data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 peop... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 38349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $11 = x^6 + \\frac{1}{x^6}$, find the value of $x^3 + \\frac{1}{x^3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 38350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be distinct real numbers such that\n\\[\n\\begin{vmatrix} 2 & 5 & 10 \\\\ 4 & x & y \\\\ 4 & y & x \\end{vmatrix}\n= 0.\\]Find $x + y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit number \\( \\mathrm{abc} \\) divided by the sum of its digits leaves a remainder of 1. The three-digit number \\( \\mathrm{cba} \\) divided by the sum of its digits also leaves a remainder of 1. If different letters represent different digits and \\( a > c \\), th... | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 38352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 38353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation? Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 38354,
"solution": "1. **Identify the centers and radii of the original and dilated circles:**\n - Original circle: Center $A(2,2)$, Radius $2$\n - Dilated circle: Center $A'(5,6)$, Radius $3$\n\n2. **Determine the dilation factor:**\n - The dilation factor $k$ can be calculated... |
deepscaler | [
{
"role": "user",
"content": "Two positive integers differ by 8 and their product is 168. What is the larger integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 38355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of six-digit palindromes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 38356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the line $8x + 5y + c = 0$, find the value of $c$ if the product of the $x$- and $y$- intercepts is $24$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8\\sqrt{15}"
} | {
"split": "train",
"index": 38357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{2}"
} | {
"split": "train",
"index": 38358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one verte... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 38359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alec must purchase 14 identical shirts and only has $\\$130$. There is a flat $\\$2$ entrance fee for shopping at the warehouse store where he plans to buy the shirts. The price of each shirt is the same whole-dollar amount. Assuming a $5\\%$ sales tax is added to the price ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 38360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the following fractions: $\\frac{1}{12} + \\frac{2}{12} + \\frac{3}{12} + \\frac{4}{12} + \\frac{5}{12} + \\frac{6}{12} + \\frac{7}{12} + \\frac{8}{12} + \\frac{9}{12} + \\frac{65}{12} + \\frac{3}{4}$.\nA) $\\frac{119}{12}$\nB) $9$\nC) $\\frac{113}{12}$\nD)... | math | {
"style": "rule",
"ground_truth": "\\frac{119}{12}"
} | {
"split": "train",
"index": 38361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during th... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 38362,
"solution": "1. **Identify the initial and final odometer readings**: Megan's odometer initially reads $15951$, a palindrome. After $2$ hours, it shows the next higher palindrome.\n\n2. **Determine the next palindrome**: To find the smallest palindrome greater than $15951$, we ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(3\\pi + \\alpha) = \\frac{3}{5}$, find the values of $\\cos(\\alpha)$, $\\cos(\\pi + \\alpha)$, and $\\sin(\\frac{3\\pi}{2} - \\alpha)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 38363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\sqrt{75x} \\cdot \\sqrt{2x} \\cdot \\sqrt{14x}$ . Express your answer in simplest radical form in terms of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10x \\sqrt{21x}"
} | {
"split": "train",
"index": 38364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $$\\lim _{A \\rightarrow+\\infty} \\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38365,
"solution": "We prove that $$\\lim _{A \\rightarrow+\\infty} \\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x=1$$ For $A>1$ the integrand is greater than 1, so $$\\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x>\\frac{1}{A} \\int_{1}^{A} 1 \\mathrm{~d} x=\\f... |
deepscaler | [
{
"role": "user",
"content": "Circle $\\Gamma$ is the incircle of $\\triangle ABC$ and is also the circumcircle of $\\triangle XYZ$. The point $X$ is on $\\overline{BC}$, point $Y$ is on $\\overline{AB}$, and the point $Z$ is on $\\overline{AC}$. If $\\angle A=40^\\circ$, $\\angle B=60^\\circ$, and $\\ang... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 38366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x,$ $y,$ and $k$ are positive real numbers such that \\[3=k^2\\left(\\dfrac{x^2}{y^2}+\\dfrac{y^2}{x^2}\\right)+k\\left(\\dfrac{x}{y}+\\dfrac{y}{x}\\right),\\]find 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": "\\frac{-1+\\sqrt7}{2}"
} | {
"split": "train",
"index": 38367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number \\(2016 * * * * 02 *\\), each of the 5 asterisks needs to be replaced by any of the digits \\(0, 2, 4, 7, 8, 9\\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 38368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola with the equation $\\frac{x^{2}}{4} - \\frac{y^{2}}{9} = 1$, where $F\\_1$ and $F\\_2$ are its foci, and point $M$ lies on the hyperbola.\n(1) If $\\angle F\\_1 M F\\_2 = 90^{\\circ}$, find the area of $\\triangle F\\_1 M F\\_2$.\n(2) If $\\angle F\\_1 M F\\... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3}"
} | {
"split": "train",
"index": 38370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kennel with 60 dogs, 9 dogs like watermelon, 48 dogs like salmon, and 5 like both salmon and watermelon. How many dogs in the kennel will not eat either? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 38371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the modular inverse of $11$, modulo $1000$?\n\nExpress your answer as an integer from $0$ to $999$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 38372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$, where $a\\_1+a\\_2=3$, $a\\_4+a\\_5=5$.\n\n(I) Find the general term formula of the sequence.\n(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\\_n=[a\\_1]+[a\\_2]+…+[a\\_n]$. Find... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 38373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \\sqrt{15}$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{33}"
} | {
"split": "train",
"index": 38374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 38375,
"solution": "Our fraction can be written in the form $\\frac{1000 - a}{a} = \\frac{1000}{a} - 1.$ Thus the fraction is reducible when $a$ divides $1000.$ We also want $500 < a < 1000.$ By PIE, the total values of $a$ that make the fraction reducible is, \\[249 + 99 - 49 = 299.\... |
deepscaler | [
{
"role": "user",
"content": "A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?\nA) 12\nB) 15\nC) 18\nD) 20\nE) 24 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $m=(2\\sin \\omega x, \\cos^2 \\omega x - \\sin^2 \\omega x)$ and $n=(\\sqrt{3}\\cos \\omega x, 1)$, where $\\omega > 0$ and $x \\in R$. If the smallest positive period of the function $f(x)=m \\cdot n$ is $\\pi$.\n\n(1) Find the value of $\\omega$;\n\n(2) In $\\t... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 38377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 38378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"split": "train",
"index": 38379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38380,
"solution": "To determine the number of teams in the BIG N conference, we need to consider that each team plays every other team exactly once. This setup is a classic example of a round-robin tournament, where the number of games played can be calculated using the combination f... |
deepscaler | [
{
"role": "user",
"content": "A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) Let's think ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38381,
"solution": "To solve this problem, we use Burnside's Lemma, which is a powerful tool in combinatorics for counting distinct objects under group actions, such as rotations and reflections. In this case, we consider the group of rotations that can be applied to a $2 \\times 2 \\... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n \\leq 20000$ have the properties that $2n$ has 64 positive divisors including 1 and $2n$, and $5n$ has 60 positive divisors including 1 and $5n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38382,
"solution": "Suppose $n=2^{r}5^{s}p_{3}^{a_{3}}p_{4}^{a_{4}}\\cdots p_{k}^{a_{k}}$. Since $2n$ has 64 divisors and $5n$ has 60 divisors, $(r+2)(s+1)\\left(a_{3}+1\\right)\\left(a_{4}+1\\right)\\cdots\\left(a_{k}+1\\right)=64$ and $(r+1)(s+2)\\left(a_{3}+1\\right)\\left(a_{4}+1\... |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 2b$ ($a, b \\in \\mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\\infty)$. Determine the minimum value of $$\\frac{a^2 + 4b^2}{a - 2b}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain product in a shopping mall sells an average of 30 units per day, with a profit of 50 yuan per unit. To reduce inventory as quickly as possible, the mall decides to take appropriate price reduction measures. It has been found that for every 1 yuan reduction in price pe... | math | {
"style": "rule",
"ground_truth": "17.5"
} | {
"split": "train",
"index": 38384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\ln x + \\ln (ax+1) - \\frac {3a}{2}x + 1$ ($a \\in \\mathbb{R}$).\n\n(1) Discuss the intervals of monotonicity for the function $f(x)$.\n\n(2) When $a = \\frac {2}{3}$, if the inequality $xe^{x-\\frac {1}{2}} + m \\geqslant f(x)$ holds true for all ... | math | {
"style": "rule",
"ground_truth": "\\ln \\frac {2}{3}"
} | {
"split": "train",
"index": 38385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum p... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"split": "train",
"index": 38386,
"solution": "Consider the cross section of the cones and sphere by a plane that contains the two axes of symmetry of the cones as shown below. The sphere with maximum radius will be tangent to the sides of each of the cones. The center of that sphere must be on the axis of symm... |
deepscaler | [
{
"role": "user",
"content": "You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\\sqrt{z}$. If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$? Express your answer as a common fraction. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{125}"
} | {
"split": "train",
"index": 38387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When \\( N \\) takes all the values from 1, 2, 3, \\ldots, 2015, how many numbers of the form \\( 3^{n} + n^{3} \\) are divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 38388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that a parabola has vertex $\\left(\\frac{1}{4},-\\frac{9}{8}\\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. The minimum possible value of $a$ can be written in the form $\\frac{p}{q}$, where $p$ and $q$ are relatively prime posi... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38389,
"solution": "To do this, we can use the formula for the minimum (or maximum) value of the $x$ coordinate at a vertex of a parabola, $-\\frac{b}{2a}$ and equate this to $\\frac{1}{4}$. Solving, we get $-\\frac{a}{2}=b$. Enter $x=\\frac{1}{4}$ to get $-\\frac{9}{8}=\\frac{a}{16}+... |
deepscaler | [
{
"role": "user",
"content": "Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 38390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n$, $0 \\le n \\le 7$, such that \\[n \\equiv -3737 \\pmod{8}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"split": "train",
"index": 38392,
"solution": "1. **Understanding the Problem**: We need to find the smallest possible value of $n$ in the expression $a!b!c! = m \\cdot 10^n$, where $a+b+c = 2006$ and $m$ is not divisible by $10$. This means we need to minimize the number of factors of $10$ in $a!b!c!$, which i... |
deepscaler | [
{
"role": "user",
"content": "The equation\n\n$$\n(x-1) \\times \\ldots \\times(x-2016) = (x-1) \\times \\ldots \\times(x-2016)\n$$\n\nis written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that n... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 38393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \\( (\\ \\ \\ ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 38394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"split": "train",
"index": 38395,
"solution": "\nConsider the scenario where three persons need to cover a distance of 70 km within 3 hours using one double-seater motorcycle. The speeds involved are pedestrian speed of 5 km/h and motorcycle speed of 50 km/h.\n\nTo determine whether this is possible, let us cal... |
deepscaler | [
{
"role": "user",
"content": "In a sequence of triangles, each successive triangle has its small triangles numbering as square numbers (1, 4, 9,...). Each triangle's smallest sub-triangles are shaded according to a pascal triangle arrangement. What fraction of the eighth triangle in the sequence will be sha... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is randomly selected from the interval $[-π, π]$. Calculate the probability that the value of the function $y = \\cos x$ falls within the range $[-\\frac{\\sqrt{3}}{2}, \\frac{\\sqrt{3}}{2}]$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 38397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a random variable $X\\sim N(2, \\sigma ^{2})$, $P(X\\leqslant 0)=0.15$, calculate $P(2\\leqslant X\\leqslant 4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"split": "train",
"index": 38398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sixteenth and seventeenth terms of an arithmetic sequence are 8 and 10, respectively. What is the second term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 38399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f$ mapping from set $M$ to set $N$, where $M=\\{a, b, c\\}$ and $N=\\{-3, -2, -1, 0, 1, 2, 3\\}$, calculate the number of mappings $f$ that satisfy the condition $f(a) + f(b) + f(c) = 0$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 38400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coordinates of a direction vector of line $l$ are $({-1,\\sqrt{3}})$, the inclination angle of line $l$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 38401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\\\(\\alpha\\\\) is an angle in the second quadrant, and \\\\(\\sin (π+α)=- \\frac {3}{5}\\\\), find the value of \\\\(\\tan 2α\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {24}{7}"
} | {
"split": "train",
"index": 38402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously \"replace intuition with concepts.\" The function named after him, $D\\left(x\... | math | {
"style": "rule",
"ground_truth": "(1)(4)"
} | {
"split": "train",
"index": 38403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 38404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spiral staircase turns $180^\\circ$ as it rises 8 feet. The radius of the staircase is 4 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "14.9"
} | {
"split": "train",
"index": 38405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \\[(-\\tfrac32, 1), \\; (0,... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt3}{3}"
} | {
"split": "train",
"index": 38406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, and $\\overrightarrow{c}$ satisfying $\\overrightarrow{a} + \\overrightarrow{b} + 2\\overrightarrow{c} = \\overrightarrow{0}$, and the magnitudes $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=3$, $|\\overrightarrow... | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 38407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x\\%$ of five-digit numbers have at least one repeated digit, then what is $x$? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69.8"
} | {
"split": "train",
"index": 38408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 38409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other pla... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A recipe calls for $4 \\frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need? Express your answer as a mixed number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\frac{1}{4}"
} | {
"split": "train",
"index": 38411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class has a total of 54 students. Now, using the systematic sampling method based on the students' ID numbers, a sample of 4 students is drawn. It is known that students with ID numbers 3, 29, and 42 are in the sample. What is the ID number of the fourth student in the sample... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the tetrahedron \\( P-ABC \\), edges \\( PA \\), \\( AB \\), and \\( AC \\) are mutually perpendicular, and \\( PA = AB = AC \\). Let \\( E \\) and \\( F \\) be the midpoints of \\( AB \\) and \\( PC \\) respectively. Find the sine of the angle \\(\\theta\\) between \\( EF \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 38413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 38414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\\triangle PF_{1}F_{2}$ is $30^{\\circ}$, then the eccentricity o... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 38415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the natural numbers from 1 to 100, find the total number of numbers that contain a digit 7 or are multiples of 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{70}"
} | {
"split": "train",
"index": 38417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_1, b_2, \\dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \\geq 1$. Find \n\\[\n\\sum_{n=1}^\\infty \\frac{b_n}{3^{n+1}}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 38418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with sides $51, 52, 53$ . Let $\\Omega$ denote the incircle of $\\bigtriangleup ABC$ . Draw tangents to $\\Omega$ which are parallel to the sides of $ABC$ . Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A, B, C, D, E, F, G, H \\) be distinct digits from 0 to 7 that satisfy the following equation:\n\n\\[ \\overrightarrow{A B C} + \\overline{D E} = \\overline{F G H} \\]\n\nFind \\(\\overline{D E}\\) if \\(\\overline{A B C} = 146\\).\n\n(Note: \\(\\overline{A B C}\\) deno... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 38421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are integers with $a > b$, what is the smallest possible positive value of $\\frac{a+b}{a-b} + \\frac{a-b}{a+b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 38422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\sqrt{2}"
} | {
"split": "train",
"index": 38423,
"solution": "1. **Define the variables**: Let the side lengths of the rectangular box be $x$, $y$, and $z$.\n\n2. **Use the given total edge length**: The sum of the lengths of all the edges of the box is given by:\n \\[\n 4(x+y+z) = 48\n \\]\n Dividing both sides by 4,... |
deepscaler | [
{
"role": "user",
"content": "A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determin... | math | {
"style": "rule",
"ground_truth": "27.1"
} | {
"split": "train",
"index": 38424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many conte... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 38425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A metal weight has a mass of 20 kg and is an alloy of four metals. The mass of the first metal in this alloy is one and a half times greater than that of the second, the mass of the second metal is in the ratio $3:4$ to the mass of the third, and the mass of the third metal is ... | math | {
"style": "rule",
"ground_truth": "4.91"
} | {
"split": "train",
"index": 38426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Hong asked Da Bai: \"Please help me calculate the result of $999 \\quad 9 \\times 999 \\quad 9$ and determine how many zeros appear in it.\"\n2019 nines times 2019 nines\nDa Bai quickly wrote a program to compute it. Xiao Hong laughed and said: \"You don't need to compute ... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 38427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let vectors $\\overrightarrow{a_{1}}=(1,5)$, $\\overrightarrow{a_{2}}=(4,-1)$, $\\overrightarrow{a_{3}}=(2,1)$, and let $\\lambda_{1}, \\lambda_{2}, \\lambda_{3}$ be non-negative real numbers such that $\\lambda_{1}+\\frac{\\lambda_{2}}{2}+\\frac{\\lambda_{3}}{3}=1$. Find the m... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 38428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three Triangles: Within triangle \\(ABC\\), a random point \\(M\\) is chosen. What is the probability that the area of one of the triangles \\(ABM, BCM,\\) and \\(CAM\\) will be greater than the sum of the areas of the other two? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 38429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side length of the regular hexagon is 10 cm. What is the number of square centimeters in the area of the shaded region? Express your answer in simplest radical form.\n\n[asy]\nsize(100);\n\npair A,B,C,D,E,F;\nA = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240)... | math | {
"style": "rule",
"ground_truth": "100\\sqrt{3}"
} | {
"split": "train",
"index": 38430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 2$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the n... | math | {
"style": "rule",
"ground_truth": "2.3"
} | {
"split": "train",
"index": 38431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determi... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 38432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ways in which the letters in \"HMMTHMMT\" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, \"MMMMTTHH\" satisfies this property, but \"HHTMMMTM\" does not. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38433,
"solution": "The final string must consist of \"blocks\" of at least two consecutive repeated letters. For example, MMMMTTHH has a block of 4 M's, a block of 2 T's, and a block of 2 H's. Both H's must be in a block, both T's must be in a block, and all M's are either in the sam... |
deepscaler | [
{
"role": "user",
"content": "Consider an infinite grid of unit squares. An $n$-omino is a subset of $n$ squares that is connected. Below are depicted examples of 8 -ominoes. Two $n$-ominoes are considered equivalent if one can be obtained from the other by translations and rotations. What is the number of ... | math | {
"style": "rule",
"ground_truth": "3426576"
} | {
"split": "train",
"index": 38434,
"solution": "We claim that there are approximately $\\frac{3^{n-1}}{4} n$-ominoes. First, we define an order on the squares in an $n$-omino, as follows: we order the squares from left to right, and within a column, we order the squares from top to bottom. We construct an $n$-om... |
deepscaler | [
{
"role": "user",
"content": "Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \\times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \\... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A king traversed a $9 \\times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\\sqrt{2}$ and the length of a move vertica... | math | {
"style": "rule",
"ground_truth": "16 + 64 \\sqrt{2}"
} | {
"split": "train",
"index": 38436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2007$ is divided by $81$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 38437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 38438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38439,
"solution": "Let's denote the number of goldfish Brent has at month $n$ as $B_n$ and the number of goldfish Gretel has at month $n$ as $G_n$. According to the problem, Brent's goldfish quadruple every month, and Gretel's goldfish double every month. We are given that at month 0... |
deepscaler | [
{
"role": "user",
"content": "We consider a white \\( 5 \\times 5 \\) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 38440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, it is known that $BC = 1$, $\\angle B = \\frac{\\pi}{3}$, and the area of $\\triangle ABC$ is $\\sqrt{3}$. The length of $AC$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 38441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint o... | math | {
"style": "rule",
"ground_truth": "184"
} | {
"split": "train",
"index": 38442,
"solution": "Draw the heptagon whose vertices are the midpoints of octagon $ABCDEFGH$ except $J$. We have a homothety since:\n 1. $J$ passes through corresponding vertices of the two heptagons.\n 2. By centroid properties, our ratio between the sidelengths is $\\frac{2}{3},$ an... |
deepscaler | [
{
"role": "user",
"content": "Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10989"
} | {
"split": "train",
"index": 38443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The radii of the excircles of a triangle are 3, 10, and 15 respectively. What is the perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 38445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product $1! \\cdot 2! \\cdot 3! \\cdots \\cdots \\cdot 99! \\cdot 100!$ has ___ consecutive zeros at the end. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1124"
} | {
"split": "train",
"index": 38446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\( 444 - 44 - 4 \\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 38447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( S = \\{1, 2, \\cdots, 2005\\} \\). If any subset of \\( S \\) containing \\( n \\) pairwise coprime numbers always includes at least one prime number, find the minimum value of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38448,
"solution": ""
} |
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