data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\\overrightarrow{AD}$ and $\\overrightarrow{BC}$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 34549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Thompson's students were asked to add two positive integers. Alex subtracted by mistake and got 4. Bella mistakenly multiplied and got 98. What was the correct answer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 34550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_n\\}_{n\\geq 1}$ be a sequence defined by $a_n=\\int_0^1 x^2(1-x)^ndx$ .\nFind the real value of $c$ such that $\\sum_{n=1}^{\\infty} (n+c)(a_n-a_{n+1})=2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 34551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{43}{14}"
} | {
"split": "train",
"index": 34552,
"solution": "Note that there are 13 prime numbers amongst the cards. We may view these as separating the remaining 29 cards into 14 groups of nonprimes - those appearing before the first prime, between the first and second, etc. Each of these groups is equally likely to appear ... |
deepscaler | [
{
"role": "user",
"content": "How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once? (Here, a word is an arbitrary sequence of letters.) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 34553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom, Sara, and Jim split $\\$1200$ among themselves to be used in investments. Each starts with a different amount. At the end of one year, they have a total of $\\$1800$. Sara triples her money by the end of the year, Jim doubles his money, whereas Tom loses $\\$200$. What was... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 34554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $$\\sqrt[3]{5488000}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "176.4"
} | {
"split": "train",
"index": 34555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$ is on $\\overline{AD}$ so that $\\frac {AN}{AD} = \\frac {17}{2009}$. Let $P$ be the point of intersection of $\\overline{AC}$ and $\\overline{MN}$. Find $\\fra... | math | {
"style": "rule",
"ground_truth": "177"
} | {
"split": "train",
"index": 34556,
"solution": "One of the ways to solve this problem is to make this parallelogram a straight line. So the whole length of the line is $APC$($AMC$ or $ANC$), and $ABC$ is $1000x+2009x=3009x.$\n$AP$($AM$ or $AN$) is $17x.$\n So the answer is $3009x/17x = \\boxed{177}$"
} |
deepscaler | [
{
"role": "user",
"content": "Square the numbers \\( a = 101 \\) and \\( b = 10101 \\). Find the square root of the number \\( c = 102030405060504030201 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10101010101"
} | {
"split": "train",
"index": 34557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 34558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers less than 900 that can be written as a product of two or more consecutive prime numbers. Find their count. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 34559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 34560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(a_{1}, a_{2}, a_{3}, \\ldots\\) is an increasing sequence of natural numbers. It is known that \\(a_{a_{k}} = 3k\\) for any \\(k\\).\n\nFind\na) \\(a_{100}\\)\n\nb) \\(a_{1983}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3762"
} | {
"split": "train",
"index": 34561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$ of length 43, 13, and 48, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle ABC$ and let $D$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"split": "train",
"index": 34563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "481"
} | {
"split": "train",
"index": 34564,
"solution": "1. **Understanding the Checkerboard Pattern**: The checkerboard is $31 \\times 31$ with alternating red and black squares. Each corner square is black. Since $31$ is odd, each row and each column starts and ends with a black square.\n\n2. **Counting Black Squares i... |
deepscaler | [
{
"role": "user",
"content": "There are 120 different five-digit numbers composed of the digits 1, 2, 3, 4, and 5. Arrange them in descending order. The 95th number is $\\quad$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21354"
} | {
"split": "train",
"index": 34565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average yield per unit area of a rice variety for five consecutive years was 9.4, 9.7, 9.8, 10.3, and 10.8 (unit: t/hm²). Calculate the variance of this sample data. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.244"
} | {
"split": "train",
"index": 34566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $g(x) = \\frac{6x^2 + 11x + 17}{7(2 + x)}$, find the minimum value of $g(x)$ for $x \\ge 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{127}{24}"
} | {
"split": "train",
"index": 34567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $1$; $5$; $9$; $\\ldots$ and $8$; $15$; $22$; $\\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2100$ terms of each sequence. How many distinct numbers are in $S$?\nA) 3800\nB) 3900\nC) 4000\nD) 4100\nE) 4200 Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "3900"
} | {
"split": "train",
"index": 34568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\\overline{AC}$, in inches, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{50}"
} | {
"split": "train",
"index": 34569,
"solution": "1. **Identify the relationship between the perimeter and the sides of the rectangle:**\n Given that the perimeter of rectangle $ABCD$ is $20$ inches, we can express this in terms of the length $l$ and width $w$ of the rectangle:\n \\[\n 2l + 2w = 20\n \\]\n... |
deepscaler | [
{
"role": "user",
"content": "Call an ordered pair $(a, b)$ of positive integers fantastic if and only if $a, b \\leq 10^{4}$ and $\\operatorname{gcd}(a \\cdot n!-1, a \\cdot(n+1)!+b)>1$ for infinitely many positive integers $n$. Find the sum of $a+b$ across all fantastic pairs $(a, b)$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "5183"
} | {
"split": "train",
"index": 34570,
"solution": "We first prove the following lemma, which will be useful later. Lemma: Let $p$ be a prime and $1 \\leq n \\leq p-1$ be an integer. Then, $n!(p-1-n)!\\equiv(-1)^{n-1}(\\bmod p)$. Proof. Write $$\\begin{aligned} n!(p-n-1)! & =(1 \\cdot 2 \\cdots n)((p-n-1) \\cdots 2 ... |
deepscaler | [
{
"role": "user",
"content": "A metro network has at least 4 stations on each line, with no more than three transfer stations per line. No transfer station has more than two lines crossing. What is the maximum number of lines such a network can have if it is possible to travel from any station to any other ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 4 are between 100 and 350? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 34572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Medians $\\overline{AD}$ and $\\overline{BE}$ of $\\triangle ABC$ intersect at an angle of $45^\\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64\\sqrt{2}"
} | {
"split": "train",
"index": 34573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. Let's t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{17}-1"
} | {
"split": "train",
"index": 34574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, calculate the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 34575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x, y)$ is randomly selected from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(0, 3)$. What is the probability that both $x < y$ and $x + y < 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 34576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The medians of a right triangle which are drawn from the vertices of the acute angles are $5$ and $\\sqrt{40}$. The value of the hypotenuse is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"split": "train",
"index": 34577,
"solution": "1. **Assign Variables to Triangle Sides**:\n Let the legs of the right triangle be $2a$ and $2b$. This choice simplifies calculations as it avoids fractions when dealing with medians.\n\n2. **Expression for Hypotenuse**:\n The hypotenuse, being the longest side... |
deepscaler | [
{
"role": "user",
"content": "Compute: $\\frac{\\cos 10^{\\circ} - 2\\sin 20^{\\circ}}{\\sin 10^{\\circ}} = \\_\\_\\_\\_\\_\\_ \\text{.}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 34578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset of $\\{1,2,3,\\ldots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "905"
} | {
"split": "train",
"index": 34579,
"solution": "We first show that we can choose at most 5 numbers from $\\{1, 2, \\ldots , 11\\}$ such that no two numbers have a difference of $4$ or $7$. We take the smallest number to be $1$, which rules out $5,8$. Now we can take at most one from each of the pairs: $[2,9]$, $... |
deepscaler | [
{
"role": "user",
"content": "Given $A = 30^\\circ$ and $B = 60^\\circ$, calculate the value of $(1+\\tan A)(1+\\tan B)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 34580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1776"
} | {
"split": "train",
"index": 34581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the function \\( g(x) \\) is defined for \\( x \\in \\{a, b, c\\} \\) such that the points \\( (a, g(a)), (b, g(b)), (c, g(c)) \\) form a triangle with an area of \\( 24 \\). The graph of \\( y = 3g(3x) \\) likewise contains just three points. Determine the area of this... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 34582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum:\n\\[\\sum_{N = 1}^{2048} \\lfloor \\log_3 N \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12049"
} | {
"split": "train",
"index": 34583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the following expression:\n\n$$\\frac { \\sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4 \\sqrt {3}"
} | {
"split": "train",
"index": 34584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number with $|z| = 2.$ Find the maximum value of\n\\[ |(z-2)^3(z+2)|. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{3}"
} | {
"split": "train",
"index": 34585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $139+27+23+11$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 34586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, and a non empty subset $A$ of $\\{1,2,...,2n\\}$, call $A$ good if the set $\\{u\\pm v|u,v\\in A\\}$ does not contain the set $\\{1,2,...,n\\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"split": "train",
"index": 34587,
"solution": "\nFor a positive integer \\( n \\), and a non-empty subset \\( A \\) of \\(\\{1, 2, \\ldots, 2n\\}\\), we call \\( A \\) good if the set \\(\\{u \\pm v \\mid u, v \\in A\\}\\) does not contain the set \\(\\{1, 2, \\ldots, n\\}\\). We aim to find the smallest real n... |
deepscaler | [
{
"role": "user",
"content": "The Happy Valley Kennel has 4 chickens, 2 dogs, and 5 cats. (Some people in Happy Valley like to keep chickens as pets!) In how many ways can the 11 animals be placed in a row of 11 cages, such that all of the animals of each type are in adjacent cages? (Two animals of the s... | math | {
"style": "rule",
"ground_truth": "34,\\!560"
} | {
"split": "train",
"index": 34588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 34589,
"solution": "We analyze the problem by considering the position of $a_3$ and how it affects the possible values of $a_1 + a_2$ and $a_4 + a_5$. We need to ensure $a_1 + a_2 < a_4 + a_5$ for the permutation to be heavy-tailed.\n\n1. **Case 1: $a_3 = 1$.**\n - The remaining num... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x + y$ if the sequence $3, ~9, ~15, \\ldots, ~x, ~y, ~39$ is an arithmetic sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 34590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of three consecutive integers is 210. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 34591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $M$ is the midpoint of line segment $AB$ in plane $\\alpha$, and point $P$ is a point outside plane $\\alpha$. If $AB = 2$ and the angles between lines $PA$, $PM$, $PB$ and plane $\\alpha$ are $30^{\\circ}$, $45^{\\circ}$, and $60^{\\circ}$ respectively, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 34592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $b^2 + c^2 - a^2 = \\sqrt{3}bc$.\n\n(1) If $\\tan B = \\frac{\\sqrt{6}}{12}$, find $\\frac{b}{a}$;\n\n(2) If $B = \\frac{2\\pi}{3}$ and $b = 2\\sqrt{3}$, fin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 34593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sports equipment store purchaser needs to purchase a total of 100 basketballs and volleyballs from the manufacturer, with a total payment not exceeding $11,815. The wholesale prices from the two types of balls and the retail prices at the store are shown in the table below. S... | math | {
"style": "rule",
"ground_truth": "2,600"
} | {
"split": "train",
"index": 34594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all four-digit numbers in which the digits $0, 3, 6, 9$ are absent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6479352"
} | {
"split": "train",
"index": 34595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "615"
} | {
"split": "train",
"index": 34596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_n$ be the Fibonacci sequence, that is, $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute\n\\[\\sum_{n=0}^{\\infty} \\frac{F_n}{10^n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{89}"
} | {
"split": "train",
"index": 34597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{9}"
} | {
"split": "train",
"index": 34598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 34599,
"solution": "1. **Identify the product of the roots using Vieta's formulas**: \n For a quadratic equation of the form $ax^2 + bx + c = 0$, the product of the roots can be given by Vieta's formulas as $\\frac{c}{a}$. \n Here, $a = 3$, $b = -4$, and $c = k$. Therefore, the ... |
deepscaler | [
{
"role": "user",
"content": "Tina is trying to solve the equation by completing the square: $$25x^2+30x-55 = 0.$$ She needs to rewrite the equation in the form \\((ax + b)^2 = c\\), where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(a > 0\\). What is the value of \\(a + b + c\\)? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "-38"
} | {
"split": "train",
"index": 34600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^2 + px + q$ and $g(x) = x^2 + rx + s$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. If both $f$ and $g$ have the same minimum val... | math | {
"style": "rule",
"ground_truth": "-200"
} | {
"split": "train",
"index": 34601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 34602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $PQR$, with $\\angle Q$ as the right angle, $\\sin{R} = \\frac{15\\sqrt{34}}{34}$. Find the length of side $PQ$.\n\n[asy]\ndraw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));\ndraw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));\nlabel(\"$P$\",(0,0)... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider three squares: $PQRS$, $TUVW$, and $WXYZ$, where each side of the squares has length $s=1$. $S$ is the midpoint of $WY$, and $R$ is the midpoint of $WU$. Calculate the ratio of the area of the shaded quadrilateral $PQSR$ to the sum of the areas of the three squares.\nA... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 34604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on ... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 34605,
"solution": "First, we notice that there are 45 tags left, after 25% of the original fish have went away/died. Then, some $x$ percent of fish have been added such that $\\frac{x}{x+75} = 40 \\%$, or $\\frac{2}{5}$. Solving for $x$, we get that $x = 50$, so the total number of f... |
deepscaler | [
{
"role": "user",
"content": "Suppose a number $m$ is randomly selected from the set $\\{7, 9, 12, 18, 21\\}$, and a number $n$ is randomly selected from $\\{2005, 2006, 2007, \\ldots, 2025\\}$. Calculate the probability that the last digit of $m^n$ is $6$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{105}"
} | {
"split": "train",
"index": 34606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "476"
} | {
"split": "train",
"index": 34607,
"solution": "The three-digit integers divisible by $17$, and their digit sum: \\[\\begin{array}{c|c} m & s(m)\\\\ \\hline 102 & 3 \\\\ 119 & 11\\\\ 136 & 10\\\\ 153 & 9\\\\ 170 & 8\\\\ 187 & 16\\\\ 204 & 6\\\\ 221 & 5\\\\ 238 & 13\\\\ 255 & 12\\\\ 272 & 11\\\\ 289 & 19\\\\ 306 ... |
deepscaler | [
{
"role": "user",
"content": "We randomly choose a function $f:[n] \\rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \\geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\\left(f^... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{n}"
} | {
"split": "train",
"index": 34608,
"solution": "Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (including 1 itself), if there is one, and zero if there is no such cycle. The problem is equivalent to finding $\\mathbb{E}(N(f)) / n$. Note that $P(N(f)=k)=\\frac{n-1}... |
deepscaler | [
{
"role": "user",
"content": "The domain of the equation $y = \\frac{3x}{2x^2 - 6x + 4}$ is the set of all real numbers with the exception of the values $x = A$ and $x = B$. What is $A + B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 34609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a_{1}+a_{2}=1$, $a_{2}+a_{3}=2$, $a_{3}+a_{4}=-3$, $a_{4}+a_{5}=-4$, $a_{5}+a_{6}=5$, $a_{6}+a_{7}=6$, $a_{7}+a_{8}=-7$, $a_{8}+a_{9}=-8$, $\\ldots $, $a_{99}+a_{100}=-99$, $a_{100}+a_{1}=-100$, calculate the value of $a_{1}+a_{2}+a_{3}+\\ldots +a_{100}$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "-50"
} | {
"split": "train",
"index": 34610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is t... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 34611,
"solution": "1. **Convert 1000 to Hexadecimal**: \n The decimal number 1000 can be converted to hexadecimal. We find that $1000_{10} = 3E8_{16}$. This conversion is done by dividing 1000 by 16 repeatedly and considering the remainders:\n - $1000 \\div 16 = 62$ remainder $8$... |
deepscaler | [
{
"role": "user",
"content": "Determine the probability that two edges selected at random from the twelve edges of a cube with side length 1 are skew lines (i.e., non-intersecting and not in the same plane). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 34612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability of drawing exactly one red ball.<br/>$(2)$ If the random variable $X$ represents the number of red ball... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 34613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\( \\sqrt{16 \\times \\sqrt{16}} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2^3"
} | {
"split": "train",
"index": 34614,
"solution": "Evaluating, \\( \\sqrt{16 \\times \\sqrt{16}} = \\sqrt{16 \\times 4} = \\sqrt{64} = 8 \\). Since \\( 8 = 2^3 \\), then \\( \\sqrt{16 \\times \\sqrt{16}} = 2^3 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34615,
"solution": "1. **Convert Etienne's euros to dollars**: Given that 1 euro is equivalent to 1.3 dollars, we can calculate the dollar value of Etienne's 400 euros as follows:\n \\[\n 400 \\text{ euros} \\times 1.3 \\frac{\\text{dollars}}{\\text{euro}} = 520 \\text{ dollars}\n... |
deepscaler | [
{
"role": "user",
"content": "If the minute hand is moved back by 5 minutes, the number of radians it has turned is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 34616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Back in 1930, Tillie had to memorize her multiplication facts from $0 \\times 0$ to $12 \\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of t... | math | {
"style": "rule",
"ground_truth": "0.21"
} | {
"split": "train",
"index": 34617,
"solution": "1. **Determine the total number of products in the table**: \n The multiplication table ranges from $0 \\times 0$ to $12 \\times 12$. Since each factor (both rows and columns) ranges from $0$ to $12$, there are $13$ options for each factor (including $0$). Theref... |
deepscaler | [
{
"role": "user",
"content": "In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of ... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{34}"
} | {
"split": "train",
"index": 34618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With the popularity of cars, the \"driver's license\" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 34619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 34620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the base-2 expression for $222_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 34621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 34622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When four positive integers are divided by $11$, the remainders are $2,$ $4,$ $6,$ and $8,$ respectively.\n\nWhen the sum of the four integers is divided by $11$, what is the remainder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 34623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the three-digit number \\( n = abc \\). If the digits \\( a \\), \\( b \\), and \\( c \\) can form an isosceles (including equilateral) triangle, how many such three-digit numbers exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 34624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In her school grade, Misha is the 30th highest and also the 40th lowest in her academic performance. How many students are there in total in Misha's grade? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 34625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 34626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the $3$ identical bags, assuming he might leave some of the bags empty? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 34627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability that the numbers 1, 1, 2, 2, 3, 3 can be arranged into two rows and three columns such that no two identical numbers appear in the same row or column. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"split": "train",
"index": 34628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{10^{6}}} \\), then the value of \\([x]\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 34630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system on the plane, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the curve C₁ is ρ²-6ρcosθ+5=0, and the parametric equa... | math | {
"style": "rule",
"ground_truth": "\\sqrt {7}"
} | {
"split": "train",
"index": 34631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "(E,O,E)"
} | {
"split": "train",
"index": 34632,
"solution": "1. **Define the sequence and initial conditions**: The sequence $\\{D_n\\}$ is defined by $D_0 = 0$, $D_1 = 0$, $D_2 = 1$, and for $n \\geq 3$, $D_n = D_{n-1} + D_{n-3}$.\n\n2. **Compute initial terms**: We calculate the first few terms of the sequence to observe a... |
deepscaler | [
{
"role": "user",
"content": "Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"split": "train",
"index": 34633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the sides are known: \\(AB = 6\\), \\(BC = 4\\), and \\(AC = 8\\). The angle bisector of \\(\\angle C\\) intersects side \\(AB\\) at point \\(D\\). A circle is drawn through points \\(A\\), \\(D\\), and \\(C\\), intersecting side \\(BC\\) at point \\(E\\)... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{15}}{2}"
} | {
"split": "train",
"index": 34634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\angle A$ and $\\angle B$ are within the interval $\\left(0, \\frac{\\pi}{2}\\right)$, and that $\\frac{\\sin A}{\\sin B} = \\sin (A+B)$, find the maximum value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 34635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a, b, c, d\\) be nonnegative real numbers such that \\(a + b + c + d = 1\\). Find the maximum value of\n\n\\[\n\\frac{ab}{a+b} + \\frac{ac}{a+c} + \\frac{ad}{a+d} + \\frac{bc}{b+c} + \\frac{bd}{b+d} + \\frac{cd}{c+d}.\n\\] Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 34636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\left\\{a_{n}\\right\\}\\) be a sequence of positive integers such that \\(a_{1}=1\\), \\(a_{2}=2009\\) and for \\(n \\geq 1\\), \\(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\\). Determine the value of \\(\\frac{a_{993}}{100 a_{991}}\\). Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "89970"
} | {
"split": "train",
"index": 34637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{63}"
} | {
"split": "train",
"index": 34638,
"solution": "I get a matching pair on the third day with probability $\\frac{1}{9}$ because there is a $\\frac{1}{9}$ probability of the second sock matching the first. Given that I already removed a matching pair of the third day, I get a matching pair on the fifth day with pr... |
deepscaler | [
{
"role": "user",
"content": "In a right triangle with legs of 5 and 12, a segment is drawn connecting the shorter leg and the hypotenuse, touching the inscribed circle and parallel to the longer leg. Find its length. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 34639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle passes through the points $(2,0)$ and $(4,0)$ and is tangent to the line $y=x$. Find the sum of all possible values for the $y$-coordinate of the center of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 34640,
"solution": "First, we see that the x coordinate must be 3. Let the y coordinate be y. Now, we see that the radius is $r=\\sqrt{1+y^{2}}$. The line from the center of the circle to the point of tangency with the line $x=y$ is perpendicular to the line $x=y$. Hence, the distance... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 34641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cells of a $5 \\times 7$ table are filled with numbers so that in each $2 \\times 3$ rectangle (either vertical or horizontal) the sum of the numbers is zero. By paying 100 rubles, you can choose any cell and find out what number is written in it. What is the smallest amoun... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 34642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player? Let's... | math | {
"style": "rule",
"ground_truth": "400,000"
} | {
"split": "train",
"index": 34643,
"solution": "1. **Identify the constraints**: Each team has 21 players, and each player must be paid at least $15,000. The total salary for all players on a team cannot exceed $700,000.\n\n2. **Calculate the minimum total salary for 20 players**: If each of the 20 players recei... |
deepscaler | [
{
"role": "user",
"content": "Does there exist a positive integer \\( m \\) such that the equation \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{abc} = \\frac{m}{a+b+c}\\) has infinitely many solutions in positive integers \\( (a, b, c) \\)? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $F(x) = \\int_{0}^{x} (t^{2} + 2t - 8) \\, dt$, where $x > 0$.\n1. Determine the intervals of monotonicity for $F(x)$.\n2. Find the maximum and minimum values of the function $F(x)$ on the interval $[1, 3]$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "-\\frac{28}{3}"
} | {
"split": "train",
"index": 34645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n$, $n \\ge 1$, such that $n \\cdot 2^{n+1}+1$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 34646,
"solution": "\nLet \\( n \\) be an integer such that \\( n \\geq 1 \\), and we want to find all values of \\( n \\) for which \\( n \\cdot 2^{n+1} + 1 \\) is a perfect square. We set:\n\n\\[\nn \\cdot 2^{n+1} + 1 = k^2\n\\]\n\nfor some integer \\( k \\). Rearranging gives:\n\n\... |
deepscaler | [
{
"role": "user",
"content": "Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with... | math | {
"style": "rule",
"ground_truth": "383"
} | {
"split": "train",
"index": 34647,
"solution": "We claim that Paul wins if and only if there are exactly 1 or 2 odd integers on the board at the start. Assuming this, the answer is $\\frac{2021+\\left(\\frac{2021}{202}\\right)}{2^{2021}}$. Since the numerator is odd, this fraction is reduced. Now, $m+n \\equiv 2... |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, the measure of $\\angle RIP$ is $45^\\circ$ and $OR=15$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5\\pi"
} | {
"split": "train",
"index": 34648,
"solution": ""
} |
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