data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle such that \\(\\frac{|BC|}{|AB| - |BC|} = \\frac{|AB| + |BC|}{|AC|}\\). Determine the ratio \\(\\angle A : \\angle C\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 : 2"
} | {
"split": "train",
"index": 31449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My grandpa has 12 pieces of art, including 4 prints by Escher and 3 by Picasso. What is the probability that all four Escher prints and all three Picasso prints will be placed consecutively? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{660}"
} | {
"split": "train",
"index": 31450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $1,000,000,000,000 - 888,777,888,777$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111,222,111,223"
} | {
"split": "train",
"index": 31451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2003"
} | {
"split": "train",
"index": 31452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alvin, Bingyi, and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games. When Alvin and Bingyi played, Alvin won \\(20\\%\\) of the game... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31453,
"solution": "Since 60 games are played and each of the 3 pairs plays the same number of games, each pair plays \\(60 \\div 3 = 20\\) games. Alvin wins \\(20\\%\\) of the 20 games that Alvin and Bingyi play, so Alvin wins \\(\\frac{20}{100} \\times 20 = \\frac{1}{5} \\times 20 =... |
deepscaler | [
{
"role": "user",
"content": "An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 31454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Greater Eighteen Hockey League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 31455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $E: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\\frac{\\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 31456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different four-digit numbers, divisible by 4, can be composed of the digits 1, 2, 3, and 4,\n\na) if each digit can occur only once?\n\nb) if each digit can occur multiple times? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 31457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction th... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each day, Jenny ate $20\\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 31459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 31460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of four different positive whole numbers is $4.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "2\\frac{1}{2}"
} | {
"split": "train",
"index": 31461,
"solution": ""
} |
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": 31462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be an even function defined on $\\mathbb{R}$, which satisfies $f(x+1) = f(x-1)$ for any $x \\in \\mathbb{R}$. If $f(x) = 2^{x-1}$ for $x \\in [0,1]$, then determine the correctness of the following statements:\n\n(1) 2 is a period of the function $f(x)$.\n(2) The fun... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What integer $n$ satisfies $0\\le n<9$ and $$-1111\\equiv n\\pmod 9~?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 20 people in my club. 8 of them are left-handed. 15 of them like jazz music. 2 of them are right-handed and dislike jazz music. How many club members are left-handed and like jazz music? Assume people are either left-handed or right-handed, but not both. Let's thi... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ratio of the area of $\\triangle BCX$ to the area of $\\triangle ACX$ in the diagram if $CX$ bisects $\\angle ACB$. Express your answer as a common fraction. [asy]\nimport markers;\n\nreal t=27/(27+30);\npair A=(-15.57,0);\npair B=(8.43,0);\npair C=(0,25.65);\npair X=t... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 31466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the perimeter of each of the nine small equilateral triangles is $6 \\mathrm{cm}$, calculate the perimeter of $\\triangle A B C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When young fishermen were asked how many fish each of them caught, the first one replied, \"I caught half the number of fish that my friend caught, plus 10 fish.\" The second one said, \"And I caught as many as my friend, plus 20 fish.\" How many fish did the fishermen catch? L... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 31468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m, n$ and $p$ are positive integers with $m+\\frac{1}{n+\\frac{1}{p}}=\\frac{17}{3}$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31469,
"solution": "Since $p$ is a positive integer, then $p \\geq 1$ and so $0<\\frac{1}{p} \\leq 1$. Since $n$ is a positive integer, then $n \\geq 1$ and so $n+\\frac{1}{p}>1$, which tells us that $0<\\frac{1}{n+\\frac{1}{p}}<1$. Therefore, $m<m+\\frac{1}{n+\\frac{1}{p}}<m+1$. Sinc... |
deepscaler | [
{
"role": "user",
"content": "Find the total number of positive four-digit integers \\( N \\) satisfying both of the following properties:\n(i) \\( N \\) is divisible by 7, and\n(ii) when the first and last digits of \\( N \\) are interchanged, the resulting positive integer is also divisible by 7. (Note th... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 31470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a taxi, one passenger can sit in the front and three in the back. In how many ways can the four passengers be seated if one of them wants to sit by the window? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD \\) to form a right dihedral angle \\( A-CD-B \\). When the length of \\( AB \\) is minimized... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 31472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is marked one quarter of the way along each side of a triangle. What fraction of the area of the triangle is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 31473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 31474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "12 + 8\\sqrt{2}"
} | {
"split": "train",
"index": 31475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\\frac{m}{n}... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 31476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $P=\\{\\mathrm{S}, \\mathrm{T}\\}$ and let $\\mathcal{P}$ be the set of all proper subsets of $P$. (A proper subset is a subset that is not the set itself.) How many ordered pairs $(\\mathcal{S}, \\mathcal{T})$ of proper subsets of $\\mathcal{P}$ are there such that (a) ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 31477,
"solution": "For ease of notation, we let $0=\\varnothing, 1=\\{\\mathrm{S}\\}, 2=\\{\\mathrm{T}\\}$. Then both $\\mathcal{S}$ and $\\mathcal{T}$ are proper subsets of $\\{0,1,2\\}$. We consider the following cases: Case 1. If $\\mathcal{S}=\\varnothing$, then $\\mathcal{S}$ is... |
deepscaler | [
{
"role": "user",
"content": "What is the least six-digit positive integer which is congruent to 7 (mod 17)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100,008"
} | {
"split": "train",
"index": 31478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( x_{1000} \\) if \\( x_{1} = 4 \\), \\( x_{2} = 6 \\), and for any natural \\( n \\geq 3 \\), \\( x_{n} \\) is the smallest composite number greater than \\( 2 x_{n-1} - x_{n-2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"split": "train",
"index": 31479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the 25th term of the arithmetic sequence $2,\n5, 8, \\ldots$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 31480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards is divided into 4 suits, with each suit containing 13 cards. Two of these suits are red, and the other two are black. The deck is shuffled, placing the cards in random order. What is the probability that the first three cards drawn from the deck are ... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{85}"
} | {
"split": "train",
"index": 31481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\\angle O_{1}PO_{2} = 90^{\\circ}$, then $AP = \\sqrt{c} + \\sqrt{d}$, where $c$ and $d$ are po... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 31482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs of integers \\((a, b)\\) with \\(1 \\leq a<b \\leq 57\\) such that \\(a^{2}\\) has a smaller remainder than \\(b^{2}\\) when divided by 57. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "738"
} | {
"split": "train",
"index": 31483,
"solution": "There are no such pairs when \\(b=57\\), so we may only consider pairs with \\(1 \\leq a<b \\leq 56\\). The key idea is that unless \\(a^{2} \\bmod 57=b^{2} \\bmod 57,(a, b)\\) can be paired with \\((57-b, 57-a)\\) and exactly one of them satisfies \\(a^{2} \\bmod ... |
deepscaler | [
{
"role": "user",
"content": "Rachelle picks a positive integer \\(a\\) and writes it next to itself to obtain a new positive integer \\(b\\). For instance, if \\(a=17\\), then \\(b=1717\\). To her surprise, she finds that \\(b\\) is a multiple of \\(a^{2}\\). Find the product of all the possible values of ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 31484,
"solution": "Suppose \\(a\\) has \\(k\\) digits. Then \\(b=a(10^{k}+1)\\). Thus \\(a\\) divides \\(10^{k}+1\\). Since \\(a \\geq 10^{k-1}\\), we have \\(\\frac{10^{k}+1}{a} \\leq 11\\). But since none of 2, 3, or 5 divide \\(10^{k}+1\\), the only possibilities are 7 and 11. The... |
deepscaler | [
{
"role": "user",
"content": "If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\sqrt{5}"
} | {
"split": "train",
"index": 31485,
"solution": "We start by simplifying the given equation:\n\\[ 4^x - 4^{x-1} = 24 \\]\n\n1. **Express $4^{x-1}$ in terms of $4^x$:**\n \\[ 4^{x-1} = \\frac{4^x}{4} = \\frac{1}{4} \\cdot 4^x \\]\n\n2. **Substitute back into the equation:**\n \\[ 4^x - \\frac{1}{4} \\cdot 4^x ... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the distinct prime factors of 315? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 31486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $\\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\\triangle ABC$ and the incenter of $\\triangle DEF... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 31487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\cos x \\cdot \\sin\\left(x + \\frac{\\pi}{3}\\right) - \\sqrt{3}\\cos^2x + \\frac{\\sqrt{3}}{4}$, where $x \\in \\mathbb{R}$.\n\n(1) Find the interval of monotonic increase for $f(x)$.\n\n(2) In an acute triangle $\\triangle ABC$, where the sides op... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 31488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $\\frac{{x}^{2}-4}{{x}^{2}-4x+4}+\\frac{x}{{x}^{2}-x}\\div \\frac{x-2}{x-1}$, then choose a value of $x$ from the integers in the range $-1\\leqslant x\\leqslant 3$ to substitute and evaluate. 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": 31489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer closest to $$\\frac{1}{\\sqrt[4]{5^{4}+1}-\\sqrt[4]{5^{4}-1}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 31490,
"solution": "Let $x=\\left(5^{4}+1\\right)^{1 / 4}$ and $y=\\left(5^{4}-1\\right)^{1 / 4}$. Note that $x$ and $y$ are both approximately 5. We have $$\\frac{1}{x-y} =\\frac{(x+y)\\left(x^{2}+y^{2}\\right)}{(x-y)(x+y)\\left(x^{2}+y^{2}\\right)}=\\frac{(x+y)\\left(x^{2}+y^{2}\\ri... |
deepscaler | [
{
"role": "user",
"content": "Given a regular triangular prism \\(ABC-A_1B_1C_1\\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \\(A_1ABC\\), \\(B_1ABC\\), and \\(C_1ABC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{36}"
} | {
"split": "train",
"index": 31491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonal \\( BD \\) of quadrilateral \\( ABCD \\) is the diameter of the circle circumscribed around this quadrilateral. Find the diagonal \\( AC \\) if \\( BD = 2 \\), \\( AB = 1 \\), and \\( \\angle ABD : \\angle DBC = 4 : 3 \\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} + \\sqrt{6}}{2}"
} | {
"split": "train",
"index": 31492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively, and the three interior angles $A$, $B$, $C$ satisfy $A+C=2B$.\n$\\text{(1)}$ If $b=2$, find the maximum value of the area of $\\triangle ABC$ and determine the... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 31493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 31494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a magic square, what is the sum \\( a+b+c \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 31495,
"solution": "Using the properties of a magic square, \\( a+b+c = 14+18+15 = 47 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the circle inscribed in a right triangle if the projections of the legs onto the hypotenuse are 9 meters and 16 meters, respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25 \\pi"
} | {
"split": "train",
"index": 31496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{2}-3x$. If for any $x_{1}$, $x_{2}$ in the interval $[-3,2]$, we have $|f(x_{1})-f(x_{2})| \\leqslant m$, then the minimum value of the real number $m$ is _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{81}{4}"
} | {
"split": "train",
"index": 31497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Liam needs to add 125 to 96 and then round the result to the nearest hundred. He plans to round the numbers before adding them. He rounds 125 to 100 and 96 to 100, resulting in an answer of $100 + 100 = 200$. Is this answer correct? What is the correct answer? Let's think step ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 31498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a game, two wheels are present. The first wheel has six segments with numbers 1 through 6. The second wheel has four segments, numbered 1, 1, 2, and 2. The game is to spin both wheels and add the resulting numbers. The player wins if the sum is a number less than 5. What is ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 31499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-eighths are there in $8\\frac{5}{3} - 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17\\frac{7}{9}"
} | {
"split": "train",
"index": 31500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**Circle $T$ has a circumference of $12\\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $45^{\\circ}$, what is the length, in inches, of segment $XZ$?** Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"split": "train",
"index": 31501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the common ratio of the infinite geometric series: $$\\frac{-3}{5}-\\frac{5}{3}-\\frac{125}{27}-\\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{9}"
} | {
"split": "train",
"index": 31502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the set consisting of all real values of $x$ such that the three numbers $2^{x}, 2^{x^{2}}, 2^{x^{3}}$ form a non-constant arithmetic progression (in that order). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\varnothing"
} | {
"split": "train",
"index": 31503,
"solution": "The empty set, $\\varnothing$. Trivially, $x=0,1$ yield constant arithmetic progressions; we show that there are no other possibilities. If these numbers do form a progression, then, by the AM-GM (arithmetic mean-geometric mean) inequality, $$2 \\cdot 2^{x^{2}}=2^{... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $16$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 31504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Mary reversed the digits of a two-digit multiplier $b$, while multiplying correctly with a positive integer $a$, and her erroneous product was $180$, determine the correct product of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 31505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\\triangle PQR$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 31506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A said: \"I am 10 years old, 2 years younger than B, and 1 year older than C.\"\nB said: \"I am not the youngest, C and I have a 3-year difference, and C is 13 years old.\"\nC said: \"I am younger than A, A is 11 years old, and B is 3 years older than A.\"\nAmong the three stat... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 31507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \\sqrt{5}c$ and $\\cos C = \\frac{3}{5}$. \n$(Ⅰ)$ Find the value of $\\sin A$. \n$(Ⅱ)$ If $b = 11$, find the area of $\\triangle ABC$. Let's... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 31508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fraction $\\frac{a}{a+27}$, where $a$ is a positive integer, equals $0.865$. What is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "173"
} | {
"split": "train",
"index": 31509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle XYZ$ have side lengths $XY=15$, $XZ=20$, and $YZ=25$. Inside $\\angle XYZ$, there are two circles: one is tangent to the rays $\\overline{XY}$, $\\overline{XZ}$, and the segment $\\overline{YZ}$, while the other is tangent to the extension of $\\overline{XY}$ be... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 31510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct, natural-number factors does $4^3 \\cdot 5^4 \\cdot 6^2$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 31511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\\in\\mathbb{N}^*$), and that $\\{a_{2n-1}\\}$ is an increasing sequence, $\\{a_{2n}\\}$ is a decreasing sequence, find the limit $$\\lim_{n\\to\\infty} \\frac{a_{2n-1}}{a_{2n}} = \\_\\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 31512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively, and it is given that $a < b < c$ and $$\\frac{a}{\\sin A} = \\frac{2b}{\\sqrt{3}}$$. \n(1) Find the size of angle $B$; \n(2) If $a=2$ and $c=3$, find the length of s... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 31513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set of data: $10.1$, $9.8$, $10$, $x$, $10.2$, the average of these data is $10$. Calculate the variance of this set of data. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.02"
} | {
"split": "train",
"index": 31514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are five positive integers that are divisors of each number in the list $$30, 90, -15, 135, 45.$$ Find the sum of these five positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 31515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $m + n$ where $m$ and $n$ are integers such that the positive difference between the roots of the equation $4x^2 - 12x - 9 = 0$ can be expressed as $\\frac{\\sqrt{m}}{n}$, with $m$ not divisible by the square of any prime number. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 31516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \\ldots + a_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2525"
} | {
"split": "train",
"index": 31517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies: \\(a_{1}=1\\), and for each \\(n \\in \\mathbf{N}^{*}\\), \\(a_{n}\\) and \\(a_{n+1}\\) are the two roots of the equation \\(x^{2}+3nx+b_{n}=0\\). Find \\(\\sum_{k=1}^{20} b_{k}\\). Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"split": "train",
"index": 31518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \\le k \\le 150$, how many $a_k$ are divisible by both 3 and 5? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the game of Mindmaster, secret codes are created by placing pegs of any of seven different colors into four slots. Colors may be repeated, and no slot may remain empty. How many secret codes are possible? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2401"
} | {
"split": "train",
"index": 31520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and th... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 31521,
"solution": "Let $Q(x) = x^2 + ax + 2$ and $R(x) = x^2 + bx + c$. We can write the following: \\[P + Q = 2x^2 + (a - 3)x - 5\\] \\[P + R = 2x^2 + (b - 3)x + (c - 7)\\] \\[Q + R = 2x^2 + (a + b)x + (c + 2)\\] Let the common root of $P+Q,P+R$ be $r$; $P+R,Q+R$ be $s$; and $P+Q,Q+... |
deepscaler | [
{
"role": "user",
"content": "At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 31522,
"solution": "1. **Identify the groups and their interactions**: We have two distinct groups at the gathering:\n - Group A: 20 people who all know each other.\n - Group B: 10 people who know no one in the gathering.\n\n2. **Calculate handshakes between Group A and Group B**:... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression: \\\\( \\frac {\\cos 40 ^{\\circ} +\\sin 50 ^{\\circ} (1+ \\sqrt {3}\\tan 10 ^{\\circ} )}{\\sin 70 ^{\\circ} \\sqrt {1+\\cos 40 ^{\\circ} }}\\\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 31523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$, $y$, and $z$ are real numbers such that\n\n$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$,\nthen $x + y + z =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31524,
"solution": "1. **Identify the equation and its implications**: We are given the equation \\((x-3)^2 + (y-4)^2 + (z-5)^2 = 0\\). Since the square of a real number is always non-negative, the sum of several non-negative numbers can only be zero if each individual number is zero.... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $23^2 + 2(23)(2) + 2^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 31525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenny wants to create all the six-letter words where the first two letters are the same as the last two letters. How many combinations of letters satisfy this property? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17576"
} | {
"split": "train",
"index": 31526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x\\_1$ is a root of the equation $x \\ln x = 2006$ and $x\\_2$ is a root of the equation $xe^{x} = 2006$, find the value of $x\\_1 \\cdot x\\_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 31527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\in \\left(0, \\frac{\\pi}{2}\\right) \\). Find the minimum value of the function \\( y = \\frac{1}{\\sin^2 x} + \\frac{12\\sqrt{3}}{\\cos x} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite directi... | math | {
"style": "rule",
"ground_truth": "9.1"
} | {
"split": "train",
"index": 31529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 31530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $\\sin A = \\frac{3}{5}$, $\\tan B = 2$. Find the value of $\\tan 2\\left(A+B\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{44}{117}"
} | {
"split": "train",
"index": 31531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices at $(-3,2),(6,-2),(3,5)$. How many square units are in the area of the triangle? 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": "25.5"
} | {
"split": "train",
"index": 31532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. It is given that $6S=a^{2}\\sin A+b^{2}\\sin B$. Express $\\cos C$ in terms of $a$, $b$, and $c$ when the maximum value of $\\frac{{a+b}}{c}$ is... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 31533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 46 ones written on the board. Each minute, Carlson 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 46 minutes? Let's think ste... | math | {
"style": "rule",
"ground_truth": "1035"
} | {
"split": "train",
"index": 31534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABC \\) be a triangle with \\( AB = 5 \\), \\( AC = 4 \\), \\( BC = 6 \\). The angle bisector of \\( \\angle C \\) intersects side \\( AB \\) at \\( X \\). Points \\( M \\) and \\( N \\) are drawn on sides \\( BC \\) and \\( AC \\), respectively, such that \\( \\overlin... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{14}}{5}"
} | {
"split": "train",
"index": 31535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( a, b, c, \\) and \\( d \\) are distinct positive integers chosen from 1 to 10 inclusive. What is the least possible value \\(\\frac{a}{b}+\\frac{c}{d}\\) could have?\n\nA) \\(\\frac{2}{10}\\)\n\nB) \\(\\frac{3}{19}\\)\n\nC) \\(\\frac{14}{45}\\)\n\nD) \\(\\frac{2... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{45}"
} | {
"split": "train",
"index": 31536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 31537,
"solution": "1. **Understanding the Ratio**: The problem states that the ratio of boys to girls in Mr. Brown's math class is $2:3$. This means for every 2 boys, there are 3 girls.\n\n2. **Setting Up Variables**: Let the number of boys be $2x$ and the number of girls be $3x$. He... |
deepscaler | [
{
"role": "user",
"content": "A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\\le i\\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\\overline{... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 31538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Calculate the value of $(1+a)(1+b)(1+c)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 31539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle? Also, determine if the triangle has any acute angles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle (right in $B$ ) inscribed in a semi-circumference of diameter $AC=10$ . Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle. Let's think step ... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"split": "train",
"index": 31541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sanatorium, there are grandmothers and grandfathers, with grandfathers being twice as many as grandmothers. The average age of the grandmothers is 5 years more than the average age of the grandfathers, and the average age of all retirees is greater than 77 years but less t... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 31542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minut... | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"split": "train",
"index": 31543,
"solution": "1. **Define the total distance and split it into two equal parts**: Let the total distance Samia had to travel be $2x$ kilometers. This means she biked for $x$ kilometers and walked for $x$ kilometers.\n\n2. **Calculate the time for each part of the journey**:\n ... |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be a point on the parabola $y = x^2 - 4x + 7,$ and let $D$ be a point on the line $y = 3x - 5.$ Find the shortest distance $CD$ and also ensure that the projection of point $C$ over line $y = 3x - 5$ lands on point $D$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{0.25}{\\sqrt{10}}"
} | {
"split": "train",
"index": 31544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(\\alpha\\\\) be an acute angle. If \\\\(\\sin \\left(\\alpha+ \\frac {\\pi}{6}\\right)= \\frac {3}{5}\\\\), then \\\\(\\cos \\left(2\\alpha- \\frac {\\pi}{6}\\right)=\\\\) ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {24}{25}"
} | {
"split": "train",
"index": 31545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\\ldots,m$ satisfying the following conditions:\n[list=1]\n[*]$B_i\\subset A,$ $|B_i|=k,$\n[*]$C_i\\... | math | {
"style": "rule",
"ground_truth": "{2^k}"
} | {
"split": "train",
"index": 31546,
"solution": "\nLet \\( A \\) be a set with \\( n \\) elements, and let \\( k < n \\) be a given positive integer. We need to find the maximum value of \\( m \\) such that it is possible to choose sets \\( B_i \\) and \\( C_i \\) for \\( i = 1, 2, \\ldots, m \\) satisfying the f... |
deepscaler | [
{
"role": "user",
"content": "For a set of five distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $M$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 31547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos \\left(a- \\frac{\\pi}{6}\\right) + \\sin a = \\frac{4 \\sqrt{3}}{5}$, find the value of $\\sin \\left(a+ \\frac{7\\pi}{6}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"split": "train",
"index": 31548,
"solution": ""
} |
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