data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a circle, parallel chords of lengths 5, 12, and 13 determine central angles of $\\theta$, $\\phi$, and $\\theta + \\phi$ radians, respectively, where $\\theta + \\phi < \\pi$. If $\\sin \\theta$, which is a positive rational number, is expressed as a fraction in lowest terms... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 5th term of an arithmetic sequence of 20 terms with first and last terms of 2 and 59, respectively? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$ . Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all\nnumbers $N$ with such pro... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 5903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R} \\to \\mathbb{R}$ such that for any two real numbers $x,y$ holds\n$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 2x \\text{ and } f(x) = 0"
} | {
"split": "train",
"index": 5904,
"solution": "\nTo solve the functional equation problem, we need to identify all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the given functional equation for any real numbers \\( x, y \\):\n\n\\[\nf(xf(y) + 2y) = f(xy) + xf(y) + f(f(y)).\n\\]\n\nWe will explo... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that ends in 3 and is divisible by 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 5905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the shaded region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\dfrac{1}{2}"
} | {
"split": "train",
"index": 5906,
"solution": "1. **Identify the equations of the lines**: \n - For the first line, using the points (0,4) and (8,1), the slope $m$ is calculated as:\n \\[\n m = \\frac{1-4}{8-0} = -\\frac{3}{8}\n \\]\n Using the point-slope form of a line, $y - y_1 = m(x - x_1)$... |
deepscaler | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy \\[\\frac{1}{x^2+1} > \\frac{3}{x} + \\frac{17}{10}.\\](Give your answer in interval notation.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2, 0)"
} | {
"split": "train",
"index": 5907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jacqueline has 40% less sugar than Liliane, and Bob has 30% less sugar than Liliane. Express the relationship between the amounts of sugar that Jacqueline and Bob have as a percentage. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14.29\\%"
} | {
"split": "train",
"index": 5908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $x$ is the expression $\\frac{2x^3+3}{x^2-20x+100}$ not defined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange 8 people, including A and B, to work for 4 days. If 2 people are arranged each day, the probability that A and B are arranged on the same day is ______. (Express the result as a fraction) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 5910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive whole number divisible by four different prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 5911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $\\angle B=30^{\\circ}$, the area of $\\triangle ABC$ is $\\frac{3}{2}$, and $\\sin A + \\sin C = 2\\sin B$. Calculate the value of $b$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}+1"
} | {
"split": "train",
"index": 5912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $54 \\times 46$ in your head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2484"
} | {
"split": "train",
"index": 5913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake w... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 5914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\sqrt{x} + 2 \\sqrt{x^2 + 7x} + \\sqrt{x + 7} = 35 - 2x.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{841}{144}"
} | {
"split": "train",
"index": 5915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A university has 120 foreign language teachers. Among them, 50 teach English, 45 teach Japanese, and 40 teach French. There are 15 teachers who teach both English and Japanese, 10 who teach both English and French, and 8 who teach both Japanese and French. Additionally, 4 teach... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\\triangle DEQ$ is greater than both $\\triangle DFQ$ and $\\triangle EFQ$? 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": 5917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set $\\{3,6,9,10\\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 5918,
"solution": "We are given the set $\\{3,6,9,10\\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal.\n\n#### Case 1: Median is $6$\nFor... |
deepscaler | [
{
"role": "user",
"content": "Two standard decks of cards are combined, making a total of 104 cards (each deck contains 13 ranks and 4 suits, with all combinations unique within its own deck). The decks are randomly shuffled together. What is the probability that the top card is an Ace of $\\heartsuit$? Let... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"split": "train",
"index": 5919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of a line is given by $Ax+By=0$. If we choose two different numbers from the set $\\{1, 2, 3, 4, 5\\}$ to be the values of $A$ and $B$ each time, then the number of different lines that can be obtained is ． Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use Horner's method to find the value of the polynomial $f(x) = -6x^4 + 5x^3 + 2x + 6$ at $x=3$, denoted as $v_3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-115"
} | {
"split": "train",
"index": 5921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Experimenters Glafira and Gavrila placed a triangle of thin wire with sides 30 mm, 40 mm, and 50 mm on a white flat surface. This wire is covered with millions of unknown microorganisms. Scientists found that when electric current is applied to the wire, these microorganisms st... | math | {
"style": "rule",
"ground_truth": "2114"
} | {
"split": "train",
"index": 5922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 5923,
"solution": "Let there be $N$ members of the committee. Suppose candidate $n$ gets $a_n$ votes. Then $a_n$ as a percentage out of $N$ is $100\\frac{a_n}{N}$. Setting up the inequality $a_n \\geq 1 + 100\\frac{a_n}{N}$ and simplifying, $a_n \\geq \\lceil(\\frac{N}{N - 100})\\rcei... |
deepscaler | [
{
"role": "user",
"content": "If $\\lceil{\\sqrt{x}}\\rceil=12$, how many possible integer values of $x$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 5924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total corn yield in centners, harvested from a certain field area, is expressed as a four-digit number composed of the digits 0, 2, 3, and 5. When the average yield per hectare was calculated, it was found to be the same number of centners as the number of hectares of the f... | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"split": "train",
"index": 5925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( 0 \\leq p \\leq 1 \\) and \\( 0 \\leq q \\leq 1 \\), define \\( H(p, q) \\) by\n\\[\n H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).\n\\]\nDefine \\( J(p) \\) to be the maximum of \\( H(p, q) \\) over all \\( q \\) (in the interval \\( 0 \\leq q \\leq 1 \\)). What is... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 5926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Randomly and without replacement, select three numbers \\(a_1, a_2, a_3\\) from the set \\(\\{1, 2, \\cdots, 2014\\}\\). Then, from the remaining 2011 numbers, again randomly and without replacement, select three numbers \\(b_1, b_2, b_3\\). What is the probability that a brick... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 5927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z=1-2 i$. Find $\\frac{1}{z}+\\frac{2}{z^{2}}+\\frac{3}{z^{3}}+\\cdots$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2i-1)/4"
} | {
"split": "train",
"index": 5928,
"solution": "Let $x=\\frac{1}{z}+\\frac{2}{z^{2}}+\\frac{3}{z^{3}}+\\cdots$, so $z \\cdot x=\\left(1+\\frac{2}{z}+\\frac{3}{z^{2}}+\\frac{4}{z^{3}}+\\cdots\\right)$. Then $z \\cdot x-x=$ $1+\\frac{1}{z}+\\frac{1}{z^{2}}+\\frac{1}{z^{3}}+\\cdots=\\frac{1}{1-1 / z}=\\frac{z}{z-1}$... |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos (2x-\\frac{\\pi }{4})$, determine the horizontal translation of the graph of the function $y=\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi }{8}"
} | {
"split": "train",
"index": 5929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Through the focus of the parabola $y^{2}=4x$, two chords $AB$ and $CD$ are drawn perpendicular to each other. Calculate $\\left( \\frac{1}{|AB|}+ \\frac{1}{|CD|} \\right)$. 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": 5930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the numbers $\\frac{x}{2}, 2x - 3, \\frac{18}{x} + 1$, taken in the specified order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2.08"
} | {
"split": "train",
"index": 5931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 5932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5933,
"solution": "1. **Calculate the total distance of the round trip**: \n The student drove $120$ miles to his home and $120$ miles back, so the total distance for the round trip is:\n \\[\n 120 \\text{ miles} + 120 \\text{ miles} = 240 \\text{ miles}\n \\]\n\n2. **Calculat... |
deepscaler | [
{
"role": "user",
"content": "The *cross* of a convex $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \\times 4$ rectangle is $\\sqrt{ \\dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \\dfrac{5}{3} \\sqrt{6}$ .\n\nSuppose $S$ is a d... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{66}}{11}"
} | {
"split": "train",
"index": 5934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\\angle BAF = 18^\\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 5935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do no... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 5936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a list of the first 12 positive integers such that for each $2\\le i\\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 5937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$, then $x$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5938,
"solution": "1. **Identify the given equation**: We are given that one minus the reciprocal of $(1-x)$ equals the reciprocal of $(1-x)$. This can be written as:\n \\[\n 1 - \\frac{1}{1-x} = \\frac{1}{1-x}\n \\]\n\n2. **Simplify the equation**: To simplify, we can add $\\fr... |
deepscaler | [
{
"role": "user",
"content": "Calculate \\(3^5 \\cdot 6^5\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1,889,568"
} | {
"split": "train",
"index": 5939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that non-negative real numbers \\( x \\) and \\( y \\) satisfy\n\\[ \nx^{2}+4y^{2}+4xy+4x^{2}y^{2}=32,\n\\]\nfind the maximum value of \\( \\sqrt{7}(x+2y)+2xy \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a randomly selected four-element subset of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ . Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\\dfrac{m}{n}$ . Find $m + n$ . Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 5941,
"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 $A > B$, $\\cos C= \\frac {5}{13}$, and $\\cos (A-B)= \\frac {3}{5}$.\n\n(1) Find the value of $\\cos 2A$;\n\n(2) If $c=15$, find the value of $a$. Let's think st... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {65}"
} | {
"split": "train",
"index": 5942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order. For example, $06 / 18 / 23$ is such a date since $6<18<23$, while today, $11 / 11 / 23$, is not. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "186"
} | {
"split": "train",
"index": 5943,
"solution": "January contains 21 such dates, February contains 20, and so on, until December contains 10. The answer is $$21+20+\\cdots+10=186$$"
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets $M={x|m\\leqslant x\\leqslant m+ \\frac {3}{4}}$ and $N={x|n- \\frac {1}{3}\\leqslant x\\leqslant n}$, both of which are subsets of ${x|0\\leqslant x\\leqslant 1}$, what is the minimum \"length\" of the set $M\\cap N$? (Note: The \"length\" of a set ${x|a\\leqsla... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 5944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can 1995 be factored as a product of two two-digit numbers? (Two factorizations of the form $a\\cdot b$ and $b\\cdot a$ are considered the same). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 5945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\\frac{S_{n+1}}{S_n}=\\boxed{?}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 5946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a 31-day month is taken at random, find \\( c \\), the probability that there are 5 Sundays in the month. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"split": "train",
"index": 5947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Madeline has 80 fair coins. She flips all the coins. Any coin that lands on tails is tossed again. Additionally, any coin that lands on heads in the first two tosses is also tossed again, but only once. What is the expected number of coins that are heads after these condi... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 5948,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord $AB$ that makes an angle of $\\frac{\\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \\frac{y^{2}}{3}=1$. \n$(1)$ Find $|AB|$; \n$(2)$ Find the perimeter of $\\triangle F_{2}AB$ ($F_{2}$ is the right focus). Let's think step b... | math | {
"style": "rule",
"ground_truth": "3+3\\sqrt{3}"
} | {
"split": "train",
"index": 5949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, find the distance from $F$ to the midpoint of segment $DE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 5950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $cos2α=-\\frac{{\\sqrt{10}}}{{10}}$, $sin({α-β})=\\frac{{\\sqrt{5}}}{5}$, and $α∈({\\frac{π}{4},\\frac{π}{2}})$, $β∈({-π,-\\frac{π}{2}})$, then $\\alpha +\\beta =$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 5951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $\\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\\frac 13$ -array is constructed as follows:\n\\begin{align*} 1 \\qquad \\frac 13\\,\\ \\qquad \\frac 19\\,\\ \\qquad \\frac 1{27} \\qquad &\\cdots\\\\ \\frac 16 \\qquad \\frac 1{18}\\,\\ \\q... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\\frac{\\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$. Let... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 5953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of $x$ is equal to $40\\%$ of $50\\%$ of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The smallest product one could obtain by multiplying two numbers in the set $\\{ -7,-5,-1,1,3 \\}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 5955,
"solution": "To find the smallest product obtainable by multiplying two numbers from the set $\\{-7, -5, -1, 1, 3\\}$, we need to consider the following cases:\n\n1. **Multiplying two negative numbers**: This will yield a positive product. The possible products are:\n - $(-7) ... |
deepscaler | [
{
"role": "user",
"content": "Find the $2 \\times 2$ matrix $\\mathbf{M}$ such that $\\mathbf{M} \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} = \\begin{pmatrix} -4 \\\\ 4 \\end{pmatrix}$ and $\\mathbf{M} \\begin{pmatrix} -3 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} -23 \\\\ 2 \\end{pmatrix}.$ Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6 & -5 \\\\ 0 & 2 \\end{pmatrix}"
} | {
"split": "train",
"index": 5956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9611"
} | {
"split": "train",
"index": 5957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 f... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 5959,
"solution": "1. **Analyze the distribution of files by size:**\n - There are 3 files of 0.8 MB each.\n - There are 12 files of 0.7 MB each.\n - The remaining files are 15 in total (since 3 + 12 = 15, and there are 30 files in total). However, the problem states that the re... |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b \\in R^{+}$ and $a + b = 1$, find the supremum of $- \\frac{1}{2a} - \\frac{2}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"split": "train",
"index": 5960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Santa Claus arrived at the house of Arnaldo and Bernaldo carrying ten distinct toys numbered from 1 to 10 and said to them: \"Toy number 1 is for you, Arnaldo, and toy number 2 is for you, Bernaldo. But this year, you may choose to keep more toys as long as you leave at least o... | math | {
"style": "rule",
"ground_truth": "6305"
} | {
"split": "train",
"index": 5961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers $1, 2, 3, \\dots, 44, 45, x$ is $50x$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1035}{2299}"
} | {
"split": "train",
"index": 5962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shift the graph of the function $f(x) = 2\\sin(2x + \\frac{\\pi}{4})$ to the right by $\\varphi (\\varphi > 0)$ units, then shrink the x-coordinate of each point on the graph to half of its original value (the y-coordinate remains unchanged), and make the resulting graph symmet... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}\\pi"
} | {
"split": "train",
"index": 5963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, w... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 5964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified, what is the value of $\\sqrt{3} \\times 3^{\\frac{1}{2}} + 12 \\div 3 \\times 2 - 4^{\\frac{3}{2}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the positive divisors of 128 are perfect squares larger than 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5966,
"solution": "Since $128=2^{7}$, its positive divisors are $2^{0}=1, 2^{1}=2, 2^{2}=4, 2^{3}=8, 2^{4}=16, 2^{5}=32, 2^{6}=64, 2^{7}=128$. Of these, the integers $1,4,16,64$ are perfect squares, which means that 128 has three positive divisors that are perfect squares larger than ... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 5967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers, $x$, satisfy $|5x - 3| \\le 7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integers $1,2, \\ldots, 64$ are written in the squares of a $8 \\times 8$ chess board, such that for each $1 \\leq i<64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals? Let's think st... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 5969,
"solution": "Our answer is $26+52+54+56+58+60+62+64$. One possible configuration: WLOG, we seek to maximize the sum of the numbers on the main diagonal (top left to bottom right). If we color the squares in a checker-board pattern and use the fact that $a$ and $a+1$ lie on diffe... |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of $\\log_{10}{(a^2)}$ and a circumference of $\\log_{10}{(b^4)}$. What is $\\log_{a}{b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 5970,
"solution": "1. **Identify the given values:** \n The radius of the circle is given as $r = \\log_{10}(a^2)$ and the circumference is given as $C = \\log_{10}(b^4)$.\n\n2. **Use logarithmic properties to simplify expressions:** \n Using the power rule of logarithms, we can... |
deepscaler | [
{
"role": "user",
"content": "Given the regression equation $y=0.75x-68.2$ and a student's height of $x=170$ cm, calculate the student's weight in kg. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59.3"
} | {
"split": "train",
"index": 5971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 5972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a point on the parabola $y = x^2 - 9x + 25,$ and let $B$ be a point on the line $y = x - 8.$ Find the shortest possible distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"split": "train",
"index": 5973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\\triangle ABC$, respectively, and $c\\cos A=5$, $a\\sin C=4$.\n\n(1) Find the length of side $c$; \n(2) If the area of $\\triangle ABC$, $S=16$, find the perimeter of $\\triangl... | math | {
"style": "rule",
"ground_truth": "13+ \\sqrt {41}"
} | {
"split": "train",
"index": 5974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 33. If one integer is 45, what is the smallest possible value of the other integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 5975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 5976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the boat is leaking water at the rate of 15 gallons per minute, the maximum time to reach the shore is 50/15 minutes. If Amy rows towards the shore at a speed of 2 mph, then every 30 minutes she increases her speed by 1 mph, find the maximum rate at which Boris can b... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probabili... | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"split": "train",
"index": 5978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 5979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "275"
} | {
"split": "train",
"index": 5980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $2 / 3$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner? Let's think s... | math | {
"style": "rule",
"ground_truth": "20/81"
} | {
"split": "train",
"index": 5981,
"solution": "If the Mathletes are to win, they must win exactly 5 out of the 7 games. One of the 5 games they win must be the 7 th game, because otherwise they would win the tournament before 7 games are completed. Thus, in the first 6 games, the Mathletes must win 4 games and l... |
deepscaler | [
{
"role": "user",
"content": "Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{912}{119}"
} | {
"split": "train",
"index": 5982,
"solution": "1. **Identify the centers and radii of the circles**: \n - Circle 1 has center $(2,4)$ and radius $4$.\n - Circle 2 has center $(14,9)$ and radius $9$.\n\n2. **Calculate the slope of the line connecting the centers**:\n - The slope of the line $L_1$ connecting... |
deepscaler | [
{
"role": "user",
"content": "All positive integers whose binary representations (excluding leading zeroes) have at least as many 1's as 0's are put in increasing order. Compute the number of digits in the binary representation of the 200th number. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5983,
"solution": "We do a rough estimation. There are 255 positive integers with at most 8 digits and a majority of them, but not more than 200, satisfy the property. Meanwhile, there are 511 positive integers with at most 9 digits, and a majority of them satisfy this property. Thus,... |
deepscaler | [
{
"role": "user",
"content": "Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?\n\nNote that the diagram is not provided, b... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George is about to get a certain amount of change less than one dollar from the cash register. If he gets the most quarters possible and the rest in pennies, he would need to receive 3 pennies to meet the amount. If he gets the most dimes possible and the rest in pennies, he wo... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 5985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors \\( \\boldsymbol{a} \\), \\( \\boldsymbol{b} \\), and \\( \\boldsymbol{c} \\) satisfying\n$$\n|\\boldsymbol{a}|:|\\boldsymbol{b}|:|\\boldsymbol{c}|=1: k: 3 \\quad \\left(k \\in \\mathbf{Z}_{+}\\right),\n$$\nand \\( \\boldsymbol{b} - \\boldsymbol{a} = 2(\\boldsymbo... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{12}"
} | {
"split": "train",
"index": 5986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $i=0,1,2, \\dots$ , there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$ ? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 5987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\\overline{AB}$, and let $E$ be the midpoint of $\\overline{AC}$. The angle bisector of $\\angle BAC$ intersects $\\overline{DE}$ and $\\overline{BC}$ at $F$ and $G$, respectively. What is the a... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 5988,
"solution": "1. **Calculate the area of $\\triangle ADE$:** \n Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = \\frac{1}{2}AB = 25$ and $AE = \\frac{1}{2}AC = 5$. The area of $\\triangle ABC$ is given as $120$. Using the formula for the area of a triangle... |
deepscaler | [
{
"role": "user",
"content": "\nGiven $\\boldsymbol{a} = (\\cos \\alpha, \\sin \\alpha)$ and $\\boldsymbol{b} = (\\cos \\beta, \\sin \\beta)$, the relationship between $\\boldsymbol{a}$ and $\\boldsymbol{b}$ is given by $|k \\boldsymbol{a} + \\boldsymbol{b}| - \\sqrt{3}|\\boldsymbol{a} - k \\boldsymbol{b}|$... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 5989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar wit... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{97}"
} | {
"split": "train",
"index": 5990,
"solution": "Note that $\\frac{5}{12}=\\frac{40.97}{97 \\cdot 96}$. Of all of the original ways we could've drawn marbles, we are adding 97 ways, namely drawing the same marble twice, all of which work. Thus, the answer is $$\\frac{40 \\cdot 97+97}{97 \\cdot 96+97}=\\frac{41}{97... |
deepscaler | [
{
"role": "user",
"content": "There exists a unique positive integer $a$ for which the sum \\[U=\\sum_{n=1}^{2023}\\left\\lfloor\\dfrac{n^{2}-na}{5}\\right\\rfloor\\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.\n (Note that $\\lfloor x\\rfloor$ denotes the greatest in... | math | {
"style": "rule",
"ground_truth": "944"
} | {
"split": "train",
"index": 5991,
"solution": "Consider the integral \\[\\int_{0}^{2023} \\dfrac{n^2-na}{5} \\, dn.\\] We hope this will give a good enough appoximation of $U$ to find $a.$ However, this integral can be easily evaluated(if you know calculus) to be \\[\\dfrac{1}{15}2023^3-\\dfrac{a}{10}2023^2=2023... |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1830"
} | {
"split": "train",
"index": 5992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5993,
"solution": "To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:\n- If one face shows 1, the opposite face shows 6.\n- If one face shows 2, the opposite face s... |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{30 \\sqrt{6}}{7}"
} | {
"split": "train",
"index": 5994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\sin\\left(\\alpha+ \\frac {\\pi}{3}\\right)=- \\frac {4}{5}$$, and $$- \\frac {\\pi}{2}<\\alpha<0$$, find the value of $\\cos\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3-4 \\sqrt {3}}{10}"
} | {
"split": "train",
"index": 5995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5996,
"solution": "1. **Calculate the sum of all numbers in the set**: The sum of the first $n$ integers is given by the formula $\\frac{n(n+1)}{2}$. For the set $\\{1, 2, 3, \\ldots, 37\\}$, we have $n = 37$. Thus, the sum is:\n \\[\n \\frac{37 \\times (37 + 1)}{2} = \\frac{37 \\... |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyrami... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 5997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Stephan has been busy at work writing all the possible rearrangements of the letters in his name. He is one bored boy. If he can write twelve rearrangements of his name every minute, how many hours does it take to write all the possible rearrangements of his name? Let's think s... | math | {
"style": "rule",
"ground_truth": "7\\text{ hours}"
} | {
"split": "train",
"index": 5998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bicycle factory plans to produce a batch of bicycles of the same model, planning to produce $220$ bicycles per day. However, due to various reasons, the actual daily production will differ from the planned quantity. The table below shows the production situation of the worker... | math | {
"style": "rule",
"ground_truth": "155080"
} | {
"split": "train",
"index": 5999,
"solution": ""
} |
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