then a number \\( b \\) is randomly selected from the same set. Calculate the probability that the last digit of \\(3^{a} + 7^{b}\\) is 8. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 21963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 21964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\log_{10} \\sin x + \\log_{10} \\cos x = -1$ and that $\\log_{10} (\\sin x + \\cos x) = \\frac{1}{2} (\\log_{10} n - 1),$ find $n.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 21966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The on-time arrival rate of bus No. 101 in a certain city is 90%. Calculate the probability that the bus arrives on time exactly 4 times out of 5 rides for a person. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.32805"
} | {
"split": "train",
"index": 21967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 21968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\\overrightarrow{m} = (2, c)$ and $\\overrightarrow{n} = (\\frac{b}{2}\\cos C - \\sin A, \\cos B)$, with $b = \\sqrt{3}$ and $\\o... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 21969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has sides $AB = 3$ , $BC = 2$ . Point $ P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$ . Find $BP$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 21970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $704_{12} + 159_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "861_{12}"
} | {
"split": "train",
"index": 21971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\(\\cos(x+y)\\) given that \\(\\cos x - \\cos y = \\frac{1}{4}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31/32"
} | {
"split": "train",
"index": 21972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that\n\n\\[a_1^2 + \\cdots + a_{2019}^2 \\ge a_1a_2 + a_2a_3 + \\cdots + a_{1008}a_{1009} + \\lambda a_{1009}a_{1010} + \\lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \\cdots + a_{2018}a_{2019}\\]\nfor all real numbers $a_1, \\ldo... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 21973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD\\), the bases are given as \\(AD = 4\\) and \\(BC = 1\\), and the angles at \\(A\\) and \\(D\\) are \\(\\arctan 2\\) and \\(\\arctan 3\\) respectively.\n\nFind the radius of the circle inscribed in triangle \\(CBE\\), where \\(E\\) is the intersection p... | math | {
"style": "rule",
"ground_truth": "\\frac{18}{25 + 2 \\sqrt{130} + \\sqrt{445}}"
} | {
"split": "train",
"index": 21974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10:15:6"
} | {
"split": "train",
"index": 21975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .\r\nFind the smallest k that:\r $S(F) \\leq k.P(F)^2$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"split": "train",
"index": 21976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular tetrahedron is inscribed in a cylinder such that two opposite edges of the tetrahedron are the diameters of the cylinder's bases. Find the ratio of the volume of the cylinder to the volume of the tetrahedron. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\pi}{2}"
} | {
"split": "train",
"index": 21977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the mean of all solutions for $x$ when $x^3 + 3x^2 - 10x = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at... | math | {
"style": "rule",
"ground_truth": "2870"
} | {
"split": "train",
"index": 21979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"split": "train",
"index": 21981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$, respectively. Find the minimum possible area of $\\triangle AI_1I_2$ ... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 21982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the graph of $|2x| + |5y| = 10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"split": "train",
"index": 21984,
"solution": "1. **Define the sides and diagonal of the rectangle**: Let the short side of the rectangle be \\(a\\) and the long side be \\(b\\). The diagonal \\(d\\) of the rectangle can be expressed using the Pythagorean Theorem as \\(d = \\sqrt{a^2 + b^2}\\).\n\n2. **Set up t... |
deepscaler | [
{
"role": "user",
"content": "The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\\overline{AB}$ and $\\overline{CD}$ are the only parallel sides. The sum of the absolute values of all po... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 21985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the angles are \\( \\angle B = 30^\\circ \\) and \\( \\angle A = 90^\\circ \\). Point \\( K \\) is marked on side \\( AC \\), and points \\( L \\) and \\( M \\) are marked on side \\( BC \\) such that \\( KL = KM \\) (point \\( L \\) lies on ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 21986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{53}{3}"
} | {
"split": "train",
"index": 21987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region defined by the inequality: \\( |y - |x - 2| + |x|| \\leq 4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 21988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many regions of the plane are bounded by the graph of $$x^{6}-x^{5}+3 x^{4} y^{2}+10 x^{3} y^{2}+3 x^{2} y^{4}-5 x y^{4}+y^{6}=0 ?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 21989,
"solution": "The left-hand side decomposes as $$\\left(x^{6}+3 x^{4} y^{2}+3 x^{2} y^{4}+y^{6}\\right)-\\left(x^{5}-10 x^{3} y^{2}+5 x y^{4}\\right)=\\left(x^{2}+y^{2}\\right)^{3}-\\left(x^{5}-10 x^{3} y^{2}+5 x y^{4}\\right)$$. Now, note that $$(x+i y)^{5}=x^{5}+5 i x^{4} y-10... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{18}{7}=31824$, $\\binom{18}{8}=43758$ and $\\binom{18}{9}=43758$, calculate $\\binom{20}{9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "163098"
} | {
"split": "train",
"index": 21990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring? Let's think step b... | math | {
"style": "rule",
"ground_truth": "802"
} | {
"split": "train",
"index": 21991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $l_{1}: x+a^{2}y+6=0$ and the line $l_{2}: (a-2)x+3ay+2a=0$ are parallel, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 21993,
"solution": "Let $e$ and $b$ be the speeds of the escalator and Bob, respectively.\n When Al was on his way down, he took $150$ steps with a speed of $3b-e$ per step. When Bob was on his way up, he took $75$ steps with a speed of $b+e$ per step. Since Al and Bob were walking th... |
deepscaler | [
{
"role": "user",
"content": "Given that $x^2+x-6$ is a factor of the polynomial $2x^4+x^3-ax^2+bx+a+b-1$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 21994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the math test scores, X, follow a normal distribution with a mean of 100 and a standard deviation of δ, and the number of students scoring between 80 and 120 points accounted for about 3/4 of the total 1600 students, calculate the number of students who scored no les... | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"split": "train",
"index": 21995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent triangles have vertices at three of the eight points in the array shown below?\n[asy]\ndot((0,0)); dot((.5,.5)); dot((.5,0)); dot((.0,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));\n[/asy] Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21997,
"solution": "To solve this problem, we need to consider the positions of the points and how they can form non-congruent triangles. The points are arranged in a grid with coordinates as follows:\n- Point 1: (0,0)\n- Point 2: (0.5,0)\n- Point 3: (1,0)\n- Point 4: (1.5,0)\n- Point... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.\n[asy]draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 21998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent v... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 21999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "662004"
} | {
"split": "train",
"index": 22000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 22001,
"solution": "1. **Identify the form of the palindrome**: A four-digit palindrome can be expressed in the form $\\overline{abba}$, where $a$ and $b$ are digits, and $a \\neq 0$ to ensure it is a four-digit number.\n\n2. **Express the number mathematically**: The number $\\overli... |
deepscaler | [
{
"role": "user",
"content": "A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \\ldots$ The smallest part is equal to $\\frac{1}{400} N$ . Find the value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 22002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 22003,
"solution": "We need to consider different seating arrangements for Alice, as her position affects the seating of the others due to her restrictions with Bob and Carla.\n\n1. **Alice sits in the center chair (3rd position):**\n - The 2nd and 4th chairs must be occupied by Der... |
deepscaler | [
{
"role": "user",
"content": "A segment \\( AB \\) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \\( \\pi / 3 \\) to the diameter \\( CD \\) of this sphere. The distance from the end \\( C \\) of the diameter to the nearest end \\( A \\) of the chord \\( AB \\) is ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 22004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $(a_n)_{n \\equal{} 1}^\\infty$ is defined on real numbers with $a_n \\not \\equal{} 0$, $a_na_{n \\plus{} 3} = a_{n \\plus{} 2}a_{n \\plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \\cdots + a_{41}a_{42}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 22005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of cubic centimeters in the volume of the cylinder formed by rotating a rectangle with side lengths 8 cm and 16 cm about its longer side. Express your answer in terms of \\(\\pi\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256\\pi"
} | {
"split": "train",
"index": 22006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that students A and B stand together among three students A, B, and C lined up in a row is what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 22007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$, for $1 \\leq n \\leq 1000$, is the number $\\frac{1}{2}\\binom{2 n}{n}$ even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 22008,
"solution": "In fact, the expression $\\binom{2 n}{n}$ is always even, and it is not a multiple of four if and only if $n$ is a power of 2, and there are 10 powers of 2 between 1 and 1000. Let $f(N)$ denote the number of factors of 2 in $N$. Thus, $$f(n!)=\\left\\lfloor\\frac{n... |
deepscaler | [
{
"role": "user",
"content": "Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 22009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \\geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{11+6\\sqrt{2}}{8}"
} | {
"split": "train",
"index": 22010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider those functions $f$ that satisfy $f(x+4)+f(x-4) = f(x)$ for all real $x$. Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 22011,
"solution": "1. **Identify the functional equation and its implications**: We are given that $f(x+4) + f(x-4) = f(x)$ for all real $x$. This equation suggests a relationship between function values at intervals of 4 units.\n\n2. **Derive additional relationships**: Substitute $... |
deepscaler | [
{
"role": "user",
"content": "Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If t... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 22012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the product of $\\frac{5}{3} \\times \\frac{6}{5} \\times \\frac{7}{6} \\times \\cdots \\times \\frac{2010}{2009}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "670"
} | {
"split": "train",
"index": 22013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2016$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 2 + e^{4 \\pi i k / n} \\right)^n - 1 \\right) \n\\]\nequal to zero? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 22014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from \\(A\\) to \\(B\\) is 999 km. Along the road, there are kilometer markers indicating the distances to \\(A\\) and \\(B\\): 0।999, 1।998, \\(\\ldots, 999।0. How many of these markers have only two different digits? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 22015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\\rho \\cos \\theta = 4$. \n$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 22016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find\n\\[\\frac{a}{bc + 1} + \\frac{b}{ac + 1} + \\frac{c}{ab + 1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{2}"
} | {
"split": "train",
"index": 22017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $30\\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\\equiv0,-1,1 \\pmod {30}$ and $(i,j)\\neq(x,y)$ . Therefore, each squ... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 22018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \\leq 1000$ satisfy the condition that $\\left[\\frac{998}{n}\\right]+\\left[\\frac{999}{n}\\right]+\\left[\\frac{1000}{n}\\right]$ is not divisible by 3? Let's thi... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 22019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sara baked 60 pies, of which one third contained chocolate, three-fifths contained berries, half contained cinnamon, and one-fifth contained poppy seeds. What is the largest possible number of pies that had none of these ingredients? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 22020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mark and Sandy are walking to the convenience store at the midpoint of their coordinates. Mark is standing at $(0,7)$, and Sandy is standing at $(-4,-1)$. At what coordinate will they meet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,3)"
} | {
"split": "train",
"index": 22021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute without using a calculator: $\\dfrac{9!}{6!3!}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 22022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school organized a Chinese traditional culture activity week to promote traditional Chinese culture. During the activity period, a Chinese traditional culture knowledge competition was held, with classes participating in the competition. Each class selected 5 represen... | math | {
"style": "rule",
"ground_truth": "\\frac{537}{125}"
} | {
"split": "train",
"index": 22023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many real solutions are there to the equation $|||| x|-2|-2|-2|=|||| x|-3|-3|-3|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 22024,
"solution": "6. The graphs of the two sides of the equation can be graphed on the same plot to reveal six intersection points."
} |
deepscaler | [
{
"role": "user",
"content": "Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "402"
} | {
"split": "train",
"index": 22025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she m... | math | {
"style": "rule",
"ground_truth": "469"
} | {
"split": "train",
"index": 22026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the numbers 1 to 6 into the six boxes in the image. The smallest result you can get is ______ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 22027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given the digits $0$, $1$, $2$, $3$, $4$, $5$. Form a four-digit number with no repeating digits.\n\n(I) How many different four-digit numbers can be formed?\n\n(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 22028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 22029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet ellipse $\\Gamma: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ have an eccentricity of $\\frac{\\sqrt{3}}{2}$. A line with slope $k (k > 0)$ passes through the left focus $F$ and intersects the ellipse $\\Gamma$ at points $A$ and $B$. If $\\overrightarrow{A F}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 22030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.\n\n\\[\\begin{tabular}{| c | c | c | c | c | c | c | c |}\n\\hline\n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \n\\hline\n9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \n\\hline\n17 & 18 & 19... | math | {
"style": "rule",
"ground_truth": "1056"
} | {
"split": "train",
"index": 22031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "182.53"
} | {
"split": "train",
"index": 22032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\\overline{BC}$, $Y$ on $\\overline{DE}$, and $Z$ on $\\overline{EF}$. Suppose that $AB=40$, and $EF=41(\\sqrt{3}-1)$. What is the side-length of the square? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "29\\sqrt{3}"
} | {
"split": "train",
"index": 22033,
"solution": "1. **Identify the relationships and congruences in the figure:**\n - Let $EZ = x$ and $\\angle XAB = \\alpha$.\n - Since $\\angle BAX = \\angle EYZ$ and $\\angle ZEY = \\angle XBA$, and $AX = YZ$, by the Angle-Angle-Side (AAS) congruence criterion, $\\triangle ... |
deepscaler | [
{
"role": "user",
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. The length of $Q R$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 22034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, let the left and right foci of the ellipse be $F_1$ and $F_2$, respectively. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If the line $AF... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 22035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $5\\cdot5! + 4\\cdot4!+4!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 22036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation $y = |x| - 3$ is translated two units to the left and three units down. What are the coordinates of the minimum point of the new graph? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,-6)"
} | {
"split": "train",
"index": 22037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many degrees are in each interior angle of a regular hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 22038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Self-Isogonal Cubics) Let $A B C$ be a triangle with $A B=2, A C=3, B C=4$. The isogonal conjugate of a point $P$, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines $P A$, $P B, P C$ across the angle bisectors of $\\angle A, \\angle B$, and $\\angl... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 22039,
"solution": "The first main insight is that all the cubics pass through the points $A, B, C, H$ (orthocenter), $O$, and the incenter and three excenters. Since two cubics intersect in at most nine points, this is all the intersections of a cubic with a cubic. On the other hand,... |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the parametric equations of the curve C₁ are $$\\begin{cases} x=2+2\\cos\\alpha \\\\ y=2\\sin\\alpha \\end{cases}$$ (where α is the parameter). Establish a polar coordinate system with the origin of the coordinate system as the pole and... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 22040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An investment of $\\$$10,000 is made in a government bond that will pay 6$\\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "13382"
} | {
"split": "train",
"index": 22041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 22042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\\overline{AB}$ and from $B$ to $\\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, an... | math | {
"style": "rule",
"ground_truth": "574"
} | {
"split": "train",
"index": 22043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\) with sides \\(BC=7\\), \\(AC=5\\), and \\(AB=3\\), an angle bisector \\(AD\\) is drawn. A circle is circumscribed around triangle \\(ABD\\), and a circle is inscribed in triangle \\(ACD\\). Find the product of their radii. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "35/32"
} | {
"split": "train",
"index": 22044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela 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": 22045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular decagon \\( ABCDEFGHIJ \\) has its center at \\( K \\). Each of the vertices and the center are to be associated with one of the digits \\( 1 \\) through \\( 10 \\), with each digit used exactly once, in such a way that the sums of the numbers on the lines \\( AKF \\), ... | math | {
"style": "rule",
"ground_truth": "3840"
} | {
"split": "train",
"index": 22046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every integers $ a,b,c$ whose greatest common divisor is $n$ , if\n\\[ \\begin{array}{l} {x \\plus{} 2y \\plus{} 3z \\equal{} a} \n{2x \\plus{} y \\minus{} 2z \\equal{} b} \n{3x \\plus{} y \\plus{} 5z \\equal{} c} \\end{array}\n\\]\nhas a solution in integers, what is th... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 22047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $n$ such that $10^n = 10^{-5}\\times \\sqrt{\\frac{10^{73}}{0.001}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 22048,
"solution": ""
} |
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