data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right triangle with hypotenuse $AC$. Let $B^{\\prime}$ be the reflection of point $B$ across $AC$, and let $C^{\\prime}$ be the reflection of $C$ across $AB^{\\prime}$. Find the ratio of $[BCB^{\\prime}]$ to $[BC^{\\prime}B^{\\prime}]$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 2600,
"solution": "Since $C, B^{\\prime}$, and $C^{\\prime}$ are collinear, it is evident that $[BCB^{\\prime}]=\\frac{1}{2}[BCC^{\\prime}]$. It immediately follows that $[BCB^{\\prime}]=[BC^{\\prime}B^{\\prime}]$. Thus, the ratio is 1."
} |
deepscaler | [
{
"role": "user",
"content": "Four people, A, B, C, and D, stand on a staircase with 7 steps. If each step can accommodate up to 3 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is (answer in digits). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "2394"
} | {
"split": "train",
"index": 2601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{QY}$? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 2602,
"solution": "1. **Calculate the total area of the octagon**: The octagon is composed of $10$ unit squares. Therefore, the total area of the octagon is $10$ square units.\n\n2. **Area bisected by $\\overline{PQ}$**: Since $\\overline{PQ}$ bisects the area of the octagon, the area... |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 10 are between 11 and 103? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 2603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "184320"
} | {
"split": "train",
"index": 2604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 2605,
"solution": "1. **Understanding the Ratio**: The problem states that the ratio of the number of games won to the number of games lost by the Middle School Middies is $\\frac{11}{4}$. This implies that for every 11 games won, there are 4 games lost.\n\n2. **Setting Up the Express... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $D$ is a point on side $\\overline{AC}$ such that $BD=DC$ and $\\angle BCD$ measures $70^\\circ$. What is the degree measure of $\\angle ADB$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 2606,
"solution": "1. **Identify Equal Angles**: Given that $BD = DC$, triangle $BDC$ is isosceles. Therefore, the base angles are equal, i.e., $\\angle DBC = \\angle DCB = 70^\\circ$.\n\n2. **Calculate $\\angle BDC$**: In triangle $BDC$, the sum of the angles in any triangle is $180^... |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability that all the rational terms are not adjacent to each other when rearranging the terms of the expansion $( \\sqrt {x}+ \\dfrac {1}{2 \\sqrt[4]{x}})^{8}$ in a list. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 2607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP(\"B\",(-8/3,16*sqrt(2)/3),W);MP(\"B'\",(8/3,16*sqrt(2)/3),E);... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 2608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the three interior angles \\( A, B, C \\) of \\(\\triangle ABC\\) satisfy \\( A = 3B = 9C \\), find the value of \\( \\cos A \\cos B + \\cos B \\cos C + \\cos C \\cos A \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"split": "train",
"index": 2609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, if $g(x) = f(x) + 2$, then $g(-1) = \\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\\{x:a \\le x \\le b \\}$.) Let\n\\[g(x) = 1 - f(x + 1).\\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$ Let's think step by... | math | {
"style": "rule",
"ground_truth": "(-1,1,0,1)"
} | {
"split": "train",
"index": 2611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\\cdots +x_5x_1$ where $x_1, x_2, \\dots, x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 2612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\triangle ABC$ is an equilateral triangle with side length $s$, determine the value of $s$ when $AP = 2$, $BP = 2\\sqrt{3}$, and $CP = 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 2613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "P.J. starts with \\(m=500\\) and chooses a positive integer \\(n\\) with \\(1 \\leq n \\leq 499\\). He applies the following algorithm to \\(m\\) and \\(n\\): P.J. sets \\(r\\) equal to the remainder when \\(m\\) is divided by \\(n\\). If \\(r=0\\), P.J. sets \\(s=0\\). If \\(r... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 2614,
"solution": "Suppose that \\(m=500\\) and \\(1 \\leq n \\leq 499\\) and \\(1 \\leq r \\leq 15\\) and \\(2 \\leq s \\leq 9\\) and \\(t=0\\). Since \\(s>0\\), then the algorithm says that \\(t\\) is the remainder when \\(r\\) is divided by \\(s\\). Since \\(t=0\\), then \\(r\\) is... |
deepscaler | [
{
"role": "user",
"content": "Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minu... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 2615,
"solution": "We could see that both $374$ and $319$ are divisible by $11$ in the outset, and that $34$ and $29$, the quotients, are relatively prime. Both are the $average$ number of minutes across the $11$ days, so we need to subtract $\\left \\lfloor{\\frac{11}{2}}\\right \\rf... |
deepscaler | [
{
"role": "user",
"content": "How many students chose Greek food if 200 students were asked to choose between pizza, Thai food, or Greek food, and the circle graph shows the results? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 2616,
"solution": "Of the 200 students, $50 \\%$ (or one-half) of the students chose Greek food. Since one-half of 200 is 100, then 100 students chose Greek food."
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\\frac ab,$ wh... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 2617,
"solution": "1. **Understanding the Problem:**\n The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (which contains points with integer coordinates $(x, y)$ where $1 \\leq x, y \\leq 30$) lie on or... |
deepscaler | [
{
"role": "user",
"content": "Pascal's Triangle's interior numbers are defined beginning from the third row. Calculate the sum of the cubes of the interior numbers in the fourth row. Following that calculation, if the sum of the cubes of the interior numbers of the fifth row is 468, find the sum of the cube... | math | {
"style": "rule",
"ground_truth": "14750"
} | {
"split": "train",
"index": 2618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \\( r \\) and \\( s \\), with \\( r < s \\). Find the ratio \\( r : s \\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "1: 3"
} | {
"split": "train",
"index": 2619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\\angle BAC = 60^{\\circ}$, $D$ is a point on side $BC$ such that $AD = \\sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\\triangle... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 2620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Income from September 2019 to December 2019 is:\n\n$$\n(55000+45000+10000+17400) * 4 = 509600 \\text{ rubles}\n$$\n\nExpenses from September 2019 to November 2019 are:\n\n$$\n(40000+20000+5000+2000+2000) * 4 = 276000 \\text{ rubles}\n$$\n\nBy 31.12.2019 the family will have sav... | math | {
"style": "rule",
"ground_truth": "1340840"
} | {
"split": "train",
"index": 2621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of lengt... | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"split": "train",
"index": 2622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 240^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 2623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a class of 50 students with exam scores following a normal distribution $N(100,10^2)$, and $P(90 ≤ ξ ≤ 100) = 0.3$, estimate the number of students who scored above 110 points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\|\\mathbf{v}\\| = 4,$ then find $\\mathbf{v} \\cdot \\mathbf{v}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the tetrahedron S-ABC, the lateral edge SA is perpendicular to the plane ABC, and the base ABC is an equilateral triangle with a side length of $\\sqrt{3}$. If SA = $2\\sqrt{3}$, then the volume of the circumscribed sphere of the tetrahedron is \\_\\_\\_\\_\\_\\_. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}\\pi"
} | {
"split": "train",
"index": 2626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some real number $r,$ the polynomial $8x^3 - 4x^2 - 42x + 45$ is divisible by $(x - r)^2.$ Find $r.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 2629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles trapezoid \\(ABCD\\), where \\(AD \\parallel BC\\), \\(BC = 2AD = 4\\), \\(\\angle ABC = 60^\\circ\\), and \\(\\overrightarrow{CE} = \\frac{1}{3} \\overrightarrow{CD}\\), calculate the value of \\(\\overrightarrow{CA} \\cdot \\overrightarrow{BE}\\). Let's thi... | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 2630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the production of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx + y + z = 8 \\\\\nxy + yz + xz = -18\n\\end{array}\\right.\n$$\n\nFind the length of the cable. Let's think ... | math | {
"style": "rule",
"ground_truth": "4\\pi \\sqrt{\\frac{59}{3}}"
} | {
"split": "train",
"index": 2631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "( Ricky Liu ) For what values of $k > 0$ is it possible to dissect a $1 \\times k$ rectangle into two similar, but incongruent, polygons? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "k \\neq 1"
} | {
"split": "train",
"index": 2632,
"solution": "We will show that a dissection satisfying the requirements of the problem is possible if and only if $k\\neq 1$ .\nWe first show by contradiction that such a dissection is not possible when $k = 1$ . Assume that we have such a dissection. The common boundary of the ... |
deepscaler | [
{
"role": "user",
"content": "Simplify $(1 + \\tan 20^\\circ)(1 + \\tan 25^\\circ).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the central cell of a $21 \\times 21$ board, there is a piece. In one move, the piece can be moved to an adjacent cell sharing a side. Alina made 10 moves. How many different cells can the piece end up in? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 2634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the equation about $x$, $x^{2}-2a\\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "344.88"
} | {
"split": "train",
"index": 2637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the angles \\(A\\) and \\(B\\) are \\(45^{\\circ}\\) and \\(30^{\\circ}\\) respectively, and \\(CM\\) is the median. The circles inscribed in triangles \\(ACM\\) and \\(BCM\\) touch segment \\(CM\\) at points \\(D\\) and \\(E\\). Find the area of triangle... | math | {
"style": "rule",
"ground_truth": "16(\\sqrt{3}+1)"
} | {
"split": "train",
"index": 2638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all solutions $(x, y) \\in \\mathbb Z^2$ of the equation\n\\[x^3 - y^3 = 2xy + 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,0),(0,-2)"
} | {
"split": "train",
"index": 2639,
"solution": "\nConsider the equation \\(x^3 - y^3 = 2xy + 8\\). We are tasked with finding all integer solutions \\((x, y) \\in \\mathbb{Z}^2\\).\n\n### Step 1: Rewrite the Equation\n\nFirst, rewrite the given equation:\n\n\\[\nx^3 - y^3 = (x - y)(x^2 + xy + y^2).\n\\]\n\nSet th... |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{6} + \\frac{1}{3} = \\frac{1}{x}$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 2640,
"solution": "Simplifying, $\\frac{1}{6} + \\frac{1}{3} = \\frac{1}{6} + \\frac{2}{6} = \\frac{3}{6} = \\frac{1}{2}$. Thus, $\\frac{1}{x} = \\frac{1}{2}$ and so $x = 2$."
} |
deepscaler | [
{
"role": "user",
"content": "Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How m... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, the sum of its first $n$ terms, $S\\_n$, satisfies $S\\_3=0$ and $S\\_5=-5$. The sum of the first 2016 terms of the sequence $\\{ \\frac{1}{a_{2n-1}a_{2n+1}} \\}$ is $\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "-\\frac{2016}{4031}"
} | {
"split": "train",
"index": 2642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111111_{(2)}"
} | {
"split": "train",
"index": 2643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andrey drove a car to the airport in a neighboring city. After one hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. He then sharply increased his speed, and as a result, covered the remaining distance at an ave... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 2644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 2645,
"solution": "Since 12 and 21 are multiples of 3 (12 = 4 \\times 3 and 21 = 7 \\times 3), the answer is not 12 or 21. 16 is a perfect square (16 = 4 \\times 4) so the answer is not 16. The sum of the digits of 26 is 8, which is not a prime number, so the answer is not 26. Since 1... |
deepscaler | [
{
"role": "user",
"content": "If the set $\\{1, a, \\frac{b}{a}\\}$ equals the set $\\{0, a^2, a+b\\}$, then find the value of $a^{2017} + b^{2017}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 2646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors such that\n\\[\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} 3 \\\\ 2 \\end{pmatrix}.\\]Compute $\\operatorname{proj}_{\\mathbf{w}} (5 \\mathbf{v}).$ Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 15 \\\\ 10 \\end{pmatrix}"
} | {
"split": "train",
"index": 2647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "(m, n) = (m, 2m), (3, 4)"
} | {
"split": "train",
"index": 2648,
"solution": "\nGiven the problem, we need to find all integer pairs \\((n, m)\\) such that \\(n > m > 2\\) and a regular \\(n\\)-sided polygon can be inscribed in a regular \\(m\\)-sided polygon. To satisfy the condition, all the vertices of the \\(n\\)-gon must lie on the sides... |
deepscaler | [
{
"role": "user",
"content": "Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, then $f(-1)=$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 2649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2650,
"solution": "Let $R A=x$. If the diagonals intersect at $X$, and the segment is $P Q$ with $P$ on $K R$, then $\\triangle P K X \\sim \\triangle R K A$ and $\\triangle O K X \\sim \\triangle R A X$ (by equal angles), giving $R A / P X=$ $A K / X K=1+A X / X K=1+A R / O K=(x+12) ... |
deepscaler | [
{
"role": "user",
"content": "How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 2651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tournament with 2017 participating teams, each round consists of three randomly chosen teams competing, with exactly one team surviving from each round. If only two teams remain, a one-on-one battle determines the winner. How many battles must take place to declare a champ... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 2652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive difference between Jo's ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 2653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Olga Ivanovna, the homeroom teacher of class 5B, is organizing a \"Mathematical Ballet\". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 2654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 2655,
"solution": "\nTo solve this problem, we need to ensure that the average after each score is entered is an integer. We are given the scores in ascending order: $71$, $76$, $80$, $82$, and $91$. The total sum of these scores is $400$. We need to determine the last score entered b... |
deepscaler | [
{
"role": "user",
"content": "Let $\\sigma (n)$ denote the sum and $\\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$ ). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$ \\frac{\\sigma (n)}{\\tau (n)}\\geq a\\sq... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 2656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $180 \\div \\left(12 + 13 \\cdot 2\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{90}{19}"
} | {
"split": "train",
"index": 2657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all of the odd divisors of $180$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 2658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polynomial \\( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \\ldots + a_1 x + a_0 \\) where each coefficient \\( a_i \\) belongs to the interval \\([100,101]\\), what is the smallest natural number \\( n \\) such that this polynomial can have a real root? Let's think step... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 2659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle PQR$ is inscribed inside $\\triangle XYZ$ such that $P, Q, R$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\\triangle PYZ, \\triangle QXR, \\triangle RQP$ have centers $O_4, O_5, O_6$, respectively. Also, $XY = 29, YZ = 35, XZ=28$, and $\\stackrel{\\frow... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 2660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{-x}+ \\frac {nx}{mx+n}$. \n$(1)$ If $m=0$, $n=1$, find the minimum value of the function $f(x)$. \n$(2)$ If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\\infty)$ is $1$, find the maximum value of $\\frac {m}{n}$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 2661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?\n\nNote that we consider the case of rotation and turn over as distinct way. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $2 \\times 2^5 - 8^{58} \\div 8^{56}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 2663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangular pyramid where two of the three lateral faces are isosceles right triangles and the third face is an equilateral triangle with a side length of 1, calculate the volume of this triangular pyramid. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"split": "train",
"index": 2664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 12)"
} | {
"split": "train",
"index": 2665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "People enter the subway uniformly from the street. After passing through the turnstiles, they end up in a small hall before the escalators. The entrance doors have just opened, and initially, the hall before the escalators was empty, with only one escalator running to go down. ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 2666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circle $C$: $(x-1)^{2}+(y-2)^{2}=25$, and line $l$: $(2m+1)x+(m+1)y-7m-4=0$, if the length of the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is _____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 2667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\\frac12$ , and no side parallel to an axis? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} + \\sqrt{2} + \\sqrt{13}"
} | {
"split": "train",
"index": 2668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference between two positive integers is 8 and their product is 56. What is the sum of these integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{2}"
} | {
"split": "train",
"index": 2669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)\n\n*Proposed by Miles Yamner and Andrew Wu*\n\n(Note: wording changed from original to clarify) Let'... | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"split": "train",
"index": 2670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is called a perfect power if it can be written in the form \\(a^b\\), where \\(a\\) and \\(b\\) are positive integers with \\(b \\geq 2\\). The increasing sequence \\(2, 3, 5, 6, 7, 10, \\ldots\\) consists of all positive integers which are not perfect powers... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 2671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At an old estate, the house is surrounded in a circle by tall trees: spruces, pines, and birches. There are a total of 96 trees. These trees have a strange property: from any coniferous tree, if you take two trees skipping one tree in between, one of them is coniferous and the ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 2672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "27.5"
} | {
"split": "train",
"index": 2673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the square of the number 22222? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 2674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest four-digit number where all digits are distinct, and which is divisible by each of its digits. Zero cannot be used. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9864"
} | {
"split": "train",
"index": 2675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest real number \\(\\lambda\\) such that the inequality\n$$\n5(ab + ac + ad + bc + bd + cd) \\leq \\lambda abcd + 12\n$$\nholds for all positive real numbers \\(a, b, c, d\\) that satisfy \\(a + b + c + d = 4\\). Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 2676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 4 students from 5 female and 4 male students to participate in a speech competition. \n(1) If 2 male and 2 female students are to be selected, how many different selections are there? \n(2) If at least 1 male and 1 female student must be selected, and male student A an... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 2677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2678,
"solution": "For positive integers $a, b$, we have $$a!\\mid b!\\quad \\Leftrightarrow a!\\leq b!\\quad \\Leftrightarrow \\quad a \\leq b$$ Thus, $$((n!)!)!\\mid(2004!)!\\Leftrightarrow(n!)!\\leq 2004!\\Leftrightarrow n!\\leq 2004 \\Leftrightarrow n \\leq 6$$"
} |
deepscaler | [
{
"role": "user",
"content": "What is the discriminant of $3x^2 - 7x - 12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"split": "train",
"index": 2679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\mathbf{A} = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0 \\end{pmatrix}.\\]Compute $\\mathbf{A}^{95}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & -1 & 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 2680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a square, 100 points are marked. The square is divided into triangles such that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside the triangle or is a ... | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 2681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Chinese mathematician Qin Jiushao (circa 1202-1261) from the Southern Song Dynasty proposed Qin Jiushao's algorithm for polynomial evaluation in his work \"Mathematical Book in Nine Chapters.\" The provided diagram illustrates an example of using Qin Jiushao's algorithm to ... | math | {
"style": "rule",
"ground_truth": "2^{5}+2^{4}+2^{3}+2^{2}+2+1"
} | {
"split": "train",
"index": 2682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, two adjacent terms $a_n$ and $a_{n+1}$ are the roots of the equation $x^2 + 3nx + b_n = 0$. Given that $a_{10} = -17$, calculate the value of $b_{51}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5840"
} | {
"split": "train",
"index": 2683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The traditional Chinese mathematical masterpiece \"Nine Chapters on the Mathematical Art\" records: \"There are 5 cows and 2 sheep, worth 19 taels of silver; 2 cows and 5 sheep, worth 16 taels of silver. How much is each cow and each sheep worth in silver?\" According to the tr... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 2684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $3^{1999}$ is divided by $13$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $x^2 + 900x + 1800$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $\\frac{c}{b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-446.\\overline{2}"
} | {
"split": "train",
"index": 2686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\theta$ be the angle between the line\n\\[\\frac{x + 1}{2} = \\frac{y}{3} = \\frac{z - 3}{6}\\]and the plane $-10x - 2y + 11z = 3.$ Find $\\sin \\theta.$\n\n[asy]\nimport three;\n\nsize(150);\ncurrentprojection = perspective(6,3,2);\n\ntriple I = (1,0,0), J = (0,1,0), K ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{21}"
} | {
"split": "train",
"index": 2687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ for which $\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}}$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6156"
} | {
"split": "train",
"index": 2688,
"solution": "The number $\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}}$ is a positive integer if and only if its square is a perfect square. We have $$(\\sqrt{100+\\sqrt{n}}+\\sqrt{100-\\sqrt{n}})^{2} =(100+\\sqrt{n})+(100-\\sqrt{n})+2 \\sqrt{(100+\\sqrt{n})(100-\\sqrt{n})} =200+2... |
deepscaler | [
{
"role": "user",
"content": "How many different ways are there to write 2004 as a sum of one or more positive integers which are all \"aproximately equal\" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only diff... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 2689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $A B C$ is a triangle with incircle $\\omega$, and $\\omega$ is tangent to $\\overline{B C}$ and $\\overline{C A}$ at $D$ and $E$ respectively. The bisectors of $\\angle A$ and $\\angle B$ intersect line $D E$ at $F$ and $G$ respectively, such that $B F=1$ and $F G=G A=... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 2690,
"solution": "Let $\\alpha, \\beta, \\gamma$ denote the measures of $\\frac{1}{2} \\angle A, \\frac{1}{2} \\angle B, \\frac{1}{2} \\angle C$, respectively. We have $m \\angle C E F=90^{\\circ}-\\gamma, m \\angle F E A=90^{\\circ}+\\gamma, m \\angle A F G=m \\angle A F E=180^{\\ci... |
deepscaler | [
{
"role": "user",
"content": "Given a positive geometric sequence $\\{a_{n}\\}$, if ${a_m}{a_n}=a_3^2$, find the minimum value of $\\frac{2}{m}+\\frac{1}{{2n}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 2691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of this region formed by six congruent squares is 294 square centimeters. What is the perimeter of the region, in centimeters?\n\n[asy]\ndraw((0,0)--(-10,0)--(-10,10)--(0,10)--cycle);\ndraw((0,10)--(0,20)--(-30,20)--(-30,10)--cycle);\ndraw((-10,10)--(-10,20));\ndraw((-... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 2692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The day before yesterday, students collected 3 kg more scrap paper than they did yesterday, and yesterday they collected 40 kg less than the total amount collected the day before yesterday and today combined. How many kilograms of scrap paper did the students collect today? Let... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 2693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\begin{align*} (5a^2 - 13a + 4)(2a - 3) \\end{align*} for $a = 1\\frac12$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 2694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are standing at a pole and a snail is moving directly away from the pole at $1 \\mathrm{~cm} / \\mathrm{s}$. When the snail is 1 meter away, you start 'Round 1'. In Round $n(n \\geq 1)$, you move directly toward the snail at $n+1 \\mathrm{~cm} / \\mathrm{s}$. When you reach... | math | {
"style": "rule",
"ground_truth": "5050"
} | {
"split": "train",
"index": 2695,
"solution": "Suppose the snail is $x_{n}$ meters away at the start of round $n$, so $x_{1}=1$, and the runner takes $\\frac{100 x_{n}}{(n+1)-1}=\\frac{100 x_{n}}{n}$ seconds to catch up to the snail. But the runner takes the same amount of time to run back to the start, so durin... |
deepscaler | [
{
"role": "user",
"content": "Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \\ge K xy$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{95}"
} | {
"split": "train",
"index": 2696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \\ge -\\frac{1}{2},$ $y \\ge -2,$ and $z \\ge -3.$ Find the maximum value of:\n\\[\n\\sqrt{4x + 2} + \\sqrt{4y + 8} + \\sqrt{4z + 12}.\n\\] Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{10}"
} | {
"split": "train",
"index": 2697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $\\Gamma$ is the incircle of $\\triangle ABC$ and is also the circumcircle of $\\triangle XYZ$. The point $X$ is on $\\overline{BC}$, the point $Y$ is on $\\overline{AB}$, and the point $Z$ is on $\\overline{AC}$. If $\\angle A=40^\\circ$, $\\angle B=60^\\circ$, and $\... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 2698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a periodic sequence $\\left\\{x_{n}\\right\\}$ that satisfies $x_{n}=\\left|x_{n-1}-x_{n-2}\\right|(n \\geqslant 3)$, if $x_{1}=1$ and $x_{2}=a \\geqslant 0$, calculate the sum of the first 2002 terms when the period of the sequence is minimized. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "1335"
} | {
"split": "train",
"index": 2699,
"solution": ""
} |
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