data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the greatest value of $a$ such that $a^2-10a+21 \\le 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 36549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the integrals:\n1) $\\int_{0}^{1} x e^{-x} \\, dx$\n2) $\\int_{1}^{2} x \\log_{2} x \\, dx$\n3) $\\int_{1}^{e} \\ln^{2} x \\, dx$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e - 2"
} | {
"split": "train",
"index": 36550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn triangle \\( \\triangle ABC \\), \\( AB = BC = 2 \\) and \\( AC = 3 \\). Let \\( O \\) be the incenter of \\( \\triangle ABC \\). If \\( \\overrightarrow{AO} = p \\overrightarrow{AB} + q \\overrightarrow{AC} \\), find the value of \\( \\frac{p}{q} \\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 36551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sixty percent of a plane's passengers are women and ten percent of those women are in first class. What is the number of women in first class if the plane is carrying 200 passengers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$ . Find the greatest real number $m$ , for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true f... | math | {
"style": "rule",
"ground_truth": "7/3"
} | {
"split": "train",
"index": 36553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that \\( x = 2a^{5} = 5b^{2} > 0 \\), where \\( a \\) and \\( b \\) are integers. What is the smallest possible value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200000"
} | {
"split": "train",
"index": 36554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $4\\sin A\\sin B-4\\cos ^{2} \\frac {A-B}{2}= \\sqrt {2}-2$.\n\n$(1)$ Find the magnitude of angle $C$; \n$(2)$ Given $\\frac {a\\sin B}{\\sin A}=4$ and the area o... | math | {
"style": "rule",
"ground_truth": "c=4"
} | {
"split": "train",
"index": 36555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $${0.027}^{− \\frac{1}{3}}−{(− \\frac{1}{7})}^{−2}+{256}^{ \\frac{3}{4}}−{3}^{−1}+{( \\sqrt{2}−1)}^{0} = $$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 36556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least common multiple of 36 and 132. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 36557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the following product:\n$$\\frac{1}{3}\\times9\\times\\frac{1}{27}\\times81\\times\\frac{1}{243}\\times729\\times\\frac{1}{2187}\\times6561\\times\\frac{1}{19683}\\times59049.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 36558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club has 20 members and needs to choose 2 members to be co-presidents and 1 member to be a treasurer. In how many ways can the club select its co-presidents and treasurer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3420"
} | {
"split": "train",
"index": 36559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an international mathematics conference in 2024, a puzzle competition involves finding distinct positive integers $A$, $B$, and $C$ such that the product $A\\cdot B\\cdot C = 2401$. Determine the largest possible value of the sum $A+B+C$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 36560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $k \\times k$ array contains each of the numbers $1, 2, \\dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \\in \\mathbb{N}^+$)? Let's think step by... | math | {
"style": "rule",
"ground_truth": "3^{n+1} - 1"
} | {
"split": "train",
"index": 36561,
"solution": "\nConsider a \\( k \\times k \\) array, where \\( k = 3^n \\) for a positive integer \\( n \\). The array contains each of the integers \\( 1, 2, \\ldots, m \\) exactly once, and the remaining entries are all zeros. We are tasked with finding the smallest possible ... |
deepscaler | [
{
"role": "user",
"content": "Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \\frac{9}{2} = 0$ is true for all values of $z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 36562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enem... | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 36563,
"solution": "We count the number of MIT-Harvard friendships. Each of the $m$ MIT students has 8 Harvard friends, for a total of $8 m$ friendships. Each of the $h$ Harvard students has $m-7$ MIT friends, for a total of $h(m-7)$ friendships. So, $8 m=h(m-7) \\Longrightarrow m h-8... |
deepscaler | [
{
"role": "user",
"content": "Two teams, Team A and Team B, are playing in a basketball finals series that uses a \"best of seven\" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as \"home, home, away,... | math | {
"style": "rule",
"ground_truth": "0.18"
} | {
"split": "train",
"index": 36564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant, $f(α)= \\frac {\\sin (π-α)\\cdot \\cos (2π-α)\\cdot \\tan (-α-π)}{\\tan (-α )\\cdot \\sin (-π -α)}$.\n(1) Simplify $f(α)$;\n(2) If $\\cos (α- \\frac {3}{2}π)= \\frac {1}{5}$, find the value of $f(α)$;\n(3) If $α=-1860^{\\circ}$,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 36565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer \\( n \\) with 1000 digits, none of which are 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "111...111211221122112211221122112211221122112211221122112211221122112211221122112"
} | {
"split": "train",
"index": 36566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a lottery game, the host randomly selects one of the four identical empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 36567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, $\\angle A$, $\\angle B$, and $\\angle C$ are right angles. If $\\angle AEB = 40^\\circ$ and $\\angle BED = \\angle BDE$, then $\\angle CDE =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95^\\circ"
} | {
"split": "train",
"index": 36568,
"solution": "1. **Identify the properties of triangle BED**: Given that $\\angle BED = \\angle BDE$ and $\\angle B = 90^\\circ$, triangle BED is an isosceles right triangle. By the triangle angle sum property, the sum of angles in any triangle is $180^\\circ$. Therefore, we can... |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\star b = ab + a + b \\) for all integers \\( a \\) and \\( b \\). Evaluate \\( 1 \\star (2 \\star (3 \\star (4 \\star \\ldots (99 \\star 100) \\ldots))) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101! - 1"
} | {
"split": "train",
"index": 36569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sqrt{2}\\sin(2\\omega x-\\frac{\\pi}{12})+1$ ($\\omega > 0$) has exactly $3$ zeros in the interval $\\left[0,\\pi \\right]$, determine the minimum value of $\\omega$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 36570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 1)(n - 3)(n - 5) \\dotsm (n - 97) < 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 36571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest va... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 36572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-3)(x+3) = 21x - 63.$$If $p > q$, what is the value of $p - q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $C$ with radius 2 has diameter $\\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. The radius of circle $D$ is three times the radius of ... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"split": "train",
"index": 36575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 4\\cos(wx+\\frac{\\pi}{6})\\sin(wx) - \\cos(2wx) + 1$, where $0 < w < 2$.\n1. If $x = \\frac{\\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.\n2. If the function $f(x)$ is increasing on the interval $[-\\frac{\\pi}{6}, \\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 36576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct triangles can be constructed by connecting three different vertices of a cube? (Two triangles are distinct if they have different locations in space.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 36577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $f(x)=x^4+ax^3+bx^2+cx+d$ has real coefficients, and $f(2i)=f(2+i)=0$. What is $a+b+c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $12_8 + 157_8.$ Express your answer in base 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "171_8"
} | {
"split": "train",
"index": 36579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36580,
"solution": "1. **Express $n$ in terms of its prime factors**: Let $n = 7^k \\cdot m$, where $m$ is an integer not divisible by 7. This ensures that all factors of 7 in $n$ are accounted for by $7^k$.\n\n2. **Use the multiplicative property of the divisor function**: The number... |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean (average) of the first $n$ positive integers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{n+1}{2}"
} | {
"split": "train",
"index": 36581,
"solution": "1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers can be calculated using the formula for the sum of an arithmetic series. The formula for the sum of the first $n$ positive integers (also known as the $n$-th tria... |
deepscaler | [
{
"role": "user",
"content": "Translate the graph of the function y = 2sin( $$\\frac {π}{3}$$ - x) - cos( $$\\frac {π}{6}$$ + x) by shifting it to the right by $$\\frac {π}{4}$$ units. Determine the minimum value of the corresponding function. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three $1 \times 1 \times 1$ cubes are joined side by side. What is the surface area of the resulting prism? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36583,
"solution": "When three $1 \times 1 \times 1$ cubes are joined together as in the diagram, the resulting prism is $3 \times 1 \times 1$. This prism has four rectangular faces that are $3 \times 1$ and two rectangular faces that are $1 \times 1$. Therefore, the surface area is $... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite series $\\sum_{k = 1}^\\infty \\frac{k^2}{2^k}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 36584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle are all integers, and the longest side is 11. Calculate the number of such triangles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 36586,
"solution": "1. **Calculate the lengths of $EB$, $ED$, $EC$, $ER$, $EP$, and $EQ$**:\n - Since $ABCD$ is a square with side $3$ cm and $AE$ is the altitude of $6$ cm, by the Pythagorean theorem in $\\triangle ABE$:\n \\[\n EB = \\sqrt{AB^2 + AE^2} = \\sqrt{3^2 + 6^2} ... |
deepscaler | [
{
"role": "user",
"content": "The area of the ring between two concentric circles is $12\\frac{1}{2}\\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 36587,
"solution": "1. **Identify the given information and the problem statement:**\n - The area of the ring between two concentric circles is $12\\frac{1}{2}\\pi$ square inches.\n - We need to find the length of a chord of the larger circle that is tangent to the smaller circle.... |
deepscaler | [
{
"role": "user",
"content": "The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\\frac{m}{18}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "555"
} | {
"split": "train",
"index": 36588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A=\\{x|1 < x < k\\}$ and the set $B=\\{y|y=2x-5, x \\in A\\}$, if $A \\cap B = \\{x|1 < x < 2\\}$, find the value of the real number $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 36589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\left\\{a_{n}\\right\\}$ with the first term $a_{1}>0$, and the following conditions:\n$$\na_{2013} + a_{2014} > 0, \\quad a_{2013} a_{2014} < 0,\n$$\n\nfind the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds... | math | {
"style": "rule",
"ground_truth": "4026"
} | {
"split": "train",
"index": 36590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Namek, each hour has 100 minutes. On the planet's clocks, the hour hand completes one full circle in 20 hours and the minute hand completes one full circle in 100 minutes. How many hours have passed on Namek from 0:00 until the next time the hour hand and the minute hand ove... | math | {
"style": "rule",
"ground_truth": "20/19"
} | {
"split": "train",
"index": 36591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, eac... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that for some $a,b,c$ we have $a+b+c = 6$, $ab+ac+bc = 5$ and $abc = -12$. What is $a^3+b^3+c^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 36593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In recent years, China's scientific and technological achievements have been remarkable. The Beidou-3 global satellite navigation system has been operational for many years. The Beidou-3 global satellite navigation system consists of 24 medium Earth orbit satellites, 3 geostati... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 36594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$? 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": 36595,
"solution": "To solve this problem, we need to first understand the positions of points $A$, $B$, $C$, and $D$ on the grid. However, the problem statement does not provide specific coordinates for these points, and the solution provided seems to assume a specific configuration ... |
deepscaler | [
{
"role": "user",
"content": "Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + \\left(b + \\frac 1b\\right)x + c = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set containing three real numbers can be represented as $\\{a, \\frac{b}{a}, 1\\}$, and also as $\\{a^2, a+b, 0\\}$. Find the value of $a^{2009} + b^{2009}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\sin (3x+ \\frac {\\pi}{3})\\cos (x- \\frac {\\pi}{6})+\\cos (3x+ \\frac {\\pi}{3})\\sin (x- \\frac {\\pi}{6})$, find the equation of one of the axes of symmetry. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{12}"
} | {
"split": "train",
"index": 36598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an equilateral triangle $ABC$ with side length of 10, a similar process of division by midpoints and shading of one of these triangles occurs. If this dividing and shading process is repeated indefinitely, and the first triangle to be shaded is the triangle involving vertex ... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 36599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\frac{1}{2}c\\sin B=(c-a\\cos B)\\sin C$. \n$(1)$ Find angle $A$; \n$(2)$ If $D$ is a point on side $AB$ such that $AD=2DB$, $AC=2$, and $BC=\\sqrt{7}$, find... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 36600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number sequence $1,1,2,3,5,8,x,21,34,55$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct non-consecutive positive integers $x$ and $y$ are factors of 48. If $x\\cdot y$ is not a factor of 48, what is the smallest possible value of $x\\cdot y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$\\lim _{x \\rightarrow \\pi} \\frac{\\sin \\left(\\frac{x^{2}}{\\pi}\\right)}{2^{\\sqrt{\\sin x+1}}-2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\ln 2}"
} | {
"split": "train",
"index": 36603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola \\( C: x^{2} = 2py \\) with \\( p > 0 \\), two tangents \\( RA \\) and \\( RB \\) are drawn from the point \\( R(1, -1) \\) to the parabola \\( C \\). The points of tangency are \\( A \\) and \\( B \\). Find the minimum area of the triangle \\( \\triangle RAB... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3}"
} | {
"split": "train",
"index": 36604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain high school physical examination for seniors, the heights (in centimeters) of 12 students are $173$, $174$, $166$, $172$, $170$, $165$, $165$, $168$, $164$, $173$, $175$, $178$. Find the upper quartile of this data set. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "173.5"
} | {
"split": "train",
"index": 36605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 4500 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19444"
} | {
"split": "train",
"index": 36606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{37}{256}"
} | {
"split": "train",
"index": 36607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A game wheel is divided into six regions labeled $A$, $B$, $C$, $D$, $E$, and $F$. The probability of the wheel stopping on region $A$ is $\\frac{1}{3}$, the probability it stops on region $B$ is $\\frac{1}{6}$, and the probability of it stopping on regions $C$ and $D$ are equa... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 36608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 1 to 6 to form the equation shown below, where different letters represent different digits, the two-digit number represented by $\\overline{A B}$ is what?\n$$\n\\overline{A B} \\times (\\overline{C D} - E) + F = 2021\n$$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the intersection of the lines given by $y=-4x$ and $y-2=12x$? Express your answer as an ordered pair, with both coordinates expressed as common fractions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(-\\frac{1}{8}, \\frac{1}{2}\\right)"
} | {
"split": "train",
"index": 36610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every dollar Ben spent on pastries, David spent $50$ cents less. Ben paid $20$ more than David. Calculate the amount Ben and David spent together in the pastry shop. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 36611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona is sequentially taking out one apple at a time from the basket. If at any point the number of green apples she has taken out is less than the number of yellow apples, and the number of yellow apples is le... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 36612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 36613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin2α + \\sinα = 0, α ∈ (\\frac{π}{2}, π)$, find the value of $\\tan(α + \\frac{π}{4})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 2"
} | {
"split": "train",
"index": 36614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Alpine School, there are 15 soccer players. Every player is enrolled in either physics or mathematics class, but not necessarily both. If 9 players are taking physics and 4 players are registered for both physics and mathematics, how many are taking mathematics? Let's think ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5)$ and the other on the sphere of radius 87 with center $(12,8,-16)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 36616,
"solution": "The distance between the two centers of the spheres can be determined via the distance formula in three dimensions: $\\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \\sqrt{14^2 + 18^2 + 21^2} = 31$. The largest possible distance would be the sum of the two ra... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\\cos B= \\frac{1}{4}$.\n(1) If $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}=4$, find the value of $b$;\n(2) If $\\sin A= \\frac{\\sqrt{6}}{4}$, find the value of... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{8}"
} | {
"split": "train",
"index": 36617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{2013 \\times 2013}{2014 \\times 2014 + 2012} = \\frac{n}{m}$ (where $m$ and $n$ are coprime natural numbers), then what is the value of $m + n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1343"
} | {
"split": "train",
"index": 36618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 36619,
"solution": "1. **Rewrite the expression using properties of exponents:** \n The given expression is $4^{16}5^{25}$. We can express $4$ as $2^2$, so:\n \\[\n 4^{16} = (2^2)^{16} = 2^{32}\n \\]\n Therefore, the expression becomes:\n \\[\n 4^{16}5^{25} = 2^{32}5^{2... |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the limit of the function:\n\n$$\\lim _{x \\rightarrow 0} \\frac{3^{5 x}-2^{x}}{x-\\sin 9 x}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8} \\ln \\frac{2}{243}"
} | {
"split": "train",
"index": 36620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a real number \\(x>1\\) satisfies \\(\\log _{2}\\left(\\log _{4} x\\right)+\\log _{4}\\left(\\log _{16} x\\right)+\\log _{16}\\left(\\log _{2} x\\right)=0\\). Compute \\(\\log _{2}\\left(\\log _{16} x\\right)+\\log _{16}\\left(\\log _{4} x\\right)+\\log _{4}\\left(\\log... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 36621,
"solution": "Let \\(A\\) and \\(B\\) be these sums, respectively. Then \\(B-A =\\log _{2}\\left(\\frac{\\log _{16} x}{\\log _{4} x}\\right)+\\log _{4}\\left(\\frac{\\log _{2} x}{\\log _{16} x}\\right)+\\log _{16}\\left(\\frac{\\log _{4} x}{\\log _{2} x}\\right) =\\log _{2}\\lef... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 36622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The density of a body is defined as the ratio of its mass to the volume it occupies. There is a homogeneous cube with a volume of $V=8 \\, m^{3}$. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change? Let's think st... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 36623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a polynomial with integer coefficients,\n\\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\\]\nfind the number of different possible rational roots of this polynomial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the side lengths of four squares with side lengths \\( 3\\sqrt{2} \\) , \\( 4\\sqrt{2} \\) , \\( 5\\sqrt{2} \\) , and \\( 6\\sqrt{2} \\) units, determine the area of the fifth square with an unknown side length. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac{1}{(\\;\\;\\;)} + \\frac{4}{(\\;\\;\\;)} + \\frac{9}{(\\;\\;\\;\\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies the functional equation $f(x) = f(x+1) - f(x+2)$ for all $x \\in \\mathbb{R}$. When $x \\in (0,3)$, $f(x) = x^2$. Express the value of $f(2014)$ using the functional equation. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-206"
} | {
"split": "train",
"index": 36628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a new operation: \\( x \\odot y = 18 + x - a \\times y \\), where \\( a \\) is a constant. For example:\n\\[ 1 \\odot 2 = 18 + 1 - a \\times 2. \\]\n\nIf \\( 2 \\odot 3 = 8 \\), then what is \\( 3 \\odot 5 \\) and \\( 5 \\odot 3 \\)? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), \\( \\angle ABC \\) is obtuse. Point \\( D \\) lies on side \\( AC \\) such that \\( \\angle ABD \\) is right, and point \\( E \\) lies on side \\( AC \\) between \\( A \\) and \\( D \\) such that \\( BD \\) bisects \\( \\angle EBC \\). Find \\( CE \\),... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store sells two suits at the same time, both priced at 168 yuan. One suit makes a 20% profit, while the other incurs a 20% loss. Calculate the net profit or loss of the store. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Initially, there are some red balls and some black balls in a box. After adding some black balls, the red balls account for one-fourth of the total number of balls. Then, after adding some red balls, the number of red balls becomes two-thirds the number of black balls. If the n... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 36632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(2 \\times 3 \\times 4)\\left(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\\right) = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 36633,
"solution": "1. **Calculate the product of the integers:** \n \\[\n 2 \\times 3 \\times 4 = 24\n \\]\n\n2. **Calculate the sum of the fractions:**\n \\[\n \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\n \\]\n To add these fractions, find a common denominator. The lea... |
deepscaler | [
{
"role": "user",
"content": "To obtain the graph of the function $y=\\cos \\left( \\frac{1}{2}x+ \\frac{\\pi}{6}\\right)$, determine the necessary horizontal shift of the graph of the function $y=\\cos \\frac{1}{2}x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 36634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\left\\{a_{n}\\right\\}$ has a period of 7 and the sequence $\\left\\{b_{n}\\right\\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying\n\\[ a_{1} = b_{1}, \\; a_{2} = b_{2}, \\; \\cdots , \\... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 36635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\\overrightarrow{a}=(0,-1,-2)$ and $\\overrightarrow{b}=(4,0,2)$, then the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$ is ______. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 36636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, ... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"split": "train",
"index": 36637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer multiples of $77$ (product of $7$ and $11$) can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0 \\leq i < j \\leq 99$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 36638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ ... | math | {
"style": "rule",
"ground_truth": "0.525"
} | {
"split": "train",
"index": 36639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(\\begin{array}{c} a \\\\ b \\\\ c \\end{array}\\) means $a+b-c$.\nFor example, \\(\\begin{array}{c} 5 \\\\ 4 \\\\ 6 \\end{array}\\) is $5+4-6 = 3$.\nThen the sum \\(\\begin{array}{c} 3 \\\\ 2 \\\\ 5 \\end{array}\\) + \\(\\begin{array}{c} 4 \\\\ 1 \\\\ 6 \\end{ar... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36640,
"solution": "1. **Interpret the given symbol**: According to the problem, the symbol \n\n \\[\n \\triangle\n \\]\n represents the operation $a + b - c$ where $a$, $b$, and $c$ are the numbers at the vertices of the triangle.\n\n2. **Apply the operation to the first tria... |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality\n\n\\[\n1 < a < b+2 < 10.\n\\]\n\n*Proposed by Lewis Chen* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 36641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 36642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{4} + \\frac{y^2}{8} = 1,\\]\nsuch that its sides are parallel to the coordinate axes. Find the area of this square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"split": "train",
"index": 36643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given lines $l_{1}$: $\\rho\\sin(\\theta-\\frac{\\pi}{3})=\\sqrt{3}$ and $l_{2}$: $\\begin{cases} x=-t \\\\ y=\\sqrt{3}t \\end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"split": "train",
"index": 36644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(\\triangle ABC\\\\) have internal angles \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) opposite to sides of lengths \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\) respectively, and it satisfies \\\\(a^{2}+c^{2}-b^{2}= \\sqrt {3}ac\\\\). \n\\\\((1)\\\\) Find the size of angle \... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 36645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy]\nsize(110);\ndraw(Arc((0,0),1,0,300));\ndraw((1,0)--(0,0)--(.5,-.5*sqrt(3)));\nlabel(\"18\",(.5,0),S); label(\"$300^\\circ$\",(0,0),NW);\n[/asy] What is the result when ... | math | {
"style": "rule",
"ground_truth": "225\\sqrt{11}"
} | {
"split": "train",
"index": 36646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A middle school cafeteria regularly purchases rice from a grain store at a price of 1500 yuan per ton. Each time rice is purchased, a transportation fee of 100 yuan is required. The cafeteria needs 1 ton of rice per day, and the storage cost for rice is 2 yuan per ton per day (... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit whole numbers are there such that the leftmost digit is a prime number, the last digit is a perfect square, and all four digits are different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 36648,
"solution": ""
} |
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