data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given square $ABCD$, points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ bisect $\\angle ADC$. Calculate the ratio of the area of $\\triangle DEF$ to the area of square $ABCD$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 19449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are then placed on top of the grid. Five of the circles have a diameter equal to a side of a small square (1 cm), and one large circle has a diameter equal to the length of ... | math | {
"style": "rule",
"ground_truth": "41.25"
} | {
"split": "train",
"index": 19450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\\frac {1}{25}$ of the probability of five heads and three tails. ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19451,
"solution": "Rewrite it as : $(P)^3$$(1-P)^5=\\frac {1}{25}$ $(P)^5$$(1-P)^3$\n This can be simplified as $24P^2 -50P + 25 = 0$\n This can be factored into $(4P-5)(6P-5)$\n This yields two solutions: $\\frac54$ (ignored because it would result in $1-p<0$ ) or $\\frac56$\n There... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b+2$ ($k \\neq 0$) intersects the positive half of the x-axis at point A and the positive half of the y-axis at point B.\n(1) Express the area $S_{\\triangle AOB}$ of triangle $AOB$ in terms of $b$... | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{10}"
} | {
"split": "train",
"index": 19452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days ... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 19453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \\times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making ... | math | {
"style": "rule",
"ground_truth": "5 + 4\\sqrt{2} + \\sqrt{5}"
} | {
"split": "train",
"index": 19454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{3}"
} | {
"split": "train",
"index": 19455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$ and $k$, let $\\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\\sum_{n=1}^{\\infty} \\sum_{k=1}^{\\infty} \\frac{\\mho(n, k)}{3^{n+k-7}}$$ Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "167"
} | {
"split": "train",
"index": 19456,
"solution": "A prime $p$ is counted in $\\mho(n, k)$ if $p \\mid n$ and $k \\leq p$. Thus, for a given prime $p$, the total contribution from $p$ in the sum is $$3^{7} \\sum_{m=1}^{\\infty} \\sum_{k=1}^{p} \\frac{1}{3^{p m+k}}=3^{7} \\sum_{i \\geq p+1} \\frac{1}{3^{i}}=\\frac{3... |
deepscaler | [
{
"role": "user",
"content": "A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all solutions to\n\\[\\sqrt{x + 2 - 2 \\sqrt{x - 4}} + \\sqrt{x + 12 - 8 \\sqrt{x - 4}} = 4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[13]"
} | {
"split": "train",
"index": 19458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\left\\{a_n\\right\\}$ is an arithmetic sequence, if $\\dfrac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, calculate the maximum value of $n$ for which $S_n > 0$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 19459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\left(m+1\\right)x^{|m|}+n-3$ with respect to $x$:<br/>$(1)$ For what values of $m$ and $n$ is the function a linear function of $x$?<br/>$(2)$ For what values of $m$ and $n$ is the function a proportional function of $x$? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "n=3"
} | {
"split": "train",
"index": 19460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $\\cos A= \\frac {3}{5},\\cos B= \\frac {5}{13}$, and $AC=3$. Find the length of $AB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {14}{5}"
} | {
"split": "train",
"index": 19461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya has a stick that is 22 cm long. He wants to break it into three parts with integer lengths and form a triangle from the resulting pieces. How many ways can he do this? (Methods that result in identical triangles are considered the same). Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the length of a diagonal of a square is $a + b$, then the area of the square is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}(a+b)^2"
} | {
"split": "train",
"index": 19463,
"solution": "1. **Identify the relationship between the diagonal and the side of the square:**\n Let the side of the square be denoted as $s$. According to the properties of a square, its diagonal splits the square into two right-angled isosceles triangles. Therefore, using t... |
deepscaler | [
{
"role": "user",
"content": "How many sequences of integers $(a_{1}, \\ldots, a_{7})$ are there for which $-1 \\leq a_{i} \\leq 1$ for every $i$, and $a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}+a_{4} a_{5}+a_{5} a_{6}+a_{6} a_{7}=4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 19464,
"solution": "For $i=1,2, \\ldots, 6$, let $b_{i}=a_{i} a_{i+1}$. From the problem condition each of $b_{1}, b_{2}, \\ldots, b_{6}$ can only be $-1,0$, or 1 . Since the sum of these six numbers is 4 , either there are five 1 s and a -1 or there are four 1 s and two 0s. In the fi... |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( x, y, z \\) are such that \\( \\operatorname{GCD}(\\operatorname{LCM}(x, y), z) \\cdot \\operatorname{LCM}(\\operatorname{GCD}(x, y), z) = 1400 \\).\n\nWhat is the maximum value that \\( \\operatorname{GCD}(\\operatorname{LCM}(x, y), z) \\) can take? Let's t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $r$, the quadratics $x^{2}+(r-1)x+6$ and $x^{2}+(2r+1)x+22$ have a common real root. The sum of the possible values of $r$ can be expressed as $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 19466,
"solution": "Let the common root be $s$. Then, $$s^{2}+(r-1)s+6=s^{2}+(2r+1)s+22$$ and $s=-\\frac{16}{r+2}$. Substituting this into $s^{2}+(r-1)s+6=0$ yields $$\\frac{256}{(r+2)^{2}}-\\frac{16(r-1)}{r+2}+6=0$$ After multiplying both sides by $(r+2)^{2}$, the equation becomes $$... |
deepscaler | [
{
"role": "user",
"content": "Calculate the infinite sum:\n\\[\n\\sum_{n=1}^\\infty \\frac{n^3 - n}{(n+3)!}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 19467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line segment begins at $(1, 3)$. It is 5 units long and ends at the point $(4, x)$ and $x > 0$. 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": "7"
} | {
"split": "train",
"index": 19468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero di... | math | {
"style": "rule",
"ground_truth": "376"
} | {
"split": "train",
"index": 19469,
"solution": "There are $\\binom{8}{5}$ ways to choose the rings, and there are $5!$ distinct arrangements to order the rings [we order them so that the first ring is the bottom-most on the first finger that actually has a ring, and so forth]. The number of ways to distribute th... |
deepscaler | [
{
"role": "user",
"content": "Let the complex number \\( z = \\cos \\frac{2\\pi}{13} + i \\sin \\frac{2\\pi}{13} \\). Find the value of \\( \\left(z^{-12} + z^{-11} + z^{-10}\\right)\\left(z^{3} + 1\\right)\\left(z^{6} + 1\\right) \\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving point P in the plane, the difference between the distance from point P to point F(1, 0) and the distance from point P to the y-axis is equal to 1.\n(Ⅰ) Find the equation of the trajectory C of the moving point P;\n(Ⅱ) Draw two lines $l_1$ and $l_2$ through point ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \\cdots - 59 + 63 - 67 + 71$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"split": "train",
"index": 19472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 5 on the tenths place is \\_\\_\\_\\_\\_ more than the 5 on the hundredths place. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.45"
} | {
"split": "train",
"index": 19473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is along side $EG$ of the triangle, and side $AB$ is now one-third the length of side $AD$. The altitude from $F$ to side $EG$ is 12 inches, and side $EG$ is 15 inches. Determine the area of re... | math | {
"style": "rule",
"ground_truth": "\\frac{10800}{289}"
} | {
"split": "train",
"index": 19474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with an eccentricity of $\\frac{\\sqrt{3}}{2}$ and a length of the minor axis of $4$. <br/>$(1)$ Find the equation of the ellipse; <br/>$(2)$ A chord passing through $P(2,1)$ divides $P$ in half. Find the equati... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 19475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights on the island than liars?\". They answered in turn such that ... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 19476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{1, a, \\frac{b}{a}\\} = \\{0, a^2, a+b\\}$, 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": 19477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 19478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all roots of the polynomial $x^3-5x^2+3x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than once, enter it as many times as its multiplicity. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1,3,3"
} | {
"split": "train",
"index": 19479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly $12$ positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 19480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product $(8)(888\\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 19481,
"solution": "\n1. **Understanding the Problem:**\n We need to find the value of $k$ such that the product of $8$ and a number consisting of $k$ eights, i.e., $(8)(888\\ldots8)$, results in a number whose digits sum up to $1000$.\n\n2. **Exploring the Pattern:**\n Let's exam... |
deepscaler | [
{
"role": "user",
"content": "Find the real root of the equation \\[\\sqrt{x} + \\sqrt{x+2} = 10.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2401}{100}"
} | {
"split": "train",
"index": 19482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fifty cards, numbered 1-50, are placed in a box. One card is randomly selected. What is the probability that the number on the card is prime and is a multiple of 7? 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}{50}"
} | {
"split": "train",
"index": 19483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"split": "train",
"index": 19484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs $(a, b)$ of positive integers satisfy $a^{2}+b^{2}=50$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19485,
"solution": "Since $b$ is a positive integer, then $b^{2} \\geq 1$, and so $a^{2} \\leq 49$, which gives $1 \\leq a \\leq 7$, since $a$ is a positive integer. If $a=7$, then $b^{2}=50-7^{2}=1$, so $b=1$. If $a=6$, then $b^{2}=50-6^{2}=14$, which is not possible since $b$ is an ... |
deepscaler | [
{
"role": "user",
"content": "Let $A, B, C$ be unique collinear points $ AB = BC =\\frac13$ . Let $P$ be a point that lies on the circle centered at $B$ with radius $\\frac13$ and the circle centered at $C$ with radius $\\frac13$ . Find the measure of angle $\\angle PAC$ in degrees. Let's thin... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 19486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of the ellipse $\\frac{x^2}{9}+\\frac{y^2}{7}=1$, $A$ is a point on the ellipse, and $\\angle AF_{1}F_{2}=45^{\\circ}$, calculate the area of triangle $AF_{1}F_{2}$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 19487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the ou... | math | {
"style": "rule",
"ground_truth": "998"
} | {
"split": "train",
"index": 19488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\sqrt{123454321}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111"
} | {
"split": "train",
"index": 19489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Given real numbers $x \\gt 0$, $y \\gt 0$, and $\\frac{2}{x}+y=1$, find the minimum value of $2x+\\frac{1}{3y}$;<br/>$(2)$ Given real numbers $x$ and $y$ satisfying $x+y=1$, $x \\gt 0$, $y \\gt 0$, find the minimum value of $\\frac{1}{2x}+\\frac{x}{y+1}$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 19490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, angle $B$ measures $110^\\circ$. What is the number of degrees in the measure of angle $C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70^\\circ"
} | {
"split": "train",
"index": 19491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:\n(i) Less than $12$ employees have a more difficult work;\n(ii) At least $30$ employees take a higher salary.\nAssuming... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 19492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 19493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of m... | math | {
"style": "rule",
"ground_truth": "618"
} | {
"split": "train",
"index": 19494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 19495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000. What is this last integer? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 19496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the distance between the points $(5, -3)$ and $(-7, 4)$. Additionally, determine if the points $(5, -3)$, $(-7, 4)$, and $(5, 4)$ form a right triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{193}"
} | {
"split": "train",
"index": 19497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{9}$ of 60 is 5, what is $\\frac{1}{20}$ of 80? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 19498,
"solution": "In base 15, 6."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\\overline{AC}$ such that $\\angle{ABP} =\n45^{\\circ}$. Given that $AP = 1$ and $CP = 2$, compute the area of $ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"split": "train",
"index": 19499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the numerator and the denominator when $0.\\overline{009}$ is expressed as a fraction in lowest terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 19500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl and André are running a race. Carl runs at a constant speed of $x \\mathrm{~m} / \\mathrm{s}$. André runs at a constant speed of $y \\mathrm{~m} / \\mathrm{s}$. Carl starts running, and then André starts running 20 s later. After André has been running for 10 s, he catches... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 19501,
"solution": "André runs for 10 seconds at a speed of $y \\mathrm{~m} / \\mathrm{s}$. Therefore, André runs $10y \\mathrm{~m}$. Carl runs for 20 seconds before André starts to run and then 10 seconds while André is running. Thus, Carl runs for 30 seconds. Since Carl runs at a sp... |
deepscaler | [
{
"role": "user",
"content": "Find the probability that the chord $\\overline{AB}$ does not intersect with chord $\\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 19502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle $ABC$ , $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \\cdot |DC|$ . Determine all possible values of the perimeter of the triangle $ABC$ . Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 19503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can a committee of three people be formed if the members are to be chosen from four married couples? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 19504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah baked 4 dozen pies for a community fair. Out of these pies:\n- One-third contained chocolate,\n- One-half contained marshmallows,\n- Three-fourths contained cayenne pepper,\n- One-eighth contained walnuts.\n\nWhat is the largest possible number of pies that had none of th... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 19506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $k$ so that\n\\[3 + \\frac{3 + k}{4} + \\frac{3 + 2k}{4^2} + \\frac{3 + 3k}{4^3} + \\dotsb = 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 19507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{\\sqrt{32}}{\\sqrt{16}-\\sqrt{2}}$. The answer can be written as $\\frac{A\\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Find the minimum possible value o... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 19508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the asymptotes of the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$ and the axis of the parabola $x^{2} = 4y$ form a triangle with an area of $2$, calculate the eccentricity of the hyperbola. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 19509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron O-ABC, where $\\angle BOC=90^\\circ$, $OA \\perpendicular$ plane BOC, and $AB= \\sqrt{10}$, $BC= \\sqrt{13}$, $AC= \\sqrt{5}$. Points O, A, B, and C are all on the surface of sphere S. Find the surface area of sphere S. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "14\\pi"
} | {
"split": "train",
"index": 19510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $(1,-5)$ and $(11,7)$ are the opposite vertices of a parallelogram. What are the coordinates of the point where the diagonals of the parallelogram intersect? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(6,1)"
} | {
"split": "train",
"index": 19511,
"solution": ""
} |
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": "453,600"
} | {
"split": "train",
"index": 19512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \\cdot q \\cdot s(r) = p\\cdot s(q) \\cdot r = s (p) \\cdot q \\cdot r.$$ Find all such numbers $p, q, r$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "(12, 36, 48), (21, 63, 84)"
} | {
"split": "train",
"index": 19513,
"solution": "\nTo find the numbers \\( p, q, r \\) that satisfy the given conditions, we follow these steps:\n\nStart by examining the conditions provided in the problem:\n1. \\( p \\cdot q \\cdot s(r) = p \\cdot s(q) \\cdot r \\)\n2. \\( p \\cdot s(q) \\cdot r = s(p) \\cdot q ... |
deepscaler | [
{
"role": "user",
"content": "The sum of 100 numbers is 1000. The largest of these numbers was doubled, while another number was decreased by 10. After these actions, the sum of all numbers remained unchanged. Find the smallest of the original numbers. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $60\\%$ of students like dancing and the rest dislike it, $80\\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students w... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 19515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We can write\n\\[\\sum_{k = 1}^{100} (-1)^k \\cdot \\frac{k^2 + k + 1}{k!} = \\frac{a}{b!} - c,\\]where $a,$ $b,$ and $c$ are positive integers. Find the smallest possible value of $a + b + c.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 19516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $40\\%$ of the birds were geese, $20\\%$ were swans, $15\\%$ were herons, and $25\\%$ were ducks, calculate the percentage of the birds that were not ducks and were geese. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53.33\\%"
} | {
"split": "train",
"index": 19517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the function \\(f(n)\\) on the positive integers such that \\(f(f(n)) = 3n\\) and \\(f(3n + 1) = 3n + 2\\) for all positive integers \\(n\\). Find \\(f(729)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 19518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set\n\\[ S=\\{1, 2, \\cdots, 12\\}, \\quad A=\\{a_{1}, a_{2}, a_{3}\\} \\]\nwhere \\( a_{1} < a_{2} < a_{3}, \\quad a_{3} - a_{2} \\leq 5, \\quad A \\subseteq S \\). Find the number of sets \\( A \\) that satisfy these conditions. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 19519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $c=\\frac{7}{2}$, the area of $\\Delta ABC$ is $\\frac{3\\sqrt{3}}{2}$, and $\\tan A+\\tan B=\\sqrt{3}(\\tan A\\tan B-1)$,\n\n(1) Find the measure of angle $C$;\n\n(2) Find t... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"split": "train",
"index": 19520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A robot invented a cipher for encoding words: it replaced certain letters of the alphabet with one-digit or two-digit numbers, using only the digits 1, 2, and 3 (different letters were replaced with different numbers). Initially, it encoded itself: ROBOT = 3112131233. After enc... | math | {
"style": "rule",
"ground_truth": "2232331122323323132"
} | {
"split": "train",
"index": 19521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point in coordinate space, where all the coordinates of $P$ are positive. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\\alpha,$ $\\beta,$ and $\\gamma,$ respectively. If $\\cos \\alpha = \\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{191}}{15}"
} | {
"split": "train",
"index": 19522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 19523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For functions $f(x)$ and $g(x)$, let $m\\in \\{x|f(x)=0\\}$, $n\\in \\{x|g(x)=0\\}$. If there exist $m$ and $n$ such that $|m-n|\\leqslant 1$, then $f(x)$ and $g(x)$ are called \"zero-point related functions\". If the functions $f(x)=e^{x-2}+\\ln(x-1)-1$ and $g(x)=x(\\ln x-ax)-... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a$ and $b$ such that\n\\[ {a^2+b\\over b^2-a}\\quad\\mbox{and}\\quad{b^2+a\\over a^2-b} \\]\nare both integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,2)(3,3)(1,2)(2,1)(2,3)(3,2)"
} | {
"split": "train",
"index": 19525,
"solution": "\nWe are tasked with finding all positive integer pairs \\((a, b)\\) such that both expressions\n\n\\[\n\\frac{a^2 + b}{b^2 - a}\n\\]\n\nand\n\n\\[\n\\frac{b^2 + a}{a^2 - b}\n\\]\n\nare integers. \n\n### Analysis and Approach\n\nTo solve this problem, we'll start b... |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) \\) satisfy the following conditions:\n(i) If \\( x > y \\), and \\( f(x) + x \\geq w \\geq f(y) + y \\), then there exists a real number \\( z \\in [y, x] \\), such that \\( f(z) = w - z \\);\n(ii) The equation \\( f(x) = 0 \\) has at least one soluti... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 19526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the su... | math | {
"style": "rule",
"ground_truth": "122"
} | {
"split": "train",
"index": 19527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of 6-digit numbers composed of the digits 0, 1, 2, 3, 4, 5 without any repetition and with alternating even and odd digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{F}$ be the set of all functions $f : (0,\\infty)\\to (0,\\infty)$ such that $f(3x) \\geq f( f(2x) )+x$ for all $x$ . Find the largest $A$ such that $f(x) \\geq A x$ for all $f\\in\\mathcal{F}$ and all $x$ . Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 19529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\alpha = \\frac{3}{5}$ and $\\cos (\\alpha - \\beta) = \\frac{12}{13}$, where $0 < \\alpha < \\beta < \\frac{\\pi}{2}$, determine the value of $\\sin \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 19530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Roll three dice once each, and let event A be \"the three numbers are all different,\" and event B be \"at least one 1 is rolled.\" Then the conditional probabilities P(A|B) and P(B|A) are respectively ( ). 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": 19531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function g defined on the set of positive rational numbers by g(x \\cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \\frac{25}{21}. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 19532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19533,
"solution": "We first show that $M+N \\geq 16$. Consider the point right before the last table is occupied. We have two cases: first suppose there exists at least one open counter seat. Then, every table must contribute at least 3 to the value of $M+N$, because no groups of 1 w... |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \\neq 0.$ Find all possible values of\n\\[\n\\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 19534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 19535,
"solution": "This is equal to the number of lattice paths from $(0,0)$ to $(7,3)$ that use only rightward and diagonal (upward+rightward) steps plus the number of lattice paths from $(0,0)$ to $(7,2)$ that use only rightward and diagonal steps, which is equal to the number of p... |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.298"
} | {
"split": "train",
"index": 19536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, \\dots, a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1 + a_2 + \\cdots + a_{2018} = 2018^{2018}$. What is the remainder when $a_1^3 + a_2^3 + \\cdots + a_{2018}^3$ is divided by $6$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19537,
"solution": "1. **Sum of Cubes Relation**: We start by expanding the cube of the sum of the sequence:\n \\[\n (a_1 + a_2 + \\cdots + a_{2018})^3 = a_1^3 + a_2^3 + \\cdots + a_{2018}^3 + 3\\sum_{i=1}^{2018} a_i^2 \\left(\\sum_{j=1}^{2018} a_j - a_i\\right) + 6\\sum_{i\\neq j... |
deepscaler | [
{
"role": "user",
"content": "Define $\\varphi^{k}(n)$ as the number of positive integers that are less than or equal to $n / k$ and relatively prime to $n$. Find $\\phi^{2001}\\left(2002^{2}-1\\right)$. (Hint: $\\phi(2003)=2002$.) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1233"
} | {
"split": "train",
"index": 19538,
"solution": "$\\varphi^{2001}\\left(2002^{2}-1\\right)=\\varphi^{2001}(2001 \\cdot 2003)=$ the number of $m$ that are relatively prime to both 2001 and 2003, where $m \\leq 2003$. Since $\\phi(n)=n-1$ implies that $n$ is prime, we must only check for those $m$ relatively prime ... |
deepscaler | [
{
"role": "user",
"content": "Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 2/5 chance of giving him a positive answer. If he asks it 5 questions, what is the probability that it gives him exactly 2 positive answers? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{216}{625}"
} | {
"split": "train",
"index": 19539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sergio recently opened a store. One day, he determined that the average number of items sold per employee to date was 75. The next day, one employee sold 6 items, one employee sold 5 items, and one employee sold 4 items. The remaining employees each sold 3 items. This made the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19540,
"solution": "Suppose that there are \\( n \\) employees at Sergio's store. After his first average calculation, his \\( n \\) employees had sold an average of 75 items each, which means that a total of \\( 75n \\) items had been sold. The next day, one employee sold 6 items, on... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $ n$, let $ c(n)$ be the largest real number such that\r\n\\[ c(n) \\le \\left| \\frac {f(a) \\minus{} f(b)}{a \\minus{} b}\\right|\\]\r\nfor all triples $ (f, a, b)$ such that\r\n\r\n--$ f$ is a polynomial of degree $ n$ taking integers to integers, a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{L_n}"
} | {
"split": "train",
"index": 19541,
"solution": "\nFor each positive integer \\( n \\), let \\( c(n) \\) be the largest real number such that\n\\[\nc(n) \\le \\left| \\frac{f(a) - f(b)}{a - b} \\right|\n\\]\nfor all triples \\( (f, a, b) \\) such that:\n- \\( f \\) is a polynomial of degree \\( n \\) taking integ... |
deepscaler | [
{
"role": "user",
"content": "On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits o... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19542,
"solution": "1. **Understanding the exchange rate and the amount exchanged**: \n Isabella exchanges her U.S. dollars to Canadian dollars at a rate where $7$ U.S. dollars yield $10$ Canadian dollars. This means for every $7$ U.S. dollars, she receives $10$ Canadian dollars.\n\... |
deepscaler | [
{
"role": "user",
"content": "The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 19543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$:\n\\[\\arcsin 3x - \\arccos (2x) = \\frac{\\pi}{6}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{\\sqrt{7}}"
} | {
"split": "train",
"index": 19544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A department needs to arrange a duty schedule for the National Day holiday (a total of 8 days) for four people: A, B, C, and D. It is known that:\n\n- A and B each need to be on duty for 4 days.\n- A cannot be on duty on the first day, and A and B cannot be on duty on the same ... | math | {
"style": "rule",
"ground_truth": "700"
} | {
"split": "train",
"index": 19545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers \\(a, b, c, x, y, z\\) that satisfy the equations \\(cy + bz = a\\), \\(az + cx = b\\), and \\(bx + ay = c\\), find the minimum value of the function \n\\[ f(x, y, z) = \\frac{x^2}{1 + x} + \\frac{y^2}{1 + y} + \\frac{z^2}{1 + z}. \\] Let's think step by ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 19546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct three-digit numbers can be written with the digits $1$, $2$, $3$ and $4$ if no digit may be used more than once in a three-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 19547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\dots < \\theta_{2n} < 360$ and angles are measured in degrees. Find th... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 19548,
"solution": ""
} |
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