data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If $x = \\frac34$ and $y = \\frac43$ , find the value of $\\frac12x^6y^7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 7700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped 9 times. What is the probability that at least 6 of the flips result in heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{256}"
} | {
"split": "train",
"index": 7701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{19}"
} | {
"split": "train",
"index": 7702,
"solution": "1. **Prime Factorization of $21!$:** \n To find the prime factorization of $21!$, we need to consider the contributions of each prime number up to 21. We count the number of times each prime divides any of the numbers from 1 to 21. This can be done using the form... |
deepscaler | [
{
"role": "user",
"content": "If $p(x) = x^4 - 3x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(p(x))^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-27"
} | {
"split": "train",
"index": 7703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arctan \\frac{2}{5} + \\arctan \\frac{5}{2}.$ Express your answer in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 7704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D$ be the circle with equation $x^2 - 8x + y^2 + 10y = -14$. A translation of the coordinate system moves the center of $D$ to $(1, -2)$. Determine $a+b+r$ after the translation, where $(a,b)$ is the center and $r$ is the radius of the translated circle. Let's think step b... | math | {
"style": "rule",
"ground_truth": "-1 + \\sqrt{27}"
} | {
"split": "train",
"index": 7705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vectors $\\overrightarrow{a} = (\\cos 25^\\circ, \\sin 25^\\circ)$, $\\overrightarrow{b} = (\\sin 20^\\circ, \\cos 20^\\circ)$, and $\\overrightarrow{u} = \\overrightarrow{a} + t\\overrightarrow{b}$, where $t\\in\\mathbb{R}$, find the minimum value of $|\\overrightarr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 7706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\\cdot CD =$ Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 7707,
"solution": "1. **Identify the Circle and Key Points**: Since $P$ is equidistant from $A$ and $B$, $P$ lies on the perpendicular bisector of $\\overline{AB}$. Given that $\\angle APB = 2\\angle ACB$, and $P$ is equidistant from $A$ and $B$, we can infer that $A$, $B$, and $C$ li... |
deepscaler | [
{
"role": "user",
"content": "A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string?... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 7708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the set \\( \\{x \\mid a \\leqslant x \\leqslant b\\} \\), we define \\( b-a \\) as its length. Let the set \\( A=\\{x \\mid a \\leqslant x \\leqslant a+1981\\} \\), \\( B=\\{x \\mid b-1014 \\leqslant x \\leqslant b\\} \\), and both \\( A \\) and \\( B \\) are subsets of th... | math | {
"style": "rule",
"ground_truth": "983"
} | {
"split": "train",
"index": 7709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\[\\left|\\left(\\frac 23+\\frac 56i\\right)^8\\right|\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2825761}{1679616}"
} | {
"split": "train",
"index": 7710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 7711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle such that $AB = 7$ , and let the angle bisector of $\\angle BAC$ intersect line $BC$ at $D$ . If there exist points $E$ and $F$ on sides $AC$ and $BC$ , respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 7712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.\n Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 7713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(-2,-2)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 2 \\sqrt{2}, \\frac{5 \\pi}{4} \\right)"
} | {
"split": "train",
"index": 7714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the perimeter of an equilateral triangle inscribed in a circle, given that a chord of this circle, equal to 2, is at a distance of 3 from its center. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{30}"
} | {
"split": "train",
"index": 7715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the grea... | math | {
"style": "rule",
"ground_truth": "89991"
} | {
"split": "train",
"index": 7716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Solve the inequality $\\frac{2x+1}{3-x} \\geq 1$.\n2. Given that $x > 0$ and $y > 0$, and $x + y = 1$, find the minimum value of $\\frac{4}{x} + \\frac{9}{y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 7717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a=4$, $b=6$, and $C=60^\\circ$:\n1. Calculate $\\overrightarrow{BC} \\cdot \\overrightarrow{CA}$;\n2. Find the projection of $\\overrightarrow{CA}$ onto... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 7718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$ Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2001}"
} | {
"split": "train",
"index": 7719,
"solution": "The problem involves calculating the probabilities $P_s$ and $P_d$ and finding their absolute difference.\n\n1. **Total number of marbles**:\n There are $1001$ red marbles and $1001$ black marbles, making a total of $1001 + 1001 = 2002$ marbles.\n\n2. **Calculatin... |
deepscaler | [
{
"role": "user",
"content": "Given a square initially painted black, with $\\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white. Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{729}{8192}"
} | {
"split": "train",
"index": 7720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn triangle \\(ABC\\), two identical rectangles \\(PQRS\\) and \\(P_1Q_1R_1S_1\\) are inscribed (with points \\(P\\) and \\(P_1\\) lying on side \\(AB\\), points \\(Q\\) and \\(Q_1\\) lying on side \\(BC\\), and points \\(R, S, R_1,\\) and \\(S_1\\) lying on side \\(AC\\)). I... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 7721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{81}"
} | {
"split": "train",
"index": 7722,
"solution": "Since the number has at least two digits, all possible combinations of first and last digits have the same number of possibilities, which is \\(\\sum_{i=0}^{8} \\frac{8!}{i!}\\). Since the first digit cannot be zero, all of the last digits have 8 possible first digi... |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\cdots, 1995\\} \\). Suppose \\( A \\) is a subset of \\( M \\) that satisfies the condition: if \\( x \\in A \\), then \\( 15x \\notin A \\). What is the maximum number of elements in \\( A \\)? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1870"
} | {
"split": "train",
"index": 7723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\\pi$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "125 - 5\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 7724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? L... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 7725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square? [asy]\nunitsize(1mm);\ndefaultpen(linewidth(0.7pt));\n\ndraw((0,0)--(20,0)--(20,20)--(0,20)--cycle);\ndraw(Circle((5,5),5));\ndraw(Circle((15,5),... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 7726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with a non-zero common difference, $S_{n}$ represents the sum of its first $n$ terms. If $S_{5}=0$, then the number of distinct values in $S_{i}$ for $i=1,2,\\ldots,100$ is ______. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 7727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six balls, numbered 2, 3, 4, 5, 6, 7, are placed in a hat. Each ball is equally likely to be chosen. If one ball is chosen, what is the probability that the number on the selected ball is a prime number? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 7728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\\overrightarrow{m}=(\\cos A,\\cos B)$ and $\\overrightarrow{n}=(b+2c,a)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n\n(1) Find the measure of angle $A$.\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 7729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Široká was expecting guests in the evening. First, she prepared 25 open-faced sandwiches. She then calculated that if each guest took two sandwiches, three of them would not have enough. She then thought that if she made 10 more sandwiches, each guest could take three, but... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ n$ and $ k$ be positive integers such that $ \\frac{1}{2} n < k \\leq \\frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \\times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccu... | math | {
"style": "rule",
"ground_truth": "4(n-k)"
} | {
"split": "train",
"index": 7732,
"solution": "\nLet \\( n \\) and \\( k \\) be positive integers such that \\( \\frac{1}{2}n < k \\leq \\frac{2}{3}n \\). Our goal is to find the least number \\( m \\) for which it is possible to place \\( m \\) pawns on an \\( n \\times n \\) chessboard such that no column or r... |
deepscaler | [
{
"role": "user",
"content": "Given sets A = {-2, 1, 2} and B = {-1, 1, 3}, calculate the probability that a line represented by the equation ax - y + b = 0 will pass through the fourth quadrant. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 7733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 : \\sqrt{2}"
} | {
"split": "train",
"index": 7734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{\\frac{x}{4}}{2}=\\frac{4}{\\frac{x}{2}}$, then $x=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pm 8"
} | {
"split": "train",
"index": 7735,
"solution": "1. Start by simplifying the given equation:\n \\[\n \\dfrac{\\frac{x}{4}}{2} = \\dfrac{4}{\\frac{x}{2}}\n \\]\n Simplify both sides:\n \\[\n \\frac{x}{4 \\times 2} = \\frac{4 \\times 2}{x}\n \\]\n \\[\n \\frac{x}{8} = \\frac{8}{x}\n \\]\n\n2. Cro... |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \\[\\{-8, -6, -4, -1, 1, 3, 5, 14\\}.\\] What is the minimum possible value of \\[\n(p+q+r+s)^2 + (t+u+v+w)^2\n\\] given that the sum $p+q+r+s$ is at least 5? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 7736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \\geq 3$). Find $a_{1000}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 7737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "28.00"
} | {
"split": "train",
"index": 7738,
"solution": "1. **Calculate the base cost of the plan:**\n The base price of Michelle's cell phone plan is $20$ dollars.\n\n2. **Calculate the cost for text messages:**\n Michelle sent $100$ text messages, and each text message costs $5$ cents. Therefore, the total cost for ... |
deepscaler | [
{
"role": "user",
"content": "Let points $A = (0,0)$, $B = (1,2)$, $C = (3,3)$, and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\\overline{CD}$ at point $\\left (\\frac{p}{q}, \\frac{r}{s} \\right )$, where these fractions are in lowes... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 7739,
"solution": "1. **Identify Coordinates and Setup**: We are given the coordinates of the quadrilateral $ABCD$ as $A = (0,0)$, $B = (1,2)$, $C = (3,3)$, and $D = (4,0)$. We need to find the point where a line through $A$ that divides the area of $ABCD$ into two equal parts interse... |
deepscaler | [
{
"role": "user",
"content": "Without using a calculator, find the largest prime factor of \\( 17^4 + 2 \\times 17^2 + 1 - 16^4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit number $\\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \\cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"split": "train",
"index": 7741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $n$, and $l$ be real numbers, and suppose that the roots of the equation \\[x^4 - 10x^3 + ax^2 - nx + l = 0\\] are four distinct positive integers. Compute $a + n + l.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 7742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((7,0),1),white); fill(circle((3,0),3),white); draw((0,0)--(8,0),black+linewidth(1)); draw((6,0)--(6,sqrt(12)),black+linewidth(1)); MP(\"A\", (0,0), W); MP(\"B\", (8,0), E); MP(\"C\"... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 7743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \\( S \\) kilometers, where \\( S \\) is a positive integer... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 7744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle XYZ$, the medians $\\overline{XM}$ and $\\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\\triangle XYZ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 7745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a fictional country where license plates are made using an extended Rotokas alphabet with fourteen letters: A, E, G, I, K, M, O, P, R, S, T, U, V, Z. License plates are six letters long and must start with G, end with M, and not contain the letters S, T, or Z. Addition... | math | {
"style": "rule",
"ground_truth": "3024"
} | {
"split": "train",
"index": 7746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If an integer \\( n \\) that is greater than 8 is a solution to the equation \\( x^2 - ax + b = 0 \\), and the representation of \\( a \\) in base \\( n \\) is 18, then the representation of \\( b \\) in base \\( n \\) is: Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 7747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ΔABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=\\sqrt{3}$, $b=\\sqrt{2}$, and $A=\\frac{\\pi}{3}$, find the value of $B=$ _______; and the area of $ΔABC=S_{ΔABC}=$ _______. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{3+ \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 7748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the base $AC$ of an isosceles triangle $ABC (AB=BC)$, a point $M$ is marked. It is known that $AM=7$, $MB=3$, and $\\angle BMC=60^{\\circ}$. Find the length of segment $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $A$ is in the interior of circle $B$. The diameter of circle $B$ is 16 cm. What is the diameter of circle $A$ for which the ratio of the shaded area to the area of circle $A$ is 3:1?\n\n[asy]size(101);\nimport graph;\npair B = (0,0);\npair A = (1,-3);\nfilldraw(Circle(B,... | math | {
"style": "rule",
"ground_truth": "8\\text{ cm}"
} | {
"split": "train",
"index": 7750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a string of $n$ $8$'s, $8888\\cdots88$, into which $+$ signs are inserted to produce an arithmetic expression. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $8000$? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 7751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (1,0)$ and $B = (5,4).$ Let $P$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible value of $AP + BP.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 7752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the circumference of the region defined by the equation $x^2+y^2 - 10 = 3y - 6x + 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi \\sqrt{73}"
} | {
"split": "train",
"index": 7753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=5$, and $|r-s|=7$. What is the sum of all possible values of $|p-s|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 7754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish th... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? 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": "\\frac{1}{6561}"
} | {
"split": "train",
"index": 7756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 7757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 chairs arranged in a circle. If \\( n \\) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \\( n \\) people, what is the minimum value of \\( n \\)? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 7758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 800 students were surveyed, and their pasta and pizza preferences were lasagna (150 students), manicotti (120 students), ravioli (180 students), spaghetti (200 students), and pizza (150 students), calculate the ratio of the number of students who preferred spaghetti ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 7759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers from 1 to 1997 have a sum of digits that is divisible by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "399"
} | {
"split": "train",
"index": 7760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cone-shaped island with a total height of 12000 feet, where the top $\\frac{1}{4}$ of its volume protrudes above the water level, determine how deep the ocean is at the base of the island. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1092"
} | {
"split": "train",
"index": 7761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose Xiao Ming's family subscribes to a newspaper. The delivery person may deliver the newspaper to Xiao Ming's home between 6:30 and 7:30 in the morning. Xiao Ming's father leaves for work between 7:00 and 8:00 in the morning. What is the probability that Xiao Ming's father... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 7762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.2 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a mathematics class of 40 ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 7763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On side \\(AD\\) of rectangle \\(ABCD\\), a point \\(E\\) is marked. On segment \\(EC\\) there is a point \\(M\\) such that \\(AB = BM\\) and \\(AE = EM\\). Find the length of side \\(BC\\), given that \\(ED = 16\\) and \\(CD = 12\\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given right triangle $ABC$ with $\\angle A = 90^\\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\\triangle ACD$ and $\\triangle ABD$ are equal, calculate the area of $\\triangle ABD$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5}"
} | {
"split": "train",
"index": 7765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\displaystyle \\sum_{n=2}^\\infty \\sum_{k=1}^{n-1} \\frac{k}{2^{n+k}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 7766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $a$ for which $\\frac{1}{\\text{log}_2a} + \\frac{1}{\\text{log}_3a} + \\frac{1}{\\text{log}_4a} = 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability tha... | math | {
"style": "rule",
"ground_truth": "831"
} | {
"split": "train",
"index": 7768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the polynomial $x^3-2$ is divided by the polynomial $x^2-2$, the remainder is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x-2"
} | {
"split": "train",
"index": 7769,
"solution": "To find the remainder when the polynomial $x^3 - 2$ is divided by $x^2 - 2$, we use polynomial long division.\n\n1. **Set up the division**: We want to divide $x^3 - 2$ by $x^2 - 2$.\n\n2. **Divide the leading terms**: The leading term of $x^3 - 2$ is $x^3$, and the... |
deepscaler | [
{
"role": "user",
"content": "In the 2017 Shanghai college entrance examination reform plan, it is required that each candidate must choose 3 subjects from 6 subjects including Physics, Chemistry, Biology, Politics, History, and Geography to take the level examination. Xiaoming decided to choose at most one... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = \\frac{x+1}{x-1}$. Then for $x^2 \\neq 1$, $f(-x)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{f(x)}"
} | {
"split": "train",
"index": 7771,
"solution": "1. **Substitute $-x$ into $f(x)$**: \n Given the function $f(x) = \\frac{x+1}{x-1}$, we substitute $-x$ for $x$:\n \\[\n f(-x) = \\frac{-x+1}{-x-1}\n \\]\n\n2. **Simplify the expression**: \n The expression $\\frac{-x+1}{-x-1}$ can be simplified by facto... |
deepscaler | [
{
"role": "user",
"content": "Find all numbers in the range of\n\\[f(x) = \\arctan x + \\arctan \\frac{1 - x}{1 + x},\\]expressed in radians. Enter all the numbers, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3 \\pi}{4}, \\frac{\\pi}{4}"
} | {
"split": "train",
"index": 7772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of recommendation plans the principal can make for a certain high school with 4 students and 3 universities can accept at most 2 students from that school is to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 7773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 7774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer \\( a \\) such that the expression\n\n\\[ \na^2 - 15a - (\\tan x - 1)(\\tan x + 2)(\\tan x + 5)(\\tan x + 8) \n\\]\n\nis less than 35 for all values of \\( x \\in (-\\pi/2, \\pi/2) \\). \n\n Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider equilateral triangle $ABC$ with side length $1$ . Suppose that a point $P$ in the plane of the triangle satisfies \\[2AP=3BP=3CP=\\kappa\\] for some constant $\\kappa$ . Compute the sum of all possible values of $\\kappa$ .\n\n*2018 CCA Math Bonanza Lightning Ro... | math | {
"style": "rule",
"ground_truth": "\\frac{18\\sqrt{3}}{5}"
} | {
"split": "train",
"index": 7776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all reals $ k$ such that\r\n\\[ a^3 \\plus{} b^3 \\plus{} c^3 \\plus{} d^3 \\plus{} 1\\geq k(a \\plus{} b \\plus{} c \\plus{} d)\r\n\\]\r\nholds for all $ a,b,c,d\\geq \\minus{} 1$ .\r\n\r\n*Edited by orl.* Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 7777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of the function \\[g(t) = \\frac{t^2+\\tfrac34 t}{t^2+1},\\]where $t$ can be any real number. (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": "[-\\tfrac18, \\tfrac98]"
} | {
"split": "train",
"index": 7778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the degree of the polynomial resulting from $(5x^6 - 4x^5 + x^2 - 18)(2x^{12} + 6x^9 - 11x^6 + 10) - (x^3 + 4)^6$ when this expression is expanded and simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos(2x- \\frac{\\pi}{6})$, find the horizontal shift required to transform the graph of $y=\\sin 2x$ into the graph of $y=\\cos(2x- \\frac{\\pi}{6})$. 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": 7780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\\frac{\\sqrt{2}}{2}$, and one of its foci is at $(\\sqrt{2}, 0)$.\n\n$(1)$ Find the equation of the ellipse $M$; \n$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 7781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.\n[as... | math | {
"style": "rule",
"ground_truth": "2 : 1"
} | {
"split": "train",
"index": 7782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar contains two red marbles, three green marbles, ten white marbles and no other marbles. Two marbles are randomly drawn from this jar without replacement. What is the probability that these two marbles drawn will both be red? Express your answer as a common fraction. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{105}"
} | {
"split": "train",
"index": 7783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$ . The minor arc $\\angle{XY}=120$ degrees with respect to circle $A$ , and $\\angle{XY}=60$ degrees with respect to circle $B$ . If $XY=2$ , \ffind the area shared by the two circles. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{10\\pi - 12\\sqrt{3}}{9}"
} | {
"split": "train",
"index": 7784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are \\( n \\) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 7785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In general, if there are $d$ doors in every room (but still only 1 correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{d\\left(d^{r}-1\\right)}{d-1}"
} | {
"split": "train",
"index": 7786,
"solution": "Let $E_{i}$ be the expected number of doors through which Mario will pass in the future if he is currently in room $i$ for $i=1,2, \\ldots, r+1$ (we will set $E_{r+1}=0$). We claim that $E_{i}=1+\\frac{d-1}{d} E_{1}+\\frac{1}{d} E_{i+1}$. This is because, as before,... |
deepscaler | [
{
"role": "user",
"content": "Add $10_7 + 163_7.$ Express your answer in base 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "203_7"
} | {
"split": "train",
"index": 7787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\\pi$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac94\\pi"
} | {
"split": "train",
"index": 7788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory ha... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 7789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+3\\sqrt{2}"
} | {
"split": "train",
"index": 7790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the first digit of a four-digit number, which is a perfect square, is decreased by 3, and the last digit is increased by 3, it also results in a perfect square. Find this number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4761"
} | {
"split": "train",
"index": 7792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first n terms of the sequence {a_n} is S_n = n^2 + n + 1, and b_n = (-1)^n a_n (n ∈ N^*). Determine the sum of the first 50 terms of the sequence {b_n}. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 7793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 7794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the series $\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}+2n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 7795,
"solution": "We know that $\\frac{1}{n^{2}+2n}=\\frac{1}{n(n+2)}=\\frac{\\frac{1}{n}-\\frac{1}{n+2}}{2}$. So, if we sum this from 1 to $\\infty$, all terms except for $\\frac{1}{2}+\\frac{\\frac{1}{2}}{2}$ will cancel out (a 'telescoping' series). Therefore, the sum will be $\\f... |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)=\\tan (\\omega x+ \\frac {\\pi}{4})$ ($\\omega > 0$) has a minimum positive period of $2\\pi$, then $\\omega=$ ______ ; $f( \\frac {\\pi}{6})=$ ______ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 7796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\\cos 40^\\circ,\\sin 40^\\circ)$, $(\\cos 60^\\circ,\\sin 60^\\circ)$, and $(\\cos t^\\circ,\\sin t^\\circ)$ is isosceles? Let's think step b... | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 7797,
"solution": "Let $A = (\\cos 40^\\circ, \\sin 40^\\circ)$, $B = (\\cos 60^\\circ, \\sin 60^\\circ)$, and $C = (\\cos t^\\circ, \\sin t^\\circ)$. We need to find the values of $t$ such that $\\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangl... |
deepscaler | [
{
"role": "user",
"content": "Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt{2}"
} | {
"split": "train",
"index": 7798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets \\( M = \\{1, 2, 3\\} \\) and \\( N = \\{1, 2, 3, 4, 5\\} \\), define the function \\( f: M \\rightarrow N \\). Let the points \\( A(1, f(1)), B(2, f(2)), C(3, f(3)) \\) form a triangle \\( \\triangle ABC \\). The circumcenter of \\( \\triangle ABC \\) is \\( D \... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7799,
"solution": ""
} |
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