data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(a, b, c,\\) and \\(d\\) are positive integers which are not necessarily distinct. If \\(a^{2}+b^{2}+c^{2}+d^{2}=70\\), what is the largest possible value of \\(a+b+c+d?\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f_1(x) = \\frac23 - \\frac3{3x+1},$ and for $n \\ge 2,$ define $f_n(x) = f_1(f_{n-1}(x)).$ Find the value of $x$ that satisfies $f_{1001}(x) = x-3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{5}{3}"
} | {
"split": "train",
"index": 32750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a square \\(ABCD\\), a line segment \\(BE\\) is drawn such that point \\(E\\) lies on the side \\(CD\\). The perimeter of triangle \\(BCE\\) is three-quarters of the perimeter of the square \\(ABCD\\). The ratio of lengths \\(CE : CD\\) is \\(\\lambda : 1\\). What is the val... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 32751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sqrt{51.11}\\approx 7.149$ and $\\sqrt{511.1}\\approx 22.608$, determine the value of $\\sqrt{511100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "714.9"
} | {
"split": "train",
"index": 32752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On sides \\( BC \\) and \\( AC \\) of triangle \\( ABC \\), points \\( D \\) and \\( E \\) are chosen respectively such that \\( \\angle BAD = 50^\\circ \\) and \\( \\angle ABE = 30^\\circ \\). Find \\( \\angle BED \\) if \\( \\angle ABC = \\angle ACB = 50^\\circ \\). Let's thi... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 32753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbola \\[-x^2+2y^2-10x-16y+1=0\\]has two foci. Find the coordinates of either of them. (Enter your answer as an ordered pair. Enter only one of the foci, not both.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-5, 1)"
} | {
"split": "train",
"index": 32754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $AC=7$. Two bugs start simultaneously from $A$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 32755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid $ABCD$ lies on the $xy$ -plane. The slopes of lines $BC$ and $AD$ are both $\\frac 13$ , and the slope of line $AB$ is $-\\frac 23$ . Given that $AB=CD$ and $BC< AD$ , the absolute value of the slope of line $CD$ can be expressed as $\\frac mn$ , wh... | math | {
"style": "rule",
"ground_truth": "1706"
} | {
"split": "train",
"index": 32756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 32757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer builds a rectangular chicken coop that leans against a wall, covering an area of 36m<sup>2</sup>. Due to geographical constraints, the length of the side of the chicken coop, $x$, cannot exceed 7m. The wall height is 2m. The cost of constructing the front of the chicke... | math | {
"style": "rule",
"ground_truth": "2760"
} | {
"split": "train",
"index": 32758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=x^{3}+ax^{2}+bx-4$ and the tangent line equation $y=x-4$ at point $P\\left(2,f\\left(2\\right)\\right)$.<br/>$(1)$ Find the values of $a$ and $b$;<br/>$(2)$ Find the extreme values of $f\\left(x\\right)$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-\\frac{58}{27}"
} | {
"split": "train",
"index": 32759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,1,1)"
} | {
"split": "train",
"index": 32760,
"solution": "\nTo solve the equation \\( 2013^x + 2014^y = 2015^z \\) for non-negative integer solutions \\((x, y, z)\\), we will explore small values manually and check if they satisfy the equation due to the rapid growth of exponential terms.\n\n1. **Initial Consideration**: ... |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word \"PERCEPTION\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "907200"
} | {
"split": "train",
"index": 32761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number such that\n\\[ |z - 2i| + |z - 5| = 7. \\]\nFind the minimum value of $|z|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{100}{29}}"
} | {
"split": "train",
"index": 32762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius o... | math | {
"style": "rule",
"ground_truth": "173"
} | {
"split": "train",
"index": 32763,
"solution": "[asy] size(10cm); pair A, B, C, D, O, P, H, L, X, Y; A = (-1, 0); B = (1, 0); H = (0, 0); C = (5, 0); D = (9, 0); L = (7, 0); O = (0, sqrt(160/13 - 1)); P = (7, sqrt(160/13 - 4)); X = (0, sqrt(160/13 - 4)); Y = (O + P) / 2; draw(A -- O -- B -- cycle); draw(C -- P -... |
deepscaler | [
{
"role": "user",
"content": "P is a polynomial. When P is divided by $x-1$, the remainder is -4 . When P is divided by $x-2$, the remainder is -1 . When $P$ is divided by $x-3$, the remainder is 4 . Determine the remainder when $P$ is divided by $x^{3}-6 x^{2}+11 x-6$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "x^{2}-5"
} | {
"split": "train",
"index": 32764,
"solution": "The remainder polynomial is simply the order two polynomial that goes through the points $(1,-4),(2,-1)$, and $(3,4): x^{2}-5$."
} |
deepscaler | [
{
"role": "user",
"content": "Every June 1, an ecologist takes a census of the number of wrens in a state park. She noticed that the number is decreasing by $40\\%$ each year. If this trend continues, in what year will the census show that the number of wrens is less than $10\\%$ of what it was on June 1, 2... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 32765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 21 through 50 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 20 and the tile from box $D$ is eithe... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{45}"
} | {
"split": "train",
"index": 32766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\theta + \\cos\\theta = \\frac{3}{4}$, where $\\theta$ is an angle of a triangle, find the value of $\\sin\\theta - \\cos\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{23}}{4}"
} | {
"split": "train",
"index": 32767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be positive integers such that $a + b + c = 30$ and $\\gcd(a,b) + \\gcd(b,c) + \\gcd(c,a) = 11$. Determine the sum of all possible distinct values of $a^2 + b^2 + c^2$.\nA) 302\nB) 318\nC) 620\nD) 391\nE) 419 Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "620"
} | {
"split": "train",
"index": 32768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 32769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T=TNFTPP$ . Points $A$ and $B$ lie on a circle centered at $O$ such that $\\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$ , $CD = 5$ , and $BD = 6$ . Determine the area of quadrilateral $ACDB$ .... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 32770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 32771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 32772,
"solution": "Suppose $n$ is such an integer. Because $n$ has $3$ proper divisors, it must have $4$ divisors,, so $n$ must be in the form $n=p\\cdot q$ or $n=p^3$ for distinct prime numbers $p$ and $q$.\n In the first case, the three proper divisors of $n$ are $1$, $p$ and $q$. ... |
deepscaler | [
{
"role": "user",
"content": "\nThere are 5 students taking a test, and each student's score ($a, b, c, d, e$) is an integer between 0 and 100 inclusive. It is known that $a \\leq b \\leq c \\leq d \\leq e$. If the average score of the 5 students is $p$, then the median score $c$ is at least $\\qquad$ . Let... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 32773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are positive integers for which $a$ has $3$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the slope of the line that is perpendicular to the line $2x + 3y = 6$. 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": 32775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $$\\sum_{n=1}^{\\infty} \\frac{3n-1}{2^n}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( M = \\{1, 3, 5, 7, 9\\} \\), find the non-empty set \\( A \\) such that:\n1. Adding 4 to each element in \\( A \\) results in a subset of \\( M \\).\n2. Subtracting 4 from each element in \\( A \\) also results in a subset of \\( M \\).\n\nDetermine the set \\... | math | {
"style": "rule",
"ground_truth": "{5}"
} | {
"split": "train",
"index": 32777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The whole numbers from 1 to 2016 inclusive are written on a blackboard. Moritz underlines all the multiples of two in red, all the multiples of three in blue, and all the multiples of four in green. How many numbers does Moritz underline exactly twice? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 32778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How m... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 32779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $sin(x- \\frac {π}{4})= \\frac {2}{3}$, find the value of $sin2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 32780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer $x$ has 12 positive factors. The numbers 12 and 15 are factors of $x$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 32781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Susie Q has 1000 dollars to invest. She invests some of the money at the Pretty Penny Bank, which compounds annually at 3 percent. She invests the rest of the money at the Five and Dime Bank, which compounds annually at 5 percent. After two years, Susie has a total of $\\$10... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 32782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\\frac{|PF|}{|PA|}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 32783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two fixed points on the plane, \\\\(A(-2,0)\\\\) and \\\\(B(2,0)\\\\), and a moving point \\\\(T\\\\) satisfying \\\\(|TA|+|TB|=2 \\sqrt {6}\\\\).\n\\\\((\\\\)I\\\\()\\\\) Find the equation of the trajectory \\\\(E\\\\) of point \\\\(T\\\\);\n\\\\((\\\\)II\\\\()\\\\) A li... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 32784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that P and Q are two points on the unit circle centered at the origin O, and they are located in the first and fourth quadrants, respectively. The x-coordinate of point P is $\\frac{4}{5}$, and the x-coordinate of point Q is $\\frac{5}{13}$. Then, $\\cos \\angle POQ... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 32785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a regular polygon with $2^n$ sides, for $n \\ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\\prod_{i=2}^{\\infty}\\frac{A_i}{A_{i+1}}$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\pi}"
} | {
"split": "train",
"index": 32786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions in nonnegative integers $(a, b, c)$ are there to the equation $2^{a}+2^{b}=c!\\quad ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32787,
"solution": "We can check that $2^{a}+2^{b}$ is never divisible by 7 , so we must have $c<7$. The binary representation of $2^{a}+2^{b}$ has at most two 1 's. Writing 0 !, 1 !, 2 !, \\ldots, 6$ ! in binary, we can check that the only possibilities are $c=2,3,4$, giving solution... |
deepscaler | [
{
"role": "user",
"content": "In each square of an $11\\times 11$ board, we are to write one of the numbers $-1$ , $0$ , or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can b... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all integers $x$ , $x \\ge 3$ , such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 32789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the solutions of the equation $45 = -x^2 - 4x?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-45"
} | {
"split": "train",
"index": 32790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x \\neq y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 32791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The moisture content of freshly cut grass is $60\\%$, and the moisture content of hay is $15\\%$. How much hay will be obtained from one ton of freshly cut grass? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "470.588"
} | {
"split": "train",
"index": 32792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\), simplify the equation \\(\\cfrac{\\frac{1}{x} + \\frac{1}{y}}{\\frac{1}{x} + 2\\frac{1}{y}} = 4\\) and express it as \\(\\frac{x+y}{x+2y}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 32793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\\frac{2}{3}$ of its games and team $B$ has won $\\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ mo... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 32794,
"solution": "1. **Understanding the problem**: \n - Team $A$ has won $\\frac{2}{3}$ of its games, so it has lost $\\frac{1}{3}$ of its games.\n - Team $B$ has won $\\frac{5}{8}$ of its games, so it has lost $\\frac{3}{8}$ of its games.\n - Team $B$ has won $7$ more games ... |
deepscaler | [
{
"role": "user",
"content": "Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \\times 6$ grid of squares. She can move the pieces in two ways: If a token has a free square in front of it, then she can move this token one square to the right. If the square imme... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 32795,
"solution": "We put a marker on $(i, j)$ when a token is on $i$th and $j$th square and $i>j$. When the token in front/behind moves one step forward to a blank square, move the marker rightward/upward one unit correspondingly. When a 'leapfrog' happens, the marker moves from $(x... |
deepscaler | [
{
"role": "user",
"content": "Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\\sin x$ and $\\cos x$, where $x$ is a real number such that $0<x<\\frac{\\pi}{2}$. The length of the hypotenuse is Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 32796,
"solution": "1. **Setup and Notation**: Consider a right triangle $ABC$ with hypotenuse $BC$. Let points $D$ and $E$ trisect $BC$. Assume without loss of generality (WLOG) that $AD = \\cos(x)$ and $AE = \\sin(x)$. The choice of $AD$ and $AE$ as $\\cos(x)$ and $\\sin(x)$ respect... |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.[asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5... | math | {
"style": "rule",
"ground_truth": "578"
} | {
"split": "train",
"index": 32797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure shown represents a plot of land and is drawn using a scale in which 1 cm equals 2 miles. One square mile is 640 acres. How large is the actual plot of land, in acres? [asy]\ndraw((0,0)--(15,0)--(10,10)--(0,10)--cycle);\ndraw((0,1)--(1,1)--(1,0));\ndraw((0,9)--(1,9)--... | math | {
"style": "rule",
"ground_truth": "320000 \\mbox{ acres}"
} | {
"split": "train",
"index": 32798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 32799,
"solution": "We start by considering the different cases based on the number of heads that appear when two fair coins are tossed. For each head, a fair die is rolled. We need to find the probability that the sum of the die rolls is odd.\n\n#### Case Analysis:\n1. **Case 1: 0 He... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\\sin A = \\sqrt{7}a$.\n(1) Find the value of $\\sin B$;\n(2) If $a$, $b$, and $c$ form an arithmetic sequence with a positive common difference, find the value of $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 32800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum $\\frac{3}{50} + \\frac{5}{500} + \\frac{7}{5000}$.\nA) $0.0714$\nB) $0.00714$\nC) $0.714$\nD) $0.0357$\nE) $0.00143$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0714"
} | {
"split": "train",
"index": 32801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(9,3)$, respectively. What is its area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45 \\sqrt{3}"
} | {
"split": "train",
"index": 32802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n$ represents a positive integer less than $100$, determine the number of integers with an odd number of positive divisors and the number of integers with an even number of divisors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 32803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"split": "train",
"index": 32804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\\sin^2\\left(\\angle CAD\\right)$.\nA) $\\frac{3}{4}$ \nB) $\\frac{1}{4}$ \n... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many such pairs of numbers \\((n, k)\\) are there, for which \\(n > k\\) and the difference between the internal angles of regular polygons with \\(n\\) and \\(k\\) sides is \\(1^{\\circ}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 32806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an $8 \\times 8$ chessboard, 6 black rooks and $k$ white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 32807,
"solution": "The answer is $k=14$. For a valid construction, place the black rooks on cells $(a, a)$ for $2 \\leq a \\leq 7$ and the white rooks on cells $(a, a+1)$ and $(a+1, a)$ for $1 \\leq a \\leq 7$. Now, we prove the optimality. As rooks can only attack opposite color roo... |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, representing equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane conta... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 32808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pedestrian walked 5.5 kilometers in 1 hour but did not reach point \\( B \\) (short by \\(2 \\pi - 5.5\\) km). Therefore, the third option is longer than the first and can be excluded.\n\nIn the first case, they need to cover a distance of 5.5 km along the alley. If they move... | math | {
"style": "rule",
"ground_truth": "11/51"
} | {
"split": "train",
"index": 32809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 32810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function f(x) = 2sinωx(cosωx + $\\sqrt{3}$sinωx) - $\\sqrt{3}$ (ω > 0) has a minimum positive period of π.\n(1) Find the interval where the function f(x) is monotonically increasing.\n(2) If the graph of function f(x) is shifted to the left by $\\frac{\\pi}{6}$ u... | math | {
"style": "rule",
"ground_truth": "\\frac{55\\pi}{4}"
} | {
"split": "train",
"index": 32811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A typical triangle is defined as one in which each angle is at least \\(30^{\\circ}\\).\n\nA triangle is identified using the following procedure. The perimeter of a circle is divided into 180 equal arcs, and 3 distinct points are randomly chosen from these division points. (An... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 32812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integer \\( n \\) such that the polynomial \\( f(x) = 3x^3 - nx - n - 2 \\) can be factored into a product of two non-constant polynomials with integer coefficients. Find the sum of all possible values of \\( n \\). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 32813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Assume two of the roots of $f(x)$ are $s + 2$ and $s + 5,$ and two of the roots of $g(x)$ are $s + 4$ and $s + 8.$ Given that:\n\\[ f(x) - g(x) = 2s \\] for all real numbers $x.$ Find $s.$ Let's... | math | {
"style": "rule",
"ground_truth": "3.6"
} | {
"split": "train",
"index": 32814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carefully observe the arrangement pattern of the following hollow circles ($○$) and solid circles ($●$): $○●○○●○○○●○○○○●○○○○○●○○○○○○●…$. If this pattern continues, a series of $○$ and $●$ will be obtained. The number of $●$ in the first $100$ of $○$ and $●$ is $\\_\\_\\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a number satisfying the equation $\\sqrt[3]{x+9}-\\sqrt[3]{x-9}=3$, then $x^2$ is between: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75 \\text{ and } 85"
} | {
"split": "train",
"index": 32816,
"solution": "1. **Define Variables**: Let $a = \\sqrt[3]{x + 9}$ and $b = \\sqrt[3]{x - 9}$. \n\n2. **Cubing the Equations**: Cubing these definitions, we get:\n \\[ a^3 = x + 9 \\quad \\text{and} \\quad b^3 = x - 9. \\]\n Subtracting these equations gives:\n \\[ a^3 - b^... |
deepscaler | [
{
"role": "user",
"content": "Today our cat gave birth to kittens! It is known that the two lightest kittens together weigh 80 g, the four heaviest kittens together weigh 200 g, and the total weight of all the kittens is 500 g. How many kittens did the cat give birth to? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find t... | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"split": "train",
"index": 32818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$.\n Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 32819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $ABCD$, points $P$ and $Q$ lie on $\\overline{AD}$ and $\\overline{AB}$, respectively. Segments $\\overline{BP}$ and $\\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 32820,
"solution": "1. **Identify Similar Triangles**: Notice that $\\triangle CRB \\sim \\triangle BAP$ by AA similarity (both have a right angle and share angle $BRP$).\n\n2. **Set Up Ratio of Sides**: From the similarity, we have the ratio of corresponding sides:\n \\[\n \\frac... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the odd integers from 21 through 65, inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "989"
} | {
"split": "train",
"index": 32821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 32822,
"solution": "Consider the vectors corresponding to the sides of the heptagon, and call them \\left[x_{i}, y_{i}\\right] for i between 1 and 7. Then since \\sum x_{i}=\\sum y_{i}=0, and a^{2} has the same parity as a, we have that \\sum x_{i}^{2}+y_{i}^{2} must be an even number... |
deepscaler | [
{
"role": "user",
"content": "Find the flux of the vector field\n\n$$\n\\vec{a}=-x \\vec{i}+2 y \\vec{j}+z \\vec{k}\n$$\n\nthrough the portion of the plane\n\n$$\nx+2 y+3 z=1\n$$\n\nlocated in the first octant (the normal forms an acute angle with the $OZ$ axis). Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"split": "train",
"index": 32823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that bag A contains 3 white balls and 5 black balls, and bag B contains 4 white balls and 6 black balls, calculate the probability that the number of white balls in bag A does not decrease after a ball is randomly taken from bag A and put into bag B, and a ball is then ra... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{44}"
} | {
"split": "train",
"index": 32824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the first eight positive integers in a $2 \\times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 32825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the remainder when $$2^{\\frac{1 \\cdot 2}{2}}+2^{\\frac{2 \\cdot 3}{2}}+\\cdots+2^{\\frac{2011 \\cdot 2012}{2}}$$ is divided by 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32826,
"solution": "We have that $2^{3} \\equiv 1(\\bmod 7)$. Hence, it suffices to consider the exponents modulo 3. We note that the exponents are the triangular number and upon division by 3 give the pattern of remainders $1,0,0,1,0,0, \\ldots$, so what we want is $$\\begin{aligned}... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 2 & 3 \\\\ 7 & -1 \\end{pmatrix} \\begin{pmatrix} 1 & -5 \\\\ 0 & 4 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 & 2 \\\\ 7 & -39 \\end{pmatrix}"
} | {
"split": "train",
"index": 32827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ship travels from Port A to Port B against the current at a speed of 24 km/h. After arriving at Port B, it returns to Port A with the current. It is known that the journey with the current takes 5 hours less than the journey against the current. The speed of the current is 3 ... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"split": "train",
"index": 32828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a round-robin tournament, every team played exactly once against each other. Each team won 9 games and lost 9 games; there were no ties. Find the number of sets of three teams {A, B, C} such that team A beat team B, team B beat team C, and team C beat team A. Let's think ste... | math | {
"style": "rule",
"ground_truth": "969"
} | {
"split": "train",
"index": 32829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two-thirds of a number plus two times another number equals 20. If one-fourth of the first number minus the second number equals 2, find these two numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{22}{7}"
} | {
"split": "train",
"index": 32830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-dig... | math | {
"style": "rule",
"ground_truth": "327"
} | {
"split": "train",
"index": 32831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos(2x+\\varphi), |\\varphi| \\leqslant \\frac{\\pi}{2}$, if $f\\left( \\frac{8\\pi}{3}-x \\right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\\sin 2x$ from the graph of $y=f(x)$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 32832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2, \\ldots, 10$ are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. $1,2,3,4,5$, or $8,9,10,1,2$). If the probability that there exists some number that was not selected by any of the four people is $p$... | math | {
"style": "rule",
"ground_truth": "3690"
} | {
"split": "train",
"index": 32833,
"solution": "The unselected numbers must be consecutive. Suppose that $\\{1,2, \\ldots, k\\}$ are the unselected numbers for some $k$. In this case, 1 cannot be selected, so there are 5 possible sets of consecutive numbers the people could have chosen. This leads to $5^{4}$ pos... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\sqrt {3}\\sin (\\omega x+\\varphi)-\\cos (\\omega x+\\varphi)$ $(\\omega > 0,0 < \\varphi < \\pi)$ is an even function, and the distance between two adjacent axes of symmetry of its graph is $\\dfrac {\\pi}{2}$, then the value of $f(- \\dfrac {\\pi}{... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 32834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Hangzhou Asian Games are underway, and table tennis, known as China's \"national sport,\" is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, fo... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 32835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the volume of a regular tetrahedron with the side of its base equal to $\\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\\circ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 32836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane quadrilateral \\(ABCD\\), points \\(E\\) and \\(F\\) are the midpoints of sides \\(AD\\) and \\(BC\\) respectively. Given that \\(AB = 1\\), \\(EF = \\sqrt{2}\\), and \\(CD = 3\\), and that \\(\\overrightarrow{AD} \\cdot \\overrightarrow{BC} = 15\\), find \\(\\over... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the tr... | math | {
"style": "rule",
"ground_truth": "253"
} | {
"split": "train",
"index": 32838,
"solution": "Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset $\\mathcal{S}$. Without loss of generality, consider any $a, b, c \\,\\in \\mathcal{S}$ with... |
deepscaler | [
{
"role": "user",
"content": "There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 32839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of the real numbers \\( x \\) for which \\(\\frac{2 x}{x^{2}+5 x+3}+\\frac{3 x}{x^{2}+x+3}=1\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 32840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"split": "train",
"index": 32841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 32842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangular pyramid \\( P-ABC \\) with a base that is an equilateral triangle of side length \\( 4 \\sqrt{3} \\), and with \\( PA=3 \\), \\( PB=4 \\), and \\( PC=5 \\). If \\( O \\) is the center of the triangle \\( ABC \\), find the length of \\( PO \\). Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 32843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle of radius 3, there are many line segments of length 4 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.\nA) $9\\pi$\nB) $\\pi$\nC) $4\\pi$\nD) $13\\pi$\nE) $16\\pi$ Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 32844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could ha... | math | {
"style": "rule",
"ground_truth": "1092"
} | {
"split": "train",
"index": 32845,
"solution": "Note that 2 and 9 are equivalent $\\bmod 7$. So we will replace the 9 with a 2 for now. Since 7 is a divisor of 21, a four digit multiple of 7 consisting of $2,0,1$, and 2 cannot have a 2 followed by a 1 (otherwise we could subtract a multiple of 21 to obtain a num... |
deepscaler | [
{
"role": "user",
"content": "Brand Z juice claims, \"We offer 30% more juice than Brand W at a price that is 15% less.\" What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{26}"
} | {
"split": "train",
"index": 32846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all nonnegative integers $a, b, c$ such that\n$$\\sqrt{a} + \\sqrt{b} + \\sqrt{c} = \\sqrt{2014}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 2014)"
} | {
"split": "train",
"index": 32847,
"solution": "\nGiven the equation: \n\n\\[\n\\sqrt{a} + \\sqrt{b} + \\sqrt{c} = \\sqrt{2014}\n\\]\n\nwhere \\(a\\), \\(b\\), and \\(c\\) are nonnegative integers, we need to find all possible combinations of \\((a, b, c)\\) that satisfy this condition.\n\n### Step 1: Analyze th... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the smallest subset of the integers with the property that $0 \\in S$ and for any $x \\in S$, we have $3 x \\in S$ and $3 x+1 \\in S$. Determine the number of non-negative integers in $S$ less than 2008. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 32848,
"solution": "Write the elements of $S$ in their ternary expansion (i.e. base 3 ). Then the second condition translates into, if $\\overline{d_{1} d_{2} \\cdots d_{k}} \\in S$, then $\\overline{d_{1} d_{2} \\cdots d_{k} 0}$ and $\\overline{d_{1} d_{2} \\cdots d_{k} 1}$ are also ... |
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