data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If $α∈(0,\\frac{π}{2})$, and $tan2α=\\frac{cosα}{2-sinα}$, determine the value of $\\sin \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 3600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $k \\geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they app... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 3601,
"solution": "Starting with a list of $66=2 \\times 33$ items, the items in the first 33 positions $1,2,3, \\ldots, 31,32,33$ are moved by an in-shuffle to the odd positions of the resulting list, namely to the positions $1,3,5, \\ldots, 61,63,65$ respectively. This means that an... |
deepscaler | [
{
"role": "user",
"content": "Two out of three independently operating components of a computing device have failed. Find the probability that the first and second components have failed, given that the failure probabilities of the first, second, and third components are 0.2, 0.4, and 0.3, respectively. Let... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 3602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $\\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.\n\nHow many $4$-digit palindromes are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 3603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AC}\\perp\\overline{CD}$. If $CD = 20$, $\\tan D = 2$, and $\\tan B = 2.5$, then what is $BC$?\n\n[asy]\npair A,B,C,D;\nC = (0,0);\nD = (20,0);\nA = (20,40);\nB= (30,... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{116}"
} | {
"split": "train",
"index": 3604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the parametric equation of line l is $$\\begin{cases} x=4+ \\frac { \\sqrt {2}}{2}t \\\\ y=3+ \\frac { \\sqrt {2}}{2}t\\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the ... | math | {
"style": "rule",
"ground_truth": "\\frac{86}{7}"
} | {
"split": "train",
"index": 3605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to both spheres and intersects the cylindrical surface, forming an ellipse, what is the sum of the lengths of the major and ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 3606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^2 - 2\\cos{\\theta}x + 1$, where $x \\in \\left[-\\frac{\\sqrt{3}}{2}, \\frac{1}{2}\\right]$.\n\n(1) When $\\theta = \\frac{\\pi}{3}$, find the maximum and minimum values of $f(x)$.\n\n(2) If $f(x)$ is a monotonous function on $x \\in \\left[-\\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{16 + 4\\sqrt{11}}{5}"
} | {
"split": "train",
"index": 3607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many digits does the smallest repeating block in the decimal expansion of $\\frac{5}{7}$ contain? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number between $0$ and $\\tfrac{\\pi}2$ such that \\[\\dfrac{\\sin^4(x)}{42}+\\dfrac{\\cos^4(x)}{75} = \\dfrac{1}{117}.\\] Find $\\tan(x)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{5}"
} | {
"split": "train",
"index": 3609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 3610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 3611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex wrote all natural divisors of a natural number \\( n \\) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 3612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac1{216}"
} | {
"split": "train",
"index": 3613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, wha... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 3614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 3615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the equation of the curve C₁ is: x²+y²=4. The curve C₁ undergoes a scaling transformation $$\\begin{cases} x′= \\frac {1}{2}x \\\\ y′=y\\end{cases}$$ to obtain the curve C₂.\n(I) Find the parametric equation of the curve C₂.\n(II) Estab... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {2}- \\frac { \\sqrt {10}}{2}"
} | {
"split": "train",
"index": 3616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The local theater has one ticket window. In how many ways can six people line up to buy a ticket? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 3617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 3618,
"solution": "\nTo solve this problem, we need to calculate the probability that a triangle formed by choosing three vertices of a regular octagon has at least one side that is also a side of the octagon.\n\n#### Total Number of Triangles:\nThe total number of ways to choose 3 ve... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $\\angle BAC=117^\\circ$ . The angle bisector of $\\angle ABC$ intersects side $AC$ at $D$ . Suppose $\\triangle ABD\\sim\\triangle ACB$ . Compute the measure of $\\angle ABC$ , in degrees. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 3619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real number \\(x\\) satisfies the equation \\(x + \\frac{1}{x} = \\sqrt{3}\\). Evaluate the expression \\(x^{7} - 5x^{5} + x^{2}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "65.31\\%"
} | {
"split": "train",
"index": 3621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = ax^7 + bx^3 + cx - 5.$ If $f(-7) = 7,$ then find $f(7).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"split": "train",
"index": 3622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win? Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{\\left( \\binom{26}{13} \\right)^2}{\\binom{52}{26}}"
} | {
"split": "train",
"index": 3623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of solutions to the equation\n\\[\\tan (7 \\pi \\cos \\theta) = \\cot (3 \\pi \\sin \\theta)\\] where $\\theta \\in (0, 3\\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 3624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoli decides which subject among history, geography, or politics to review during tonight's self-study session based on the outcome of a mathematical game. The rules of the game are as follows: in the Cartesian coordinate system, starting from the origin $O$, and then ending ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{10}"
} | {
"split": "train",
"index": 3625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 3626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any finite sequence of positive integers \\pi, let $S(\\pi)$ be the number of strictly increasing subsequences in \\pi with length 2 or more. For example, in the sequence $\\pi=\\{3,1,2,4\\}$, there are five increasing sub-sequences: $\\{3,4\\},\\{1,2\\},\\{1,4\\},\\{2,4\\}... | math | {
"style": "rule",
"ground_truth": "8287"
} | {
"split": "train",
"index": 3627,
"solution": "For each subset of Joy's set of cards, we compute the number of orders of cards in which the cards in the subset are arranged in increasing order. When we sum over all subsets of Joy's cards, we will obtain the desired sum. Consider any subset of $k$ cards. The prob... |
deepscaler | [
{
"role": "user",
"content": "The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}<x<\\frac{1}{2}"
} | {
"split": "train",
"index": 3628,
"solution": "1. **Rewrite the inequality**: Start by rewriting the given inequality:\n \\[\n 6x^2 + 5x < 4\n \\]\n Subtract 4 from both sides to set the inequality to zero:\n \\[\n 6x^2 + 5x - 4 < 0\n \\]\n\n2. **Factor the quadratic expression**: Next, factor the ... |
deepscaler | [
{
"role": "user",
"content": "If $\\displaystyle\\prod_{i=6}^{2021} (1-\\tan^2((2^i)^\\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$ .\n\n*Proposed by Deyuan Li and Andrew Milas* Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 3629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? 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": 3630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 600$ whose value can be uniquely determined when the values of $\\left\\lfloor \\frac n4\\right\\rfloor$, $\\left\\lfloor\\frac n5\\right\\rfloor$, and $\\left\\lfloor\\frac n6\\right\\rfloor$ are given, where $\\lfloor x \\rfloor$ d... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 3631,
"solution": "1. For $n$ to be uniquely determined, $n$ AND $n + 1$ both need to be a multiple of $4, 5,$ or $6.$ Since either $n$ or $n + 1$ is odd, we know that either $n$ or $n + 1$ has to be a multiple of $5.$ We can state the following cases:\n 1. $n$ is a multiple of $4$ an... |
deepscaler | [
{
"role": "user",
"content": "Add \\(53_8 + 27_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": "102_8"
} | {
"split": "train",
"index": 3632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\\cos B$ in terms of $a$ and $r$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3}{4}"
} | {
"split": "train",
"index": 3633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack? Let's think step b... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $i^{11} + i^{111}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2i"
} | {
"split": "train",
"index": 3635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\sqrt{7+4\\sqrt{3}} - \\sqrt{7-4\\sqrt{3}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 3636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $ABCD$ is a trapezoid such that $\\overline{AB}\\parallel \\overline{CD}$ and $\\overline{AC}\\perp\\overline{CD}$. If $CD = 15$, $\\tan D = 2$, and $\\tan B = 2.5$, then what is $BC$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{261}"
} | {
"split": "train",
"index": 3637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. Determine the different possible pairs of people Cara could be sitting between on this table. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 3638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spinner with seven congruent sectors numbered from 1 to 7 is used. If Jane and her brother each spin the spinner once, and Jane wins if the absolute difference of their numbers is less than 4, what is the probability that Jane wins? Express your answer as a common fraction. L... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{49}"
} | {
"split": "train",
"index": 3639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two sides of a triangle are $8 \\mathrm{dm}$ and $5 \\mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "7.8"
} | {
"split": "train",
"index": 3640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the curve $C$: $\\begin{cases}x=2\\cos \\alpha \\\\ y= \\sqrt{3}\\sin \\alpha\\end{cases}$ ($\\alpha$ is a parameter) and the fixed point $A(0, \\sqrt{3})$, $F_1$ and $F_2$ are the left and right foci of this curve, respectively. Establish a polar coordinate system with t... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{3}}{13}"
} | {
"split": "train",
"index": 3641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 3642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively. It is given that $c \\cos B = (2a - b) \\cos C$.\n1. Find the magnitude of angle $C$.\n2. If $AB = 4$, find the maximum value of the area $S$ of $\\triangle... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 3643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_{n+1}^2=a_na_{n+2}$, and $a_1= \\frac{1}{3}$, $a_4= \\frac{1}{81}$. \n(1) Find the general formula for the sequence $\\{a_n\\}$. \n(2) Let $f(x)=\\log_3x$, $b_n=f(a_1)+f(a_2)+\\ldots+f(a_n)$, $T_n= \\frac{1}{b_1}+ \\frac{1}{b_2}+... | math | {
"style": "rule",
"ground_truth": "\\frac{-2017}{1009}"
} | {
"split": "train",
"index": 3644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)=\\ln (e^{x}+a+1)$ ($a$ is a constant) is an odd function on the set of real numbers $R$.\n(1) Find the value of the real number $a$;\n(2) If the equation $\\frac{\\ln x}{f(x)}=x^{2}-2ex+m$ has exactly one real root, find the value of $m$. Let's thi... | math | {
"style": "rule",
"ground_truth": "e^{2}+\\frac{1}{e}"
} | {
"split": "train",
"index": 3645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the island of Mumble, the Mumblian alphabet has only $5$ letters, and every word in the Mumblian language has no more than $3$ letters in it. How many words are possible? (A word can use a letter more than once, but $0$ letters does not count as a word.) Let's think step by ... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 3646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100t+10u+1"
} | {
"split": "train",
"index": 3647,
"solution": "1. **Identify the original number**: A two-digit number with tens' digit $t$ and units' digit $u$ can be expressed as $10t + u$.\n\n2. **Placing the digit 1 after the number**: When the digit $1$ is placed after this two-digit number, it effectively shifts the origi... |
deepscaler | [
{
"role": "user",
"content": "A workshop produces a type of instrument with a fixed cost of $7500. Each additional unit of the instrument requires an additional investment of $100. The total revenue function is given by: $H(x) = \\begin{cases} 400x - x^2, (0 \\leq x \\leq 200) \\\\ 40000, (x > 200) \\end{ca... | math | {
"style": "rule",
"ground_truth": "15000"
} | {
"split": "train",
"index": 3648,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is a geometric sequence, and $S_n$ is the sum of its first $n$ terms. If $a_2 \\cdot a_3 = 2a_1$, and the arithmetic mean of $a_4$ and $2a_7$ is $\\frac{5}{4}$, then $S_6 = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{2}"
} | {
"split": "train",
"index": 3649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 3650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A steak initially at a temperature of 5°C is put into an oven. After 15 minutes, its temperature reaches 45°C. After another 15 minutes, its temperature is 77°C. The oven maintains a constant temperature. The steak changes temperature at a rate proportional to the difference be... | math | {
"style": "rule",
"ground_truth": "205"
} | {
"split": "train",
"index": 3651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of children, numbering between 50 and 70, attended a spring math camp. To celebrate Pi Day (March 14), they decided to give each other squares if they were just acquaintances and circles if they were friends. Andrey noted that each boy received 3 circles and 8 squares, ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 3652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brave NiuNiu (a milk drink company) organizes a promotion during the Chinese New Year: one gets a red packet when buying a carton of milk of their brand, and there is one of the following characters in the red packet \"虎\"(Tiger), \"生\"(Gain), \"威\"(Strength). If one collects t... | math | {
"style": "rule",
"ground_truth": "7 \\frac{1}{3}"
} | {
"split": "train",
"index": 3653,
"solution": "The answer is B. We can use Poisson process to get the explicit formula for the general case. Suppose that there are in total $n$ characters. The probability for the character $i$ is $p_{i}$, and we aim to collect $k_{i}$ copies of the character $i$. We denote by $N... |
deepscaler | [
{
"role": "user",
"content": "A dealer sold 200 cars, and the data for some of those sales are recorded in this table. If the rest of the cars she sold were Hondas, how many Hondas did she sell?\n\n\\begin{tabular}{ |c | c|}\n\\hline \\textbf{Type of Car} & \\textbf{$\\%$ of Total Cars Sold} \\\\ \\hline\nA... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 3654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shirt regularly priced at $\\$30$ is on sale for $20\\%$ off. Mary has a coupon that will take another $25\\%$ off of the sale price. What single percent discount would give the same final price as the two successive discounts? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"split": "train",
"index": 3655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $8 \\cdot 5\\frac{2}{5} - 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40.2"
} | {
"split": "train",
"index": 3656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the numbers 1 to 16 into a $4 \\times 4$ grid such that each number in a row is larger than the number to its left and each number in a column is larger than the number above it. Given that the numbers 4 and 13 are already placed in the grid, determine the number of differ... | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 3657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain circle, the chord of a $d$-degree arc is $22$ centimeters long, and the chord of a $2d$-degree arc is $20$ centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of the chord of a $3d$-degree arc is $- m + \\sqrt {n}$ centimeters, where... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 3658,
"solution": "Let $z=\\frac{d}{2}$, $R$ be the circumradius, and $a$ be the length of 3d degree chord. Using the extended sine law, we obtain: \\[22=2R\\sin(z)\\] \\[20+a=2R\\sin(2z)\\] \\[a=2R\\sin(3z)\\] Dividing the second from the first we get $\\cos(z)=\\frac{20+a}{44}$ By t... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $∠B≠\\frac{π}{2}$, $cos2B=\\sqrt{3}cosB-1$. \n$(1)$ Find $\\angle B$; \n$(2)$ Choose one of the three conditions, condition 1, condition 2, or condition 3, as known to ensure the existence and uniqueness of $\\triangle ABC$, and find the area of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}+2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 3659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be three positive real numbers whose sum is 1. If no one of these numbers is more than three times any other, find the minimum value of the product $abc.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{343}"
} | {
"split": "train",
"index": 3660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the equations \\( a x^{2} - b x + c = 0 \\) and \\( c x^{2} - a x + b = 0 \\) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equati... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 3661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 3662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "h"
} | {
"split": "train",
"index": 3663,
"solution": "1. **Define the operation $\\otimes$:** Given $x \\otimes y = x^3 - y$.\n\n2. **Calculate $h \\otimes h$:** Using the definition, substitute $x = h$ and $y = h$:\n \\[\n h \\otimes h = h^3 - h.\n \\]\n\n3. **Calculate $h \\otimes (h \\otimes h)$:** Now, substi... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( z_{1} \\) and \\( z_{2} \\) are complex numbers with \\( \\left|z_{1}\\right|=3 \\), \\( \\left|z_{2}\\right|=5 \\), and \\( \\left|z_{1}+z_{2}\\right|=7 \\), find the value of \\( \\arg \\left(\\left(\\frac{z_{2}}{z_{1}}\\right)^{3}\\right) \\). Let's think step... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 3664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a necessary and sufficient condition on the natural number $ n$ for the equation\n\\[ x^n \\plus{} (2 \\plus{} x)^n \\plus{} (2 \\minus{} x)^n \\equal{} 0\n\\]\nto have a integral root. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n=1"
} | {
"split": "train",
"index": 3665,
"solution": "\nTo solve the problem and find the necessary and sufficient condition for the natural number \\( n \\) such that the equation\n\n\\[\nx^n + (2 + x)^n + (2 - x)^n = 0\n\\]\n\nhas an integral root, we proceed as follows:\n\n### Step 1: Analyze the Case \\( n = 1 \\)\... |
deepscaler | [
{
"role": "user",
"content": "Let $h(4x-1) = 2x + 7$. For what value of $x$ is $h(x) = x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression:\n\n$$\n\\frac{\\frac{1}{\\sqrt{3+x} \\cdot \\sqrt{x+2}}+\\frac{1}{\\sqrt{3-x} \\cdot \\sqrt{x-2}}}{\\frac{1}{\\sqrt{3+x} \\cdot \\sqrt{x+2}}-\\frac{1}{\\sqrt{3-x} \\cdot \\sqrt{x-2}}} ; \\quad x=\\sqrt{6} \\text {. }\n$$ Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 3667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measures of angles \\( X \\) and \\( Y \\) are both positive, integer numbers of degrees. The measure of angle \\( X \\) is a multiple of the measure of angle \\( Y \\), and angles \\( X \\) and \\( Y \\) are supplementary angles. How many measures are possible for angle \\... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 3668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $r$ if $3(r-7) = 4(2-2r) + 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 3669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\\overrightarrow{AB}$ in the direction of $\\overrightarrow{CD}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{5}"
} | {
"split": "train",
"index": 3670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit numbers are divisible by 5 and have 75 as their last two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 3671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 3672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\\overline{AP}$ is drawn so that it is tangent to $\\overline{AM}$ and $\\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positi... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 3673,
"solution": "Let the circle intersect $\\overline{PM}$ at $B$. Then note $\\triangle OPB$ and $\\triangle MPA$ are similar. Also note that $AM = BM$ by power of a point. Using the fact that the ratio of corresponding sides in similar triangles is equal to the ratio of their peri... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that, if there are initially $n+1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1\\%$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3674,
"solution": "By a similar inductive argument, the probability for a given $n$ is $$p_{n}=\\frac{n!}{(2n+1)!!}$$ Clearly this is decreasing in $n$. It is easy to see that $$p_{5}=\\frac{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5}{3 \\cdot 5 \\cdot 7 \\cdot 9 \\cdot 11}=\\frac{8}{693}>... |
deepscaler | [
{
"role": "user",
"content": "In the triangle shown, what is the positive difference between the greatest and least possible integral values of $x$?\n\n[asy]\n\ndefaultpen(linewidth(0.7));\npair a,b,c;\nb = (1,2); c = (3,0);\ndraw(a--b--c--a);\nlabel(\"$x$\",a--b,dir(135)); label(\"5\",b--c,dir(45)); label(... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 3675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha, \\beta$, and $\\gamma$ be three real numbers. Suppose that $\\cos \\alpha+\\cos \\beta+\\cos \\gamma =1$ and $\\sin \\alpha+\\sin \\beta+\\sin \\gamma =1$. Find the smallest possible value of $\\cos \\alpha$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{-1-\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 3676,
"solution": "Let $a=\\cos \\alpha+i \\sin \\alpha, b=\\cos \\beta+i \\sin \\beta$, and $c=\\cos \\gamma+i \\sin \\gamma$. We then have $a+b+c=1+i$ where $a, b, c$ are complex numbers on the unit circle. Now, to minimize $\\cos \\alpha=\\operatorname{Re}[a]$, consider a triangle ... |
deepscaler | [
{
"role": "user",
"content": "Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the ... | math | {
"style": "rule",
"ground_truth": "\\frac{431}{480}"
} | {
"split": "train",
"index": 3677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{128}"
} | {
"split": "train",
"index": 3678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $a^{2}+c^{2}=ac+b^{2}$, $b= \\sqrt{3}$, and $a\\geqslant c$. The minimum value of $2a-c$ is ______. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 3679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\\equiv 4\\pmod p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the roundness of 1,728,000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 3681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, 2009 numbers are placed, each of which is equal to 1 or -1. Not all numbers are the same. Consider all possible groups of ten consecutive numbers. Find the product of the numbers in each group of ten and sum them up. What is the largest possible sum? Let's think st... | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 3682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 3683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $ \\tan \\left( \\alpha + \\frac{\\pi}{4} \\right) = -\\frac{1}{2} $ and $\\frac{\\pi}{2} < \\alpha < \\pi$, calculate the value of $ \\frac{\\sin 2\\alpha - 2\\cos^2 \\alpha}{\\sin \\left(\\alpha - \\frac{\\pi}{4} \\right)} $. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "-\\frac{2 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 3684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\displaystyle\\frac{1-i}{2+3i}$, where $i^2 = -1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{13} - \\frac{5}{13}i"
} | {
"split": "train",
"index": 3685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $k$ such that \\[x_1^2 + x_2^2 + \\dots + x_{11}^2 \\geq kx_6^2\\] whenever $x_1, x_2, \\ldots, x_{11}$ are real numbers such that $x_1 + x_2 + \\cdots + x_{11} = 0$ and $x_6$ is the median of $x_1, x_2, \\ldots, x_{11}$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{66}{5}"
} | {
"split": "train",
"index": 3686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $AB=4$, $AC=6$, $BC= \\sqrt{7}$, and the center of its circumcircle is $O$, find $\\overset{⇀}{AO}· \\overset{⇀}{BC} =$ ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 3688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a four-digit natural number $\\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\\overline{ab}-\\overline{bc}=\\overline{cd}$, then this four-digit number is called a \"decreasing number\". For example, the four-digit number $4129$, sinc... | math | {
"style": "rule",
"ground_truth": "8165"
} | {
"split": "train",
"index": 3689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. Find the maximum value of $a \\cos \\theta + b \\sin \\theta$ in terms of $a$ and $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{a^2 + b^2}"
} | {
"split": "train",
"index": 3690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{48}"
} | {
"split": "train",
"index": 3691,
"solution": "Suppose we fix the time at which Train B arrives at Intersection X; then call the interval during which Train A could arrive (given its schedule) and collide with Train B the 'disaster window.' We consider two cases: (i) Train B enters Intersection $X$ between 9:30 ... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $AB = 6$, $BC = 8$, and $AC = 10$. Point $D$ is on line $\\overline{BC}$ such that $\\overline{AD}$ bisects angle $BAC$. The inscribed circles of $\\triangle ADB$ and $\\triangle ADC$ have radii $r_1$ and $r_2$, respectively. What is $r_1... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{35}"
} | {
"split": "train",
"index": 3692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer \\( x \\) that satisfies both \\( x + 7219 \\equiv 5305 \\pmod{17} \\) and \\( x \\equiv 4 \\pmod{7} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 3693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nTwo adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane contai... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 3694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{4^x}{2^{x+y}}=8$ and $\\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 3695,
"solution": "1. **Simplify the first equation:**\n \\[\n \\frac{4^x}{2^{x+y}} = 8\n \\]\n Since $4^x = (2^2)^x = 2^{2x}$, we can rewrite the equation as:\n \\[\n \\frac{2^{2x}}{2^{x+y}} = 8\n \\]\n Simplifying the left side using the properties of exponents:\n ... |
deepscaler | [
{
"role": "user",
"content": "A piece of paper with a thickness of $0.1$ millimeters is folded once, resulting in a thickness of $2 \\times 0.1$ millimeters. Continuing to fold it $2$ times, $3$ times, $4$ times, and so on, determine the total thickness after folding the paper $20$ times and express it as t... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 3696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\frac {πx}{6}$, and the set $M={0,1,2,3,4,5,6,7,8}$, calculate the probability that $f(m)f(n)=0$ for randomly selected different elements $m$, $n$ from $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 3697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the polynomial $x^{3}+x^{10}=a_{0}+a_{1}(x+1)+...+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, then $a_{9}=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 3698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain group of men, 70% have brown eyes, 70% have dark hair, 85% are taller than 5 feet 8 inches, and 90% weigh more than 140 pounds. What percentage of men definitely possess all four of these characteristics? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3699,
"solution": ""
} |
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