data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to the angles \\( A \\), \\( B \\), and \\( C \\) are of lengths \\( a \\), \\( b \\), and \\( c \\) respectively. Point \\( G \\) satisfies\n$$\n\\overrightarrow{GA} + \\overrightarrow{GB} + \\overrightarrow{GC} = \\mathbf... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine 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": "9000"
} | {
"split": "train",
"index": 28350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the definite integral $\\int_{-\\sqrt{2}}^{\\sqrt{2}} \\sqrt{4-x^2}dx =$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi + 2"
} | {
"split": "train",
"index": 28351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of \n\\[\\left(\\left(\\left((3+2)^{-1}+1\\right)^{-1}+2\\right)^{-1}+1\\right)^{-1}+1.\\]\nA) $\\frac{40}{23}$\nB) $\\frac{17}{23}$\nC) $\\frac{23}{17}$\nD) $\\frac{23}{40}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40}{23}"
} | {
"split": "train",
"index": 28352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A five-digit number is called a \"pit\" if its first three digits are in descending order and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number 22222? Le... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 28353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, $b=2$, $B=\\frac{\\pi }{3}$, $\\sin 2A+\\sin (A-C)-\\sin B=0$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 28354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence $\\left\\{a_{n}\\right\\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\\frac{1}{2}$. Let $\\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\\Pi_{n}$ maximized? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $y$: $50^4 = 10^y$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.79588"
} | {
"split": "train",
"index": 28356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression $200(200-7)-(200\\cdot 200-7)$, evaluate the expression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1393"
} | {
"split": "train",
"index": 28357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself? Let's think step b... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 28358,
"solution": "Let the people be $A, B, C, D, E$ so that their heights are in that order, with $A$ the tallest and $E$ the shortest. We will do casework based on the position of $C$. - Case 1: $C$ is in the middle. Then, $B$ must be on one of the two ends, for two choices. This l... |
deepscaler | [
{
"role": "user",
"content": "Given the equations $3x + 2y = 6$ and $2x + 3y = 7$, find $14x^2 + 25xy + 14y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 28359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a regular \\(n\\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \\(m\\) steps where \\(m\\) is coprime with \\(n\\) and \\(m\\) is not a multiple of \\(3\\). How many non-similar regular 120-pointed stars ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 28360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a *trio* if one of the following is true:\n\n\n\n- A shakes hands with B, and B shakes hands with C, or\n- A doesn't shake hands with B, and B doesn't shake hand... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An up-right path from $(a, b) \\in \\mathbb{R}^{2}$ to $(c, d) \\in \\mathbb{R}^{2}$ is a finite sequence $(x_{1}, y_{1}), \\ldots,(x_{k}, y_{k})$ of points in $\\mathbb{R}^{2}$ such that $(a, b)=(x_{1}, y_{1}),(c, d)=(x_{k}, y_{k})$, and for each $1 \\leq i<k$ we have that eit... | math | {
"style": "rule",
"ground_truth": "0.2937156494680644"
} | {
"split": "train",
"index": 28362,
"solution": "Note that any up-right path must pass through exactly one point of the form $(n,-n)$ (i.e. a point on the upper-left to lower-right diagonal), and the number of such paths is $\\binom{800}{400-n}^{2}$ because there are $\\binom{800}{400-n}$ up-right paths from $(-4... |
deepscaler | [
{
"role": "user",
"content": "Given that $w$ and $z$ are complex numbers such that $|w+z|=1$ and $\\left|w^{2}+z^{2}\\right|=14$, find the smallest possible value of $\\left|w^{3}+z^{3}\\right|$. Here, $|\\cdot|$ denotes the absolute value of a complex number, given by $|a+b i|=\\sqrt{a^{2}+b^{2}}$ whenever... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{2}"
} | {
"split": "train",
"index": 28363,
"solution": "We can rewrite $\\left|w^{3}+z^{3}\\right|=|w+z|\\left|w^{2}-w z+z^{2}\\right|=\\left|w^{2}-w z+z^{2}\\right|=\\left|\\frac{3}{2}\\left(w^{2}+z^{2}\\right)-\\frac{1}{2}(w+z)^{2}\\right|$$ By the triangle inequality, $\\left|\\frac{3}{2}\\left(w^{2}+z^{2}\\right)-\\... |
deepscaler | [
{
"role": "user",
"content": "Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train? Let's think s... | math | {
"style": "rule",
"ground_truth": "233.33"
} | {
"split": "train",
"index": 28364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\\frac{1}{2}$, a circle $\\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\\sqrt{6}=0$. <br/>$(1)$ Find the equati... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{6}}{11}"
} | {
"split": "train",
"index": 28365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $c$ such that $$ \\sum_{i=1}^{n}\\sum_{j=1}^{n}(n-|i-j|)x_ix_j \\geq c\\sum_{j=1}^{n}x^2_i $$ for any positive integer $n $ and any real numbers $x_1,x_2,\\dots,x_n.$ 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": 28366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ such that $a^{2}+b^{2}+c^{2}+d^{2} \\geq a b+\\lambda b c+c d$ for all real numbers $a, b, c, d$. 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": 28367,
"solution": "Let $f(a, b, c, d)=\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)-(a b+\\lambda b c+c d)$. For fixed $(b, c, d), f$ is minimized at $a=\\frac{b}{2}$, and for fixed $(a, b, c), f$ is minimized at $d=\\frac{c}{2}$, so simply we want the largest $\\lambda$ such that $f\\left(... |
deepscaler | [
{
"role": "user",
"content": "In recent years, handicrafts made by a certain handicraft village have been very popular overseas. The villagers of the village have established a cooperative for exporting handicrafts. In order to strictly control the quality, the cooperative invites 3 experts to inspect each ... | math | {
"style": "rule",
"ground_truth": "\\frac{13100}{27}"
} | {
"split": "train",
"index": 28368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 28369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = 3x^2 + 2x + 1$, if $\\int_{-1}^{1} f(x)\\,dx = 2f(a)$, then $a = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 28370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 28371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\phi$ be the smallest acute angle for which $\\cos \\phi,$ $\\cos 2 \\phi,$ $\\cos 3 \\phi$ form an arithmetic progression, in some order. Find $\\sin \\phi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 28372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $a=2$, $c=3$, and it satisfies $(2a-c)\\cdot\\cos B=b\\cdot\\cos C$. Find the value of $\\overrightarrow{AB}\\cdot\\overrightarrow{BC}$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 28373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the odd function $f(x)$ is an increasing function defined on $\\mathbb{R}$, and the sequence $x_n$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, then the value of $x_{2011}$ is equal to. Let's think... | math | {
"style": "rule",
"ground_truth": "4003"
} | {
"split": "train",
"index": 28374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(5, 6, 7\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 28375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $4 \\times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends whe... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 28376,
"solution": "We claim that the answer is 6. On Aziraphale's first two turns, it is always possible for him to take 2 adjacent squares from the central four; without loss of generality, suppose they are the squares at $(1,1)$ and $(1,2)$. If allowed, Aziraphale's next turn will ... |
deepscaler | [
{
"role": "user",
"content": "A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=7$, find the distance from the midpoint $M$ of $AB$ to the directrix of the parabola. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 28377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=( \\sqrt {3}\\sin x,m+\\cos x)$ and $\\overrightarrow{b}=(\\cos x,-m+\\cos x)$, and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$\n(1) Find the analytical expression for the function $f(x)$;\n(2) When $x\\in\\left[-\\frac{... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 28378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(x+15)+(100x+15)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101x+30"
} | {
"split": "train",
"index": 28379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$. \n(1) Determine the range of the real number $a$ and the expression for $S_n$; \n(2) Does there exist a real number $a$ such that $S_1$, $S_3$,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{2 + 3\\sqrt{2}}"
} | {
"split": "train",
"index": 28381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's consider the number 2023. If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 28382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder has a radius of 2 inches and a height of 3 inches. What is the radius of a sphere that has the same volume as this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{9}"
} | {
"split": "train",
"index": 28383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit palindrome is defined as any four-digit natural number that has the same digit in the units place as in the thousands place, and the same digit in the tens place as in the hundreds place. How many pairs of four-digit palindromes exist whose difference is 3674? Let'... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 28384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of thes... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 28385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the limit of the numerical sequence:\n\n\\[\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{n^{2}-6 n+5}{n^{2}-5 n+5}\\right)^{3 n+2}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^{-3}"
} | {
"split": "train",
"index": 28386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold ... | math | {
"style": "rule",
"ground_truth": "1614"
} | {
"split": "train",
"index": 28387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A poll shows that $70\\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "0.189"
} | {
"split": "train",
"index": 28389,
"solution": "1. **Define the Events:**\n Let $Y$ represent the event that a voter approves of the mayor's work, and $N$ represent the event that a voter does not approve. According to the problem, the probability that a voter approves of the mayor's work is $P(Y) = 0.7$. Cons... |
deepscaler | [
{
"role": "user",
"content": "Teams A and B are playing a volleyball match. Currently, Team A needs to win one more game to become the champion, while Team B needs to win two more games to become the champion. If the probability of each team winning each game is 0.5, then the probability of Team A becoming ... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 28390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 9\\frac{2}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65\\frac{4}{5}"
} | {
"split": "train",
"index": 28391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\ln (3x)+8x+1$, find the value of $\\underset{\\Delta x\\to 0}{{\\lim }}\\,\\dfrac{f(1-2\\Delta x)-f(1)}{\\Delta x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 28392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the largest real solution to the equation\n$\\frac{3}{x-3}+\\frac{5}{x-5}+\\frac{17}{x-17}+\\frac{19}{x-19}=x^2-11x-4$\n There are positive integers $a$, $b$, and $c$ such that $m=a+\\sqrt{b+\\sqrt{c}}$. Find $a+b+c$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 28393,
"solution": "The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to $\\frac{3}{x-3}$, then the fraction becomes of the form $\\frac{x}{x - 3}$. A similar cancellation happens with the other four terms. I... |
deepscaler | [
{
"role": "user",
"content": "In the given figure, $ABCD$ is a parallelogram. We know that $\\angle D = 60^\\circ$, $AD = 2$ and $AB = \\sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"split": "train",
"index": 28394,
"solution": "\nWe are given a parallelogram \\(ABCD\\) with \\(\\angle D = 60^\\circ\\), \\(AD = 2\\), and \\(AB = \\sqrt{3} + 1\\). Point \\(M\\) is the midpoint of \\(AD\\), and segment \\(CK\\) is the angle bisector of \\(\\angle C\\). We need to find \\(\\angle CKB\\).\n\n#... |
deepscaler | [
{
"role": "user",
"content": "The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and agai... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 28395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\\frac{r^2-5q^2}{p^2-1}=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 2, 6)"
} | {
"split": "train",
"index": 28396,
"solution": "\nTo find all triples \\((p, q, r)\\) of positive integers, where \\(p, q\\) are also primes, such that:\n\n\\[\n\\frac{r^2 - 5q^2}{p^2 - 1} = 2,\n\\]\n\nwe start by rearranging the equation:\n\n\\[\nr^2 - 5q^2 = 2(p^2 - 1).\n\\]\n\nThis can be further rewritten as... |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability that in a deck of 52 cards, the second card has a different suit than the first, and the third card has a different suit than the first and second. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{169}{425}"
} | {
"split": "train",
"index": 28397,
"solution": "The probability that the second card has a different suit than the first is $\\frac{3 \\cdot 13}{52-1}$. The probability that the third card has a different suit than the first and second card is $\\frac{2 \\cdot 13}{52-2}$. Since the two events need to both occur,... |
deepscaler | [
{
"role": "user",
"content": "In a circle with a radius of 5 units, \\( CD \\) and \\( AB \\) are mutually perpendicular diameters. A chord \\( CH \\) intersects \\( AB \\) at \\( K \\) and has a length of 8 units, calculate the lengths of the two segments into which \\( AB \\) is divided. Let's think step ... | math | {
"style": "rule",
"ground_truth": "8.75"
} | {
"split": "train",
"index": 28398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of \\(\\frac{\\cos 30.5^{\\circ}+\\cos 31.5^{\\circ}+\\ldots+\\cos 44.5^{\\circ}}{\\sin 30.5^{\\circ}+\\sin 31.5^{\\circ}+\\ldots+\\sin 44.5^{\\circ}}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(\\sqrt{2}-1)(\\sqrt{3}+\\sqrt{2})=2-\\sqrt{2}-\\sqrt{3}+\\sqrt{6}"
} | {
"split": "train",
"index": 28399,
"solution": "Consider a 360-sided regular polygon with side length 1, rotated so that its sides are at half-degree inclinations (that is, its sides all have inclinations of \\(0.5^{\\circ}, 1.5^{\\circ}, 2.5^{\\circ}\\), and so on. Go to the bottom point on this polygon and the... |
deepscaler | [
{
"role": "user",
"content": "If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31 < p \\le 41"
} | {
"split": "train",
"index": 28400,
"solution": "1. **Identify the requirement for integer roots**: The quadratic equation given is $x^2 + px - 444p = 0$. For the roots of this equation to be integers, the discriminant must be a perfect square. The discriminant $\\Delta$ of a quadratic equation $ax^2 + bx + c = 0... |
deepscaler | [
{
"role": "user",
"content": "How many even perfect square factors does $2^4 \\cdot 7^9$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $2^{4}-2^{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2^{3}"
} | {
"split": "train",
"index": 28402,
"solution": "We note that $2^{2}=2 \\times 2=4,2^{3}=2^{2} \\times 2=4 \\times 2=8$, and $2^{4}=2^{2} \\times 2^{2}=4 \\times 4=16$. Therefore, $2^{4}-2^{3}=16-8=8=2^{3}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given that line $l\\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\\_1$ is parallel to $l\\_2$ or $l\\_1$ is perpendicular to $l\\_2$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"split": "train",
"index": 28403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=\\sin^{2}x+2 \\sqrt {3}\\sin x\\cos x- \\frac {1}{2}\\cos 2x$$, where $x\\in\\mathbb{R}$. \n(I) Find the smallest positive period and the range of $f(x)$. \n(II) If $$x_{0}(0\\leq x_{0}\\leq \\frac {\\pi}{2})$$ is a zero of $f(x)$, find the value of ... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {15}- \\sqrt {3}}{8}"
} | {
"split": "train",
"index": 28404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three generous friends, each with some money, redistribute the money as followed:\nAmy gives enough money to Jan and Toy to double each amount has.\nJan then gives enough to Amy and Toy to double their amounts. \nFinally, Toy gives enough to Amy and Jan to double their amounts.... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 28405,
"solution": "1. **Initial Setup**: Let's denote the initial amounts of money that Amy, Jan, and Toy have as $a$, $j$, and $t$ respectively. According to the problem, Toy starts with $t = 36$ dollars.\n\n2. **After Amy's Redistribution**: Amy gives enough money to Jan and Toy to... |
deepscaler | [
{
"role": "user",
"content": "Given two numbers, a and b, are randomly selected within the interval (-π, π), determine the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{4}"
} | {
"split": "train",
"index": 28406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $2 \\times 12 \\times 22 \\times 32 \\times \\ldots \\times 72 \\times 82 \\times 92$ is divided by $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 28407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are positive integers and the equation \\( ab - 8a + 7b = 395 \\) holds true, what is the minimal possible value of \\( |a - b| \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 28408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABC is a triangle. D is the midpoint of AB, E is a point on the side BC such that BE = 2 EC and ∠ADC = ∠BAE. Find ∠BAC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 28409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____. Let's... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 28410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 28411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''\... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 28412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ordered pairs $(a,b)$ of integers such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 28413,
"solution": "Let $k=\\log_a b$. Then our equation becomes $k+\\frac{6}{k}=5$. Multiplying through by $k$ and solving the quadratic gives us $k=2$ or $k=3$. Hence $a^2=b$ or $a^3=b$.\n For the first case $a^2=b$, $a$ can range from 2 to 44, a total of 43 values. For the second c... |
deepscaler | [
{
"role": "user",
"content": "Given $a\\in \\mathbb{R}$, $b\\in \\mathbb{R}$, if the set $\\{a, \\frac{b}{a}, 1\\} = \\{a^{2}, a-b, 0\\}$, calculate the value of $a^{2019}+b^{2019}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A barn with a roof is rectangular in shape, $12$ yd. wide, $15$ yd. long, and $6$ yd. high. Calculate the total area to be painted. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "828"
} | {
"split": "train",
"index": 28415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For \\(0 \\leq x \\leq 1\\) and positive integer \\(n\\), let \\(f_0(x) = |1 - 2x|\\) and \\(f_n(x) = f_0(f_{n-1}(x))\\). How many solutions are there to the equation \\(f_{10}(x) = x\\) in the range \\(0 \\leq x \\leq 1\\)? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 28416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $|\\omega^2+6\\omega+58|$ if $\\omega=9+2i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 28417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? Let's think step by... | math | {
"style": "rule",
"ground_truth": "3a - 2b"
} | {
"split": "train",
"index": 28418,
"solution": "\nLet's denote:\n- $x$ as the weight of the empty bucket.\n- $y$ as the weight of the water when the bucket is full.\n\nFrom the problem, we have two equations based on the given conditions:\n1. When the bucket is two-thirds full, the total weight is $a$ kilograms:... |
deepscaler | [
{
"role": "user",
"content": "Points \\( C \\) and \\( D \\) have the same \\( y \\)-coordinate of 25, but different \\( x \\)-coordinates. What is the difference between the slope and the \\( y \\)-intercept of the line containing both points? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 28419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 28420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 2002 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7456"
} | {
"split": "train",
"index": 28421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The units digit of $3^{1001} 7^{1002} 13^{1003}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 28422,
"solution": "1. **Finding the units digit of $3^{1001}$:**\n - We observe the pattern of the units digits of powers of $3$ modulo $10$:\n - $3^1 \\equiv 3 \\pmod{10}$\n - $3^2 \\equiv 9 \\pmod{10}$\n - $3^3 \\equiv 27 \\equiv 7 \\pmod{10}$\n - $3^4 \\equiv 81 ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\frac {1}{3 - \\frac {1}{3 - \\frac {1}{3 - \\frac{1}{4}}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{29}"
} | {
"split": "train",
"index": 28423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:\n\\[ M \\leq a < b \\leq c < d \\leq M+49, \\qquad ad=bc \\]\nthen we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number. Let's think... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 28424,
"solution": "\nFor a positive integer \\( M \\), we need to determine if it is a GOOD or BAD number based on the existence of integers \\( a, b, c, \\) and \\( d \\) such that:\n\\[ M \\leq a < b \\leq c < d \\leq M + 49, \\qquad ad = bc. \\]\n\nWe aim to find the greatest GOOD... |
deepscaler | [
{
"role": "user",
"content": "Given an acute angle \\( \\theta \\), the equation \\( x^{2} + 4x \\cos \\theta + \\cot \\theta = 0 \\) has a double root. Find the radian measure of \\( \\theta \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 28425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $\\sin {15}^{{}^\\circ }+\\cos {15}^{{}^\\circ }$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 28426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the increments of the argument and the function for \\( y = 2x^2 + 1 \\) when the argument \\( x \\) changes from 1 to 1.02. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0808"
} | {
"split": "train",
"index": 28427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pentagon $J A M E S$ is such that $A M=S J$ and the internal angles satisfy $\\angle J=\\angle A=\\angle E=90^{\\circ}$, and $\\angle M=\\angle S$. Given that there exists a diagonal of $J A M E S$ that bisects its area, find the ratio of the shortest side of $J A M E S$ to the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 28428,
"solution": "Since $\\angle J=\\angle A=90^{\\circ}$ and $A M=J S, J A M S$ must be a rectangle. In addition, $\\angle M+\\angle S=270^{\\circ}$, so $\\angle M=\\angle S=135^{\\circ}$. Therefore, $\\angle E S M=\\angle E M S=45^{\\circ}$, which means $M E S$ is an isosceles rig... |
deepscaler | [
{
"role": "user",
"content": "The sum of the non-negative numbers \\(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\\) is 1. Let \\(M\\) be the maximum of the quantities \\(a_{1} + a_{2} + a_{3}, a_{2} + a_{3} + a_{4}, a_{3} + a_{4} + a_{5}, a_{4} + a_{5} + a_{6}, a_{5} + a_{6} + a_{7}\\).\n\nHow small can... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 28429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in quadrilateral $EFGH$, tangent to $\\overline{EF}$ at $R$ and to $\\overline{GH}$ at $S$. Given that $ER=25$, $RF=35$, $GS=40$, and $SH=20$, and that the circle is also tangent to $\\overline{EH}$ at $T$ with $ET=45$, find the square of the radius of the... | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"split": "train",
"index": 28430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. 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": 28431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), given \\( a^{2} + b^{2} + c^{2} = 2\\sqrt{3} \\, ab \\, \\sin C \\), find \\( \\cos \\frac{A}{2} \\cos \\frac{B}{2} \\cos \\frac{C}{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{8}"
} | {
"split": "train",
"index": 28432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The right focus of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\\vec{OF} - \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 28433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has altitudes of lengths 15, 21, and 35. Find its area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 28434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jarris the triangle is playing in the \\((x, y)\\) plane. Let his maximum \\(y\\) coordinate be \\(k\\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \\(x\\)-axis, find the minimum possible value of \\(k\\). Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "24/5"
} | {
"split": "train",
"index": 28435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(2, 0)$, $B(0, 2)$, $C(\\cos\\alpha, \\sin\\alpha)$ and $0 < \\alpha < \\pi$:\n1. If $|\\vec{OA} + \\vec{OC}| = \\sqrt{7}$, find the angle between $\\vec{OB}$ and $\\vec{OC}$.\n2. If $\\vec{AC} \\perp \\vec{BC}$, find the value of $\\cos\\alpha$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{7}}{4}"
} | {
"split": "train",
"index": 28436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all positive integers $n$, let $g(n)=\\log_{3003} n^3$. Find $g(7)+g(11)+g(13)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 28437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common point... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 28438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The repeating decimal for $\\frac{5}{13}$ is $0.cdc\\ldots$ What is the value of the sum $c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 28440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man has part of $4500 invested at 4% and the rest at 6%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.8\\%"
} | {
"split": "train",
"index": 28441,
"solution": "1. **Identify the amounts invested at different rates:**\n Let $x$ be the amount invested at $6\\%$. Then, the amount invested at $4\\%$ is $(4500 - x)$.\n\n2. **Set up the equation based on equal annual returns:**\n The annual return from the investment at $4\... |
deepscaler | [
{
"role": "user",
"content": "Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next d... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 28442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a survey, 100 students were asked if they like lentils and were also asked if they like chickpeas. A total of 68 students like lentils. A total of 53 like chickpeas. A total of 6 like neither lentils nor chickpeas. How many of the 100 students like both lentils and chickpeas... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 28443,
"solution": "Suppose that $x$ students like both lentils and chickpeas. Since 68 students like lentils, these 68 students either like chickpeas or they do not. Since $x$ students like lentils and chickpeas, then $x$ of the 68 students that like lentils also like chickpeas and s... |
deepscaler | [
{
"role": "user",
"content": "Given the line $l: \\lambda x-y-\\lambda +1=0$ and the circle $C: x^{2}+y^{2}-4y=0$, calculate the minimum value of $|AB|$. 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": 28444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the ratio of the legs of a right triangle is $3: 4$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:\nA) $\\frac{16}{9}$\nB) $\\frac{9}{16}$\nC) $\\frac{3}{4}$\nD) $\\frac{4}{3}$ Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 28445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $S$, and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = S$.\n(1) Find the values of $\\sin A$, $\\cos A$, and $\\tan 2A$.\n(2) If $B = \\frac{\\pi}{4}, \\; |\\overrightarrow{CA} - \\overrightarrow{CB}| = 6$, find the area $S$ of $... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathbf{A} = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}\\) and \\(\\mathbf{B} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\\) be two matrices such that \\(\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A}\\). Assuming \\(4b \\neq c\\), find \\(\\frac{a - 2d}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 28447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 28448,
"solution": ""
} |
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