data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Diane has one 1-cent stamp, two identical 2-cent stamps, and so on, up to nine identical 9-cent stamps. In how many different arrangements can Diane paste exactly 10 cents worth of postage in a row across the top of an envelope? (Note, however, that simply rotating or inverting... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 30949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.3\\overline{45}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{110}"
} | {
"split": "train",
"index": 30950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 points on a coordinate plane. Let \\( N \\) be the number of triplets \\((A, B, C)\\) that satisfy the following conditions: the vertices are chosen from these 100 points, \\( A \\) and \\( B \\) have the same y-coordinate, and \\( B \\) and \\( C \\) have the sam... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"split": "train",
"index": 30951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, medians $\\overline{AM}$ and $\\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"split": "train",
"index": 30952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A table consisting of 1861 rows and 1861 columns is filled with natural numbers from 1 to 1861 such that each row contains all numbers from 1 to 1861. Find the sum of the numbers on the diagonal that connects the top left and bottom right corners of the table if the filling of ... | math | {
"style": "rule",
"ground_truth": "1732591"
} | {
"split": "train",
"index": 30953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a sequence of numbers \\(a_{1}, a_{2}, \\cdots\\) satisfies, for any positive integer \\(n\\),\n$$\na_{n}=\\frac{n^{2}+n-2-\\sqrt{2}}{n^{2}-2},\n$$\n\nthen what is the value of \\(a_{1} a_{2} \\cdots a_{2016}\\)? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2016\\sqrt{2} - 2015"
} | {
"split": "train",
"index": 30954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 30955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral \\( MNLQ \\), the angles at vertices \\( N \\) and \\( L \\) are right angles, and \\(\\operatorname{tg} \\angle QMN = \\frac{2}{3}\\). Find the diagonal \\( NQ \\), given that the side \\( LQ \\) is half the length of side \\( MN \\) and is 2 units ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"split": "train",
"index": 30956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \\frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as a common fraction in terms of $\\pi$. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{289\\pi}{4}"
} | {
"split": "train",
"index": 30957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If one-fourth of $2^{30}$ is equal to $2^x$, what is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 30958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following constraints are imposed: - Every person must get a positive integer number of shares, and all 16 shares must be given out. - No one person can have more sh... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 30959,
"solution": "We are finding the number of integer solutions to $a+b+c+d=16$ with $1 \\leq a, b, c, d \\leq 8$. We count the number of solutions to $a+b+c+d=16$ over positive integers, and subtract the number of solutions in which at least one variable is larger than 8. If at le... |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-3+5-\\left(-2\\right)$;<br/>$(2)-6÷\\frac{1}{4}×(-4)$;<br/>$(3)(\\frac{5}{6}-\\frac{3}{4}+\\frac{1}{3})×(-24)$;<br/>$(4)-1^{2023}-[4-(-3)^2]÷(\\frac{2}{7}-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 30960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "581"
} | {
"split": "train",
"index": 30961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{{\\begin{array}{l}{x=t}\\\\{y=-1+\\sqrt{3}t}\\end{array}}\\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}+1}{3}"
} | {
"split": "train",
"index": 30962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set of four-ordered real number pairs \\((a, b, c, d)\\), where \\(a, b, c, d \\in \\{0, 1, 2, 3\\}\\) and \\(a, b, c, d\\) can be the same, calculate how many such pairs exist so that \\(ad - bc\\) is odd. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 30963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular decagon, calculate the number of distinct points in the interior of the decagon where two or more diagonals intersect. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 30964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 30965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pascal has a triangle. In the $n$th row, there are $n+1$ numbers $a_{n, 0}, a_{n, 1}, a_{n, 2}, \\ldots, a_{n, n}$ where $a_{n, 0}=a_{n, n}=1$. For all $1 \\leq k \\leq n-1, a_{n, k}=a_{n-1, k}-a_{n-1, k-1}$. What is the sum of all numbers in the 2018th row? Let's think step by... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30966,
"solution": "In general, the sum of the numbers on the $n$th row will be $$\\sum_{k=0}^{n} a_{n, k}=a_{n, 0}+\\sum_{k=1}^{n-1}\\left(a_{n-1, k}-a_{n-1, k-1}\\right)+a_{n, n}=a_{n, 0}+\\left(a_{n-1, n-1}-a_{n-1,0}\\right)+a_{n, n}=2$$"
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum $2345 + 3452 + 4523 + 5234$ and then subtract $1234$ from the result. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14320"
} | {
"split": "train",
"index": 30967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square ABCD with a side length of 2, points M and N are the midpoints of sides BC and CD, respectively. If vector $\\overrightarrow {MN}$ = x $\\overrightarrow {AB}$ + y $\\overrightarrow {AD}$, find the values of xy and $\\overrightarrow {AM}$ • $\\overrightarrow {MN}$... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "4.24"
} | {
"split": "train",
"index": 30969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 30970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence $\\sqrt{2}, \\sqrt{5}, 2\\sqrt{2}, \\sqrt{11}, \\ldots$. Determine the position of $\\sqrt{41}$ in this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 30971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number \\( n \\) has a sum of digits equal to 100, while \\( 44n \\) has a sum of digits equal to 800. Find the sum of the digits of \\( 3n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 30972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 29 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 29 minutes? Let's think step ... | math | {
"style": "rule",
"ground_truth": "406"
} | {
"split": "train",
"index": 30973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parametric graph is given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{2}, \\\\\ny &= \\sin t.\n\\end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all m and n satisfying \\( 1 \\leq n \\leq m \\leq 5 \\), the polar equation \\( \\rho = \\frac{1}{1 - C_{m}^{n} \\cos \\theta} \\) represents how many different hyperbolas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square\n\n$\\begin{tabular}{|c|c|c|} \\hline 50 & \\textit{b} & \\textit{c} \\\\ \\hline \\textit{d} & \\textit{e} & \\textit{f} \\\\ \\hline \\textit{g} & \\textit{h} & 2 \\\\ \\hline \\end{tabular}$\nis a multiplicative magic square. That is, the product of the numbers in... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 30976,
"solution": "1. **Identify the products of rows, columns, and diagonals:** Given the multiplicative magic square, we know that the product of the numbers in each row, column, and diagonal must be the same. Let's denote this common product by $P$. Thus, we have:\n - $50 \\cdot... |
deepscaler | [
{
"role": "user",
"content": "For real numbers $a$ and $b$, define $a * b=(a-b)^2$. What is $(x-y)^2*(y-x)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 30977,
"solution": "1. **Understanding the operation**: Given the operation $a * b = (a-b)^2$, we need to evaluate $(x-y)^2 * (y-x)^2$.\n\n2. **Symmetry in Squaring**: Note that squaring a real number results in the same value whether the number is positive or negative. Therefore, $(-... |
deepscaler | [
{
"role": "user",
"content": "Given that the perimeter of a sector is 10cm, and its area is 4cm<sup>2</sup>, find the radian measure of the central angle $\\alpha$. 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": 30978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equations of curve C as $$\\begin{cases} x=2\\cos\\theta \\\\ y= \\sqrt {3}\\sin\\theta \\end{cases}(\\theta\\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\\beg... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}}{5}"
} | {
"split": "train",
"index": 30979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is $\frac{1}{2}$. After she wins a game, the probability ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 30980,
"solution": "We want to determine the probability that Carina wins 3 games before she loses 2 games. This means that she either wins 3 and loses 0, or wins 3 and loses 1. If Carina wins her first three games, we do not need to consider the case of Carina losing her fourth game,... |
deepscaler | [
{
"role": "user",
"content": "A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consec... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{210}"
} | {
"split": "train",
"index": 30981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The rectangle $ABCD$ below has dimensions $AB = 12 \\sqrt{3}$ and $BC = 13 \\sqrt{3}$. Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\\overline{AP}$ and $\\overline{BP}$ are joined, and the figure is then crea... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"split": "train",
"index": 30983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$$\\lim _{x \\rightarrow 0}\\left(\\frac{\\arcsin ^{2} x}{\\arcsin ^{2} 4 x}\\right)^{2 x+1}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"split": "train",
"index": 30984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a polynomial with real, nonnegative coefficients. If $f(5) = 25$ and $f(20) = 1024$, find the largest possible value of $f(10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $\\angle ACB$ as its right angle, $m\\angle ABC = 60^\\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\\overline{BP}$ to meet $\\overline{AC}$ at $D$. What is the probability that $BD > 5\\sqrt2$? Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt3}{3}"
} | {
"split": "train",
"index": 30986,
"solution": "1. **Identify the lengths of sides in triangle $ABC$**:\n Given that $\\angle ACB = 90^\\circ$ and $\\angle ABC = 60^\\circ$, triangle $ABC$ is a 30-60-90 triangle. In such triangles, the sides are in the ratio $1:\\sqrt{3}:2$. Since $AB = 10$ (hypotenuse), the o... |
deepscaler | [
{
"role": "user",
"content": "A map of the town that Annie, Barbara, and Charlie live in can be represented by the Cartesian plane. Annie is located at $(6,-20)$ and Barbara is located at $(1, 14)$. They agree to meet at the closest point that is equidistant from their current locations and walk upwards tog... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=\\cos\\omega x\\cdot \\sin(\\omega x- \\frac {\\pi}{3})+ \\sqrt {3}\\cos^{2}\\omega x- \\frac { \\sqrt {3}}{4}(\\omega＞0，x\\in\\mathbb{R})$$, and the distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $$\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac {3+2 \\sqrt {6}}{3}"
} | {
"split": "train",
"index": 30988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and $2\\sin \\frac {7\\pi}{6}\\sin \\left( \\frac {\\pi}{6}+C\\right)+\\cos C= - \\frac {1}{2}$. \n$(1)$ Find $C$; \n$(2)$ If $c= \\sqrt {13}$ and the area of $\\triangl... | math | {
"style": "rule",
"ground_truth": "\\frac {7 \\sqrt {39}}{26}"
} | {
"split": "train",
"index": 30989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(C_4+C_4^1x+C_4^2x^2+C_4^3x^3)^2$, calculate the sum of the coefficients of all terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 30990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex quadrilateral is drawn in the coordinate plane such that each of its vertices $(x, y)$ satisfies the equations $x^{2}+y^{2}=73$ and $x y=24$. What is the area of this quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 30991,
"solution": "The vertices all satisfy $(x+y)^{2}=x^{2}+y^{2}+2 x y=73+2 \\cdot 24=121$, so $x+y= \\pm 11$. Similarly, $(x-y)^{2}=x^{2}+y^{2}-2 x y=73-2 \\cdot 24=25$, so $x-y= \\pm 5$. Thus, there are four solutions: $(x, y)=(8,3),(3,8),(-3,-8),(-8,-3)$. All four of these solut... |
deepscaler | [
{
"role": "user",
"content": "Let $(p_n)$ and $(q_n)$ be sequences of real numbers defined by the equation\n\\[ (3+i)^n = p_n + q_n i\\] for all integers $n \\geq 0$, where $i = \\sqrt{-1}$. Determine the value of\n\\[ \\sum_{n=0}^\\infty \\frac{p_n q_n}{9^n}.\\]\nA) $\\frac{5}{8}$\nB) $\\frac{7}{8}$\nC) $\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 30992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two lines $l' $ and $m'$ have equations $y = -3x + 9$, and $y = -5x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l'$ will fall between $l'$ and $m'$? Express your answer as a decimal to the nearest hundredth. Let's think step by... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 30993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $P(x)$ denotes a polynomial of degree $n$ such that $P(k)=\\frac{k}{k+1}$ for $k=0,1,2,\\ldots,n$, determine $P(n+1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{n+1}{n+2}"
} | {
"split": "train",
"index": 30994,
"solution": "We are given that $P(x)$ is a polynomial of degree $n$ such that $P(k) = \\frac{k}{k+1}$ for $k = 0, 1, 2, \\ldots, n$. We need to find $P(n+1)$.\n\n1. **Using Lagrange Interpolation Formula:**\n The Lagrange Interpolation Formula for a polynomial $P(x)$ that tak... |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 30995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $QCD$ given that $Q$ is the intersection of the line through $B$ and the midpoint of $AC$ with the plane through $A, C, D$ and $N$ is the midpoint of $CD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{20}"
} | {
"split": "train",
"index": 30996,
"solution": "We place the points in the coordinate plane. We let $A=\\left(0,0, \\frac{\\sqrt{6}}{3}\\right), B=\\left(0, \\frac{\\sqrt{3}}{3}, 0\\right)$, $C=\\left(-\\frac{1}{2},-\\frac{\\sqrt{3}}{6}, 0\\right)$, and $D=\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{6}, 0\\right)$. T... |
deepscaler | [
{
"role": "user",
"content": "Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the ... | math | {
"style": "rule",
"ground_truth": "0.64"
} | {
"split": "train",
"index": 30997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of \\(\\sin \\left(-\\frac{5 \\pi}{3}\\right) + \\cos \\left(-\\frac{5 \\pi}{4}\\right) + \\tan \\left(-\\frac{11 \\pi}{6}\\right) + \\cot \\left(-\\frac{4 \\pi}{3}\\right)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} - \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 30998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ is linear in each of the $x_{i}$ and $f\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)=\\frac{1}{x_{1} x_{2} \\cdots x_{n}}$ when $x_{i} \\in\\{3,4\\}$ for all $i$. In terms of $n$, what is $f(5,5, \\ldots, 5)$? Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6^{n}}"
} | {
"split": "train",
"index": 30999,
"solution": "Let $f_{n}\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ denote the $n$-variable version of the function. We will prove that $f_{n}(5, \\ldots, 5)=\\frac{1}{6^{n}}$ by induction. The base case was done in the two previous problems. Suppose we know that $f_{n-1}(5,5, ... |
deepscaler | [
{
"role": "user",
"content": "Find the sum: $(-39) + (-37) + \\cdots + (-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-400"
} | {
"split": "train",
"index": 31000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(a, b)$ such that $a \\le 50$ is it true that $x^2 - ax + b$ has integer roots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 31001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many consecutive \"0\"s are there at the end of the product \\(5 \\times 10 \\times 15 \\times 20 \\times \\cdots \\times 2010 \\times 2015\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "398"
} | {
"split": "train",
"index": 31002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From 5 cards labeled with the numbers $1, 2, 3, 4, 5$, 3 cards are randomly drawn to form a three-digit number. How many such three-digit numbers cannot be divided by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 31003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 31004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be positive real numbers such that $a \\leq b \\leq c \\leq 2 a$. Find the maximum possible value of $$\\frac{b}{a}+\\frac{c}{b}+\\frac{a}{c}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 31005,
"solution": "Fix the values of $b, c$. By inspecting the graph of $$f(x)=\\frac{b}{x}+\\frac{x}{c}$$ we see that on any interval the graph attains its maximum at an endpoint. This argument applies when we fix any two variables, so it suffices to check boundary cases in which $b... |
deepscaler | [
{
"role": "user",
"content": "A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \\cdots < a_k $$ such that $$ \\dfrac{1}{\\sqrt{a_1}} + \\dfrac{1}{\\sqrt{a_2}} + \\cdots + \\dfrac{1}{\\sqrt{a_k}} = 1 $$ .\n\nLet $f(n)$ be the sum of the first $n$ *[good* numb... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x$, $y \\in \\mathbb{R}^{+}$ and $2x+3y=1$, find the minimum value of $\\frac{1}{x}+ \\frac{1}{y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5+2 \\sqrt{6}"
} | {
"split": "train",
"index": 31007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\\circ}$ are: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7: 23 and 7: 53"
} | {
"split": "train",
"index": 31008,
"solution": "1. **Initial Position at 7:00**: \n - The hour hand at 7:00 is at $\\frac{7}{12} \\times 360^\\circ = 210^\\circ$ from the 12 o'clock position.\n - The minute hand at 7:00 is at $0^\\circ$ (at the 12 o'clock position).\n\n2. **Relative Position of Minute Hand t... |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y= \\sqrt {x}$ at $x= \\frac {1}{4}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{4}"
} | {
"split": "train",
"index": 31009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several consecutive natural numbers. If we select 4 different numbers from them and add them together, we can obtain 385 different sums. How many such natural numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 31010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are some bullfinches in a pet store. One of the children exclaimed, \"There are more than fifty bullfinches!\" Another replied, \"Don't worry, there are fewer than fifty bullfinches.\" The mother added, \"At least there is one!\" The father concluded, \"Only one of your s... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 31011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train departs from point $A$ and travels towards point $B$ at a uniform speed. 11 minutes later, another train departs from point $B$ traveling towards point $A$ at a constant speed on the same route. After their meeting point, it takes the trains 20 minutes and 45 minutes to... | math | {
"style": "rule",
"ground_truth": "9/5"
} | {
"split": "train",
"index": 31012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two boards, one five inches wide and the other eight inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are separated, what is the area of the unpainted region on the five-inch board? (Neglect ... | math | {
"style": "rule",
"ground_truth": "40 \\sqrt{2}"
} | {
"split": "train",
"index": 31013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer $x$ such that $\\frac{x}{3}+\\frac{4}{5} < \\frac{5}{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m=\\underbrace{22222222}_{\\text{8 digits}}$ and $n=\\underbrace{444444444}_{\\text{9 digits}}$.\n\nWhat is $\\gcd(m,n)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ ar... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 31016,
"solution": "Let $OP=x$.\n Proceed as the first solution in finding that quadrilateral $EPFO$ has side lengths $OE=20$, $OF=24$, $EP=\\sqrt{x^2-20^2}$, and $PF=\\sqrt{x^2-24^2}$, and diagonals $OP=x$ and $EF=12$.\n We note that quadrilateral $EPFO$ is cyclic and use Ptolemy's t... |
deepscaler | [
{
"role": "user",
"content": "The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\times 10^{-5}"
} | {
"split": "train",
"index": 31017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31018,
"solution": "1. **Identify the polynomial and its degree**:\n Given the polynomial equation:\n \\[\n 4x^3 - 8x^2 - 63x - 9 = 0\n \\]\n This is a cubic polynomial of degree 3.\n\n2. **Simplify the polynomial**:\n Factor out the common coefficient of the highest degre... |
deepscaler | [
{
"role": "user",
"content": "1. There are 5 different books, and we need to choose 3 books to give to 3 students, one book per student. There are a total of different ways to do this.\n2. There are 5 different books, and we want to buy 3 books to give to 3 students, one book per student. There are a to... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 31019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of every third odd integer between $200$ and $500$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17400"
} | {
"split": "train",
"index": 31020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My soccer team has 16 players. I have to choose a starting lineup of a goalie and 10 regular players (the regular players are interchangeable). How many different starting lineups can I choose? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48,\\!048"
} | {
"split": "train",
"index": 31021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"split": "train",
"index": 31022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13837"
} | {
"split": "train",
"index": 31023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time? ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{25}"
} | {
"split": "train",
"index": 31024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = x^2$ and $g(x) = 3x + 4$, what is $f(g(-3))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4:00"
} | {
"split": "train",
"index": 31026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different integral solutions \\((x, y)\\) does \\(3|x| + 5|y| = 100\\) have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 31027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangular region of 2x1 unit squares at the center of a large grid of unit squares. Each subsequent ring forms around this rectangle by one unit thickness. Determine the number of unit squares in the $50^{th}$ ring around this central rectangle. Let's think step by... | math | {
"style": "rule",
"ground_truth": "402"
} | {
"split": "train",
"index": 31028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a granary records the arrival of 30 tons of grain as \"+30\", determine the meaning of \"-30\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 31029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 31030,
"solution": "We do casework on $R$, the number of red vertices. Let the cube be called $A B C D E F G H$, with opposite faces $A B C D$ and $E F G H$, such that $A$ is directly above $E$. - $\\underline{R=0}$ : There is one such coloring, which has only blue vertices. - $\\unde... |
deepscaler | [
{
"role": "user",
"content": "Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \\geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 31031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Frank's Fruit Market, 6 bananas cost as much as 4 apples, and 5 apples cost as much as 3 oranges. In addition, 4 oranges cost as much as 7 pears. How many pears cost as much as 36 bananas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of $2367 + 3672 + 6723 + 7236$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19998"
} | {
"split": "train",
"index": 31033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. What is the largest $k$ for which there exist $n \\times n$ matrices $M_1, \\dots, M_k$ and $N_1, \\dots, N_k$ with real entries such that for all $i$ and $j$, the matrix product $M_i N_j$ has a zero entry somewhere on its diagonal if and only if ... | math | {
"style": "rule",
"ground_truth": "n^n"
} | {
"split": "train",
"index": 31034,
"solution": "The largest such $k$ is $n^n$. We first show that this value can be achieved by an explicit construction. Let $e_1,\\dots,e_n$ be the standard basis of $\\RR^n$. For $i_1,\\dots,i_n \\in \\{1,\\dots,n\\}$, let $M_{i_1,\\dots,i_n}$ be the matrix with row vectors $e_... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 0^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 31036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probabil... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{702}"
} | {
"split": "train",
"index": 31037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 10 are between 9 and 101? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest p... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{27}"
} | {
"split": "train",
"index": 31039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the arithmetic mean of two numbers is $6$ and their geometric mean is $10$, then an equation with the given two numbers as roots is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 - 12x + 100 = 0"
} | {
"split": "train",
"index": 31040,
"solution": "1. **Identify the Arithmetic Mean (AM) and Geometric Mean (GM):**\n Given that the arithmetic mean of two numbers $\\eta$ and $\\zeta$ is $6$, we can write:\n \\[\n \\frac{\\eta + \\zeta}{2} = 6\n \\]\n Multiplying both sides by $2$ gives:\n \\[\n \\e... |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered triples of positive integers $(a, b, c)$ such that $6a+10b+15c=3000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4851"
} | {
"split": "train",
"index": 31041,
"solution": "Note that $6a$ must be a multiple of 5, so $a$ must be a multiple of 5. Similarly, $b$ must be a multiple of 3, and $c$ must be a multiple of 2. Set $a=5A, b=3B, c=2C$. Then the equation reduces to $A+B+C=100$. This has $\\binom{99}{2}=4851$ solutions."
} |
deepscaler | [
{
"role": "user",
"content": "Determine the constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2 - 64x + 975}{x^2 + 99x - 2200}$. What is $\\alpha+\\beta$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 31042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a sequence's sum of the first $n$ terms, $S_n = 1 - 5 + 9 - \\ldots + (-1)^{n+1}(4n - 3)$ for $n \\in \\mathbb{N}^*$, find the value of $S_{15} - S_{22} + S_{31}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 31043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cindy leaves school at the same time every day. If she cycles at \\(20 \\ \\text{km/h}\\), she arrives home at 4:30 in the afternoon. If she cycles at \\(10 \\ \\text{km/h}\\), she arrives home at 5:15 in the afternoon. Determine the speed, in \\(\\text{km/h}\\), at which she m... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)= \\begin{cases} a^{x}, x<0 \\\\ ( \\frac {1}{4}-a)x+2a, x\\geq0\\end{cases}$$ such that for any $x\\_1 \\neq x\\_2$, the inequality $$\\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is sliced into five equal-height sections by planes parallel to its base. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"split": "train",
"index": 31046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 31047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) fr... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 31048,
"solution": "1. **Calculate the number of gaps between the poles**: \n Since there are 41 poles, there are \\(41 - 1 = 40\\) gaps between the poles.\n\n2. **Calculate the total number of strides and leaps**:\n - Elmer takes 44 strides per gap, so for 40 gaps, he takes \\(44... |
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