data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A number is reduced by 5 times and then increased by 20 times to get 40. What is this number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Into how many parts do the planes of the faces of a tetrahedron divide the space? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers with up to six digits contain the digit 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "468559"
} | {
"split": "train",
"index": 24151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numb... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 24152,
"solution": "1. **Understanding the Problem:**\n - We have three balls numbered $1, 2, 3$.\n - Each ball is equally likely to be drawn.\n - The drawing process is repeated three times with replacement.\n - We need to find the probability that the ball numbered $2$ was d... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be integers not all the same with $a, b, c\\ge 4$ that satisfy $$ 4abc = (a + 3) (b + 3) (c + 3). $$ Find the numerical value of $a + b + c$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 24153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a survey of $150$ employees at a tech company, it is found that:\n- $90$ employees are working on project A.\n- $50$ employees are working on project B.\n- $30$ employees are working on both project A and B.\n\nDetermine what percent of the employees surveyed are not working... | math | {
"style": "rule",
"ground_truth": "26.67\\%"
} | {
"split": "train",
"index": 24154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A venture capital firm is considering investing in the development of a new energy product, with estimated investment returns ranging from 100,000 yuan to 10,000,000 yuan. The firm is planning to establish a reward scheme for the research team: the reward amount $y$ (in units o... | math | {
"style": "rule",
"ground_truth": "328"
} | {
"split": "train",
"index": 24155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 24156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acqua... | math | {
"style": "rule",
"ground_truth": "\\binom{n-1}{2}"
} | {
"split": "train",
"index": 24157,
"solution": "\nTo solve this problem, we need to determine the maximum number of pairs of participants who are not acquainted but share a common acquaintance at the party. This problem can be approached as a combinatorial optimization problem using graph theory.\n\n1. **Graph R... |
deepscaler | [
{
"role": "user",
"content": "A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical form. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 24158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ be a diameter in a circle of radius $5\\sqrt{2}.$ Let $\\overline{CD}$ be a chord in the circle that intersects $\\overline{AB}$ at a point $E$ such that $BE=2\\sqrt{5}$ and $\\angle AEC = 45^{\\circ}.$ What is $CE^2+DE^2?$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 24159,
"solution": "1. **Identify the Geometry and Given Information:**\n - Let $O$ be the center of the circle.\n - $\\overline{AB}$ is a diameter, so $A$, $O$, and $B$ are collinear, and $OA = OB = 5\\sqrt{2}$.\n - $\\overline{CD}$ intersects $\\overline{AB}$ at $E$ such that ... |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $$\\sqrt[3]{3-\\frac{1}{x}}=-4$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x=\\frac{1}{67}"
} | {
"split": "train",
"index": 24160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $\\dfrac{1}{2} + \\dfrac{1}{x} = \\dfrac{5}{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given functions $y_1=\\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\\neq 0$).<br/>$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.<br/>$(2)$ If poin... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 24162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 24163,
"solution": "To find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$, we can substitute $a = 1$ and $b = 1$ into the binomial. This simplifies the expression to just the sum of the coefficients, as each term in the expansion will have the form $... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(ABCD\\) is a rectangle with \\(AD = 13\\), \\(DE = 5\\), and \\(EA = 12\\). The area of \\(ABCD\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 24164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = x^2 + 12x + 30 \\). Find the largest real root of the equation \\( f(f(f(x))) = 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6 + \\sqrt[8]{6}"
} | {
"split": "train",
"index": 24165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 24166,
"solution": "To solve this problem, we need to consider all possible configurations of green and red squares in the $2 \\times 2$ grid such that no green square shares its top or right side with a red square. We will analyze each case based on the number of green squares.\n\n##... |
deepscaler | [
{
"role": "user",
"content": "The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from $A$ to $B$?\n\n[asy]\n\nsize(4cm,4cm);\n\npair a1, b1, c1, d1;\n\na1=(1,1);\nb1=(0,1);\nc1=(1.6,1.4);\nd1=(1,0);\n\npair e1... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 24167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$ is $S_{n}$, and $a_{1}=4$, $a_{n}+a_{n+1}=4n+2$ for $n\\in \\mathbb{N}^{*}$, calculate the maximum value of $n$ that satisfies $S_{n} \\lt 2023$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nOn the image, there are several circles connected by segments. Kostya chooses a natural number \\( n \\) and places different natural numbers not exceeding \\( n \\) in the circles so that for all the placed numbers the following property is satisfied: if the numbers \\( a \\... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 24169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that $abc + ab + a$ is divisible by $4$, where positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3, \\dots, 1024\\}$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 24170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $P$ is a point on $BC$ so that $\\angle APD=90^{\\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. Find ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{28}{3}"
} | {
"split": "train",
"index": 24171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A new definition: $\\left[a,b,c\\right]$ represents the \"graph number\" of a quadratic function $y=ax^{2}+bx+c$ (where $a\\neq 0$, and $a$, $b$, $c$ are real numbers). For example, the \"graph number\" of $y=-x^{2}+2x+3$ is $\\left[-1,2,3\\right]$.\n\n$(1)$ The \"graph number\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 24172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $WXYZ$ is a trapezoid where $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. Given $YZ = 15$, $\\tan Z = 2$, and $\\tan X = 2.5$, what is the length of $XY$? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{261}"
} | {
"split": "train",
"index": 24173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly drawn and then a white ball is put back. What is the probability that all red balls are drawn exactly at the 4th draw? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0434"
} | {
"split": "train",
"index": 24174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Someone, when asked for the number of their ticket, replied: \"If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number.\" Determine the ticket number. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 24175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four le... | math | {
"style": "rule",
"ground_truth": "372"
} | {
"split": "train",
"index": 24176,
"solution": "There are 7 different characters that can be picked, with 0 being the only number that can be repeated twice.\nIf $0$ appears 0 or 1 times amongst the sequence, there are $\\frac{7!}{(7-5)!} = 2520$ sequences possible.\nIf $0$ appears twice in the sequence, there a... |
deepscaler | [
{
"role": "user",
"content": "Which of the following is equal to $2017 - \\frac{1}{2017}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2018 \\times 2016}{2017}"
} | {
"split": "train",
"index": 24177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\\frac{|MO|}{|MF|}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 24178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: \\( \\frac {\\tan 150^{\\circ} \\cos (-210^{\\circ}) \\sin (-420^{\\circ})}{\\sin 1050^{\\circ} \\cos (-600^{\\circ})} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 24179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\\frac{m+\\sqrt{n}}r$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 24180,
"solution": "Notice the original expression can be written as $2000x^6+100x^5-200x^4+200x^4+10x^3-20x^2+20x^2+x-2$.\n Which equals to $(20x^2+x-2)(100x^4+10x^2+1)=0$\n So our solution is to find what is the root for $20x^2+x-2=0$ since the determinant of $100t^2+10t+1<0$(Let $x... |
deepscaler | [
{
"role": "user",
"content": "An object is moving towards a converging lens with a focal length of \\( f = 10 \\ \\mathrm{cm} \\) along the line defined by the two focal points at a speed of \\( 2 \\ \\mathrm{m/s} \\). What is the relative speed between the object and its image when the object distance is \... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 24181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. 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": 24182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct prime factors does 56 have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \\cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "(2, 3) \\text{ and } (6, 15)"
} | {
"split": "train",
"index": 24184,
"solution": "\nLet us determine all pairs of positive integers \\( (a, b) \\) such that:\n\n\\[\na + b + (\\gcd(a, b))^2 = \\mathrm{lcm}(a, b) = 2 \\cdot \\mathrm{lcm}(a-1, b)\n\\]\n\nwhere \\(\\mathrm{lcm}(a, b)\\) is the least common multiple and \\(\\gcd(a, b)\\) is the grea... |
deepscaler | [
{
"role": "user",
"content": "Let \\(x, y, z\\) be positive real numbers such that \\(xyz = 1\\). Find the maximum value of \n\\[\n\\frac{x^2y}{x+y} + \\frac{y^2z}{y+z} + \\frac{z^2x}{z+x}.\n\\] 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": 24185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Erin has 4 sisters and 6 brothers, determine the product of the number of sisters and the number of brothers of her brother Ethan. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 24186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the binomial coefficients of the first two terms of the expansion of \\\\({(2x+\\frac{1}{\\sqrt{x}})}^{n}\\\\) is \\\\(10\\\\).\n\n\\\\((1)\\\\) Find the value of \\\\(y' = 2x\\\\).\n\n\\\\((2)\\\\) Find the constant term in this expansion. Let's think ste... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 24187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\\sin \\angle PF_{1}F_{2} = \\frac {1}{3}$, dete... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 24188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\\cos B+b\\sin A$.\n(1) Find $A$;\n(2) If $a=2$ and $b=c$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+1"
} | {
"split": "train",
"index": 24189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers at a minimum must be taken to ensure that among the selected numbers, there will definitely be a pair whose sum is a multiple of ten? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 24190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression \\(5 * 4 * 3 * 2 * 1 = 0\\), replace the asterisks with arithmetic operators \\(+, -, \\times, \\div\\), using each operator exactly once, so that the equality holds true (note: \\(2 + 2 \\times 2 = 6\\)). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "5 - 4 \\times 3 : 2 + 1"
} | {
"split": "train",
"index": 24191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin(\\omega x + \\phi)$ ($\\omega > 0$, $0 \\leq \\phi \\leq \\pi$) is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\\sqrt{4+\\pi^2}$. \n(1) Find the analytical expression of the function ... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"split": "train",
"index": 24192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Explain how any unit fraction $\\frac{1}{n}$ can be decomposed into other unit fractions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2n}+\\frac{1}{3n}+\\frac{1}{6n}"
} | {
"split": "train",
"index": 24193,
"solution": "$\\frac{1}{2n}+\\frac{1}{3n}+\\frac{1}{6n}$"
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $|\\omega^2 + 4\\omega + 34|$ if $\\omega = 5 + 3i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6664}"
} | {
"split": "train",
"index": 24194,
"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 $n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24195,
"solution": "Using the properties of logarithms, we can simplify the first equation to $\\log_{10} \\sin x + \\log_{10} \\cos x = \\log_{10}(\\sin x \\cos x) = -1$. Therefore, \\[\\sin x \\cos x = \\frac{1}{10}.\\qquad (*)\\]\n Now, manipulate the second equation. \\begin{align... |
deepscaler | [
{
"role": "user",
"content": "The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?\n$\\begin{array}{c@{}c@{}c@{}c@{}c}\n& & 6 & 4 & 1 \\\\\n& & 8 & 5 & 2 \\\\\n& + & 9 & 7 & 3 \\\\\n\\hline\n& 2 & 4 & 5 & 6\n\\end{array}$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 24196,
"solution": "1. **Verify the given addition**: First, we add the numbers as they are given:\n \\[\n 641 + 852 + 973 = 2466\n \\]\n However, the sum provided in the problem is $2456$. This indicates an error in the addition or in the digits themselves.\n\n2. **Identify t... |
deepscaler | [
{
"role": "user",
"content": "Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Deter... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{39}"
} | {
"split": "train",
"index": 24197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 24198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be nonzero real numbers such that $a+b+c=0$ and $a^{3}+b^{3}+c^{3}=a^{5}+b^{5}+c^{5}$. Find the value of $a^{2}+b^{2}+c^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"split": "train",
"index": 24199,
"solution": "Let $\\sigma_{1}=a+b+c, \\sigma_{2}=ab+bc+ca$ and $\\sigma_{3}=abc$ be the three elementary symmetric polynomials. Since $a^{3}+b^{3}+c^{3}$ is a symmetric polynomial, it can be written as a polynomial in $\\sigma_{1}, \\sigma_{2}$ and $\\sigma_{3}$. Now, observe t... |
deepscaler | [
{
"role": "user",
"content": "The three-digit positive integer $N$ has a ones digit of 3. What is the probability that $N$ is divisible by 3? 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{1}{3}"
} | {
"split": "train",
"index": 24200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to $n$ and let $a_{i}$ be the number of friends of the $i$-th resident. Suppose that $\\sum_{i=1}^{n} a_{i}^{2}=n^{2}-n$. Let $... | math | {
"style": "rule",
"ground_truth": "k = 5"
} | {
"split": "train",
"index": 24201,
"solution": "Let us define the simple, undirected graph $G$ so that the vertices of $G$ are the town's residents and the edges of $G$ are the friendships between the residents. Let $V(G)=\\{v_{1}, v_{2}, \\ldots, v_{n}\\}$ denote the vertices of $G ; a_{i}$ is degree of $v_{i}$... |
deepscaler | [
{
"role": "user",
"content": "If a restaurant offers 15 different dishes, and Yann and Camille each decide to order either one or two different dishes, how many different combinations of meals can they order? Assume that the dishes can be repeated but the order in which each person orders the dishes matters... | math | {
"style": "rule",
"ground_truth": "57600"
} | {
"split": "train",
"index": 24202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive real number \\( x \\) is such that\n$$\n\\sqrt[3]{1-x^{3}} + \\sqrt[3]{1+x^{3}} = 1.\n$$\n\nFind \\( x^{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt[3]{28}}{3}"
} | {
"split": "train",
"index": 24203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let n be the number of real values of $p$ for which the roots of \n$x^2-px+p=0$\nare equal. Then n equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24204,
"solution": "1. **Identify the condition for equal roots:** For a quadratic equation $Ax^2 + Bx + C = 0$ to have equal roots, the discriminant must be zero. The discriminant is given by $\\Delta = B^2 - 4AC$.\n\n2. **Apply the condition to the given equation:** The given quadra... |
deepscaler | [
{
"role": "user",
"content": "Complex numbers \\(a\\), \\(b\\), \\(c\\) form an equilateral triangle with side length 24 in the complex plane. If \\(|a + b + c| = 48\\), find \\(|ab + ac + bc|\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 24205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the volume of cone $A$ to the volume of cone $B$? Express your answer as a common fraction. [asy]\nsize(260);\ndraw(ellipse((0,0),14.8,6),S);\nlabel(\"Cone $A$\", (0,-6), S);\ndraw((0,28.3)--(0,0),dashed);\nlabel(\"$28.3$\",(0,14),SW);\ndraw((-14.8,0)--(0,2... | math | {
"style": "rule",
"ground_truth": "\\frac{148}{283}"
} | {
"split": "train",
"index": 24206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $n$ is the four-digit number $712n$, with units digit $n$, divisible by 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 24207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When fitting a set of data with the model $y=ce^{kx}$, in order to find the regression equation, let $z=\\ln y$ and transform it to get the linear equation $z=0.3x+4$. Then, the values of $c$ and $k$ are respectively \\_\\_\\_\\_\\_\\_ and \\_\\_\\_\\_\\_\\_. Let's think step b... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 24208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Antarctican language has an alphabet of just 16 letters. Interestingly, every word in the language has exactly 3 letters, and it is known that no word's first letter equals any word's last letter (for instance, if the alphabet were $\\{a, b\\}$ then $a a b$ and aaa could no... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 24209,
"solution": "1024 Every letter can be the first letter of a word, or the last letter of a word, or possibly neither, but not both. If there are $a$ different first letters and $b$ different last letters, then we can form $a \\cdot 16 \\cdot b$ different words (and the desired c... |
deepscaler | [
{
"role": "user",
"content": "Given the power function $f(x)=kx^{\\alpha}$, its graph passes through the point $(\\frac{1}{2}, \\frac{\\sqrt{2}}{2})$. Find the value of $k+\\alpha$. 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": 24210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $100$th digit after the decimal point when $\\frac{3}{26}$ is expressed as a decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common factor of 90, 135, and 225? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 24212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 24213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a \\) and \\( b \\) are real numbers, and the following system of inequalities in terms of \\( x \\):\n\\[\n\\left\\{\\begin{array}{l}\n20x + a > 0, \\\\\n15x - b \\leq 0\n\\end{array}\\right.\n\\]\nhas integer solutions of only 2, 3, and 4, find the maximum valu... | math | {
"style": "rule",
"ground_truth": "-1200"
} | {
"split": "train",
"index": 24214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $6$, $8$, and $10$ so that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in the same triangle, but one side of ... | math | {
"style": "rule",
"ground_truth": "\\frac{111}{175}"
} | {
"split": "train",
"index": 24215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.\n\n[asy]\nunitsize(1 cm);\n\nreal func (real x) {\n return (x^2 + 3/4);\n}\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 24216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the vertices of the hyperbola $9x^2 + 54x - y^2 + 10y + 55 = 0.$ 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": 24217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there are 1000 villages, each with 99 inhabitants. Each inhabitant is either a knight, who always tells the truth, or a liar, who always lies. It is known that the island has exactly 54,054 knights. One day, each inhabitant was asked the question: \"Are there more... | math | {
"style": "rule",
"ground_truth": "638"
} | {
"split": "train",
"index": 24218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(\\sqrt{6} + 2) : 1"
} | {
"split": "train",
"index": 24219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, 10\\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all s... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"split": "train",
"index": 24220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \\minus{} 3^b$ divides $ a^4 \\plus{} b^2$.\n\n[i]Author: Stephan Wagner, Austria[/i] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 4)"
} | {
"split": "train",
"index": 24221,
"solution": "\nTo find all pairs of natural numbers \\((a, b)\\) such that \\(7^a - 3^b\\) divides \\(a^4 + b^2\\), we proceed as follows:\n\n1. Let \\(d = 7^a - 3^b\\). We need to ensure \\(d \\mid a^4 + b^2\\). This implies that \\(a^4 + b^2 = k \\cdot (7^a - 3^b)\\) for some... |
deepscaler | [
{
"role": "user",
"content": "Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\\%$ profit and then buys it back from Mr. $Y$ at a $15\\%$ loss. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 24222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g_0(x) = x + |x - 150| - |x + 150|$, and for $n \\geq 1$, let $g_n(x) = |g_{n-1}(x)| - 2$. For how many values of $x$ is $g_{100}(x) = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "299"
} | {
"split": "train",
"index": 24223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $x$ satisfy the equation $(x^2-x-1)^{x+2}=1?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 24224,
"solution": "To find the integers $x$ that satisfy $(x^2-x-1)^{x+2}=1$, we need to consider different cases where the expression can equal $1$. These cases include:\n\n1. **$x^2-x-1 = 1$:** This simplifies to $x^2 - x - 2 = 0$. Factoring gives:\n \\[\n (x-2)(x+1) = 0\n \\... |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 24225,
"solution": "1. **Identify the area of the square**: Given that square $ABCD$ has side lengths of 3, the area of the square is calculated as:\n \\[\n \\text{Area} = \\text{side}^2 = 3^2 = 9.\n \\]\n\n2. **Determine the area of each part**: Since segments $CM$ and $CN$ div... |
deepscaler | [
{
"role": "user",
"content": "Find the area in the plane contained by the graph of\n\\[|2x + 3y| + |2x - 3y| \\le 12.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric progression \\( b_{1}, b_{2}, \\ldots \\) is such that \\( b_{25} = 2 \\tan \\alpha \\) and \\( b_{31} = 2 \\sin \\alpha \\) for some acute angle \\( \\alpha \\). Find the term number \\( n \\) for which \\( b_{n} = \\sin 2\\alpha \\). Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 24227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $$A=\\frac{1}{6}\\left(\\left(\\log _{2}(3)\\right)^{3}-\\left(\\log _{2}(6)\\right)^{3}-\\left(\\log _{2}(12)\\right)^{3}+\\left(\\log _{2}(24)\\right)^{3}\\right)$$ Compute $2^{A}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 24228,
"solution": "Let $a=\\log _{2}(3)$, so $2^{a}=3$ and $A=\\frac{1}{6}\\left[a^{3}-(a+1)^{3}-(a+2)^{3}+(a+3)^{3}\\right]$. But $(x+1)^{3}-x^{3}=3 x^{2}+3 x+1$, so $A=\\frac{1}{6}\\left[3(a+2)^{2}+3(a+2)-3 a^{2}-3 a\\right]=\\frac{1}{2}[4 a+4+2]=2 a+3$. Thus $2^{A}=\\left(2^{a}\\r... |
deepscaler | [
{
"role": "user",
"content": "An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ Let's... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 24230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y = 2x + 7$ is to be parameterized using vectors. Which of the following options are valid parameterizations?\n\n(A) $\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}$\n\n(B) $\\begin{pma... | math | {
"style": "rule",
"ground_truth": "\\text{B,E}"
} | {
"split": "train",
"index": 24231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "First $a$ is chosen at random from the set $\\{1,2,3,\\cdots,99,100\\}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 24232,
"solution": "1. **Identify the Cyclic Nature of Units Digits for Powers of 3 and 7**:\n - The units digit of powers of $3$ cycles through $1, 3, 9, 7$. This can be verified by calculating $3^1, 3^2, 3^3, 3^4$, etc., and observing the units digits.\n - Similarly, the units d... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sin x, \\sqrt{3}\\cos x)$, $\\overrightarrow{b}=(-1,1)$, and $\\overrightarrow{c}=(1,1)$, where $x \\in [0, \\pi]$.\n\n(1) If $(\\overrightarrow{a} + \\overrightarrow{b}) \\parallel \\overrightarrow{c}$, find the value of $x$;\n\n(2) If $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 24233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\frac{16^{24}}{64^{8}}$.\nA) $16^2$\nB) $16^4$\nC) $16^8$\nD) $16^{16}$\nE) $16^{24}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16^8"
} | {
"split": "train",
"index": 24234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all values of $x$ that are solutions to the equation $x^2 = 7x - 12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 24235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence \\(\\left\\{a_{n}\\right\\}\\), \\(a_{1} = -1\\), \\(a_{2} = 1\\), \\(a_{3} = -2\\). Given that for all \\(n \\in \\mathbf{N}_{+}\\), \\(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\\), and \\(a_{n+1} a_{n+2} a_{n+3} \\neq 1\\), find the s... | math | {
"style": "rule",
"ground_truth": "-4321"
} | {
"split": "train",
"index": 24236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle ABC$ and $\\triangle DBC$ share $BC$. $AB = 5\\ \\text{cm}$, $AC = 12\\ \\text{cm}$, $DC = 8\\ \\text{cm}$, and $BD = 20\\ \\text{cm}$. What is the least possible integral number of centimeters in $BC$?\n\n[asy]\nsize(100); import graph; currentpen = fontsize(10pt);... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 24237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) Le... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24238,
"solution": "To solve this problem, we need to determine the number of ways to schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses are taken in consecutive periods.\n\n#### Step 1: Calculate the total numb... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of $\\sin^4 x + \\cos^4 x.$ 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": 24239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$ , such that for any initial configuration of p... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 24240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest composite number that has no prime factors less than 15. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "323"
} | {
"split": "train",
"index": 24241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different three-letter sets of initials are possible using the letters $A$ through $G$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 24242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the six positive integer factors of 30? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 24243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\{a, b, c, d\\}\\) be a subset of \\(\\{1, 2, \\ldots, 17\\}\\). If 17 divides \\(a - b + c - d\\), then \\(\\{a, b, c, d\\}\\) is called a \"good subset.\" Find the number of good subsets. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "476"
} | {
"split": "train",
"index": 24244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many prime numbers are divisible by $39$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 24245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $x$ that satisfies $cd = x-3i$ given $|c|=3$ and $|d|=5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{6}"
} | {
"split": "train",
"index": 24246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a shooting competition, nine clay targets are set up in three hanging columns. The first column contains four targets, the second column contains three targets, and the third column contains two targets. A sharpshooter must break all the targets under the following condition... | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 24247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{n} = \\frac{1}{3} + \\frac{1}{12} + \\frac{1}{30} + \\frac{1}{60} + \\cdots + \\frac{2}{n(n-1)(n-2)} + \\frac{2}{(n+1) n(n-1)} \\), find \\( \\lim_{n \\rightarrow \\infty} a_{n} \\). 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": 24248,
"solution": ""
} |
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