data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 38049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tower is $45 \\mathrm{~m}$ away from the bank of a river. If the width of the river is seen at an angle of $20^{\\circ}$ from a height of $18 \\mathrm{~m}$ in the tower, how wide is the river? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.38"
} | {
"split": "train",
"index": 38050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If we write $\\sqrt{5}+\\frac{1}{\\sqrt{5}} + \\sqrt{7} + \\frac{1}{\\sqrt{7}}$ in the form $\\dfrac{a\\sqrt{5} + b\\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 38051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\\frac{m}{n}$, where $m$ and $n$ ... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 38052,
"solution": "It is easier to think of the dice as $21$ sided dice with $6$ sixes, $5$ fives, etc. Then there are $21^2=441$ possible rolls. There are $2\\cdot(1\\cdot 6+2\\cdot 5+3\\cdot 4)=56$ rolls that will result in a seven. The odds are therefore $\\frac{56}{441}=\\frac{8}... |
deepscaler | [
{
"role": "user",
"content": "Fill the numbers $1, 2, \\cdots, 36$ into a $6 \\times 6$ grid with each cell containing one number, such that each row is in ascending order from left to right. What is the minimum possible sum of the six numbers in the third column? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 38053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 38054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\tan \\frac{\\alpha+\\beta}{2}=\\frac{\\sqrt{6}}{2}\\) and \\(\\cot \\alpha \\cdot \\cot \\beta=\\frac{7}{13}\\), find the value of \\(\\cos (\\alpha-\\beta)\\). 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": 38055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure shown consists of a right triangle and two squares. If the figure's total area equals 850 square inches, what is the value of $x$ in inches? [asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\n\ndraw((0,5)--(0,-2)--(-2,-2)--(-2,0)--(5,0)--(5,5)--cycle... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 38056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a plane passing through the origin. When $\\begin{pmatrix} 5 \\\\ 3 \\\\ 5 \\end{pmatrix}$ is projected onto plane $P,$ the result is $\\begin{pmatrix} 3 \\\\ 5 \\\\ 1 \\end{pmatrix}.$ When $\\begin{pmatrix} 4 \\\\ 0 \\\\ 7 \\end{pmatrix}$ is projected onto plane $... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 1 \\\\ 3 \\\\ 1 \\end{pmatrix}"
} | {
"split": "train",
"index": 38057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\\overarc {AC}$ and $\\overarc{BC}$, and to $\\overline{AB}$. If the length of $\\overarc{BC}$ is $12$, then the circumference of the circle is Let's think step... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 38058,
"solution": "\n1. **Identify the Geometry of the Problem:**\n Since the centers of the arcs $AC$ and $BC$ are at $B$ and $A$ respectively, and each arc is part of a circle with radius equal to $AB$, triangle $ABC$ is equilateral. This is because all sides $AB$, $BC$, and $CA$... |
deepscaler | [
{
"role": "user",
"content": "At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22\\frac {1}{2}^{\\circ}"
} | {
"split": "train",
"index": 38059,
"solution": "To find the angle between the hour and minute hands of a clock at a given time, we can use the formula:\n\\[\n\\text{Angle} = \\left|\\frac{60h - 11m}{2}\\right|\n\\]\nwhere $h$ is the hour and $m$ is the minute.\n\n1. **Identify the values of $h$ and $m$:**\n At... |
deepscaler | [
{
"role": "user",
"content": "Let $r(x)$ be a monic quintic polynomial such that $r(1) = 3$, $r(2) = 7$, $r(3) = 13$, $r(4) = 21$, and $r(5) = 31$. Find $r(6)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 38060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $z$ is a complex number such that\n\\[\n z + z^{-1} = \\sqrt{3},\n\\]what is the value of\n\\[\n z^{2010} + z^{-2010} \\, ? \n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 38061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"split": "train",
"index": 38062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kindergarten's junior group, there are two (small) Christmas trees and five children. The caregivers want to divide the children into two round dances around each of the trees, with each round dance having at least one child. The caregivers distinguish between children but... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 38064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38065,
"solution": "1. **Calculate the cost price of the candy bars:**\n The troop buys candy bars at a rate of 5 bars for $2 dollars. Therefore, the cost per bar is:\n \\[\n \\frac{2 \\text{ dollars}}{5 \\text{ bars}} = 0.4 \\text{ dollars per bar}\n \\]\n Since they buy 10... |
deepscaler | [
{
"role": "user",
"content": "Three not necessarily integer numbers are given. If each of these numbers is increased by 1, the product of the numbers also increases by 1. If each of these numbers is increased by 2, their product also increases by 2. Find these numbers. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( n \\in \\mathbf{N} \\) for which there exist a number \\( m \\in \\mathbf{N} \\), a triangle \\( ABC \\) with sides \\( AB = 33 \\), \\( AC = 21 \\), \\( BC = n \\), and points \\( D \\), \\( E \\) on sides \\( AB \\), \\( AC \\) respectively, satisfying ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the complex number $z$ satisfying the equation $2z-3\\bar{z}=-2-30i$. Note that $\\bar{z}$ denotes the conjugate of $z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2-6i"
} | {
"split": "train",
"index": 38068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In recent years, the food delivery industry in China has been developing rapidly, and delivery drivers shuttling through the streets of cities have become a beautiful scenery. A certain food delivery driver travels to and from $4$ different food delivery stores (numbered $1, 2,... | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"split": "train",
"index": 38069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-similar quadrilaterals have angles whose degree measures are distinct positive integers in an arithmetic progression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 38070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\(\\sqrt{114 + 44\\sqrt{6}}\\) and express it in the form \\(x + y\\sqrt{z}\\), where \\(x\\), \\(y\\), and \\(z\\) are integers and \\(z\\) has no square factors other than 1. Find \\(x + y + z\\). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 38071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{84}{144}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 38072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with radius 4 cm is tangent to three sides of a rectangle, as shown. The area of the rectangle is twice the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\\pi$.\n\n[asy]\nimport graph;\ndraw... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 38073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $46x^3-115x^7.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-23x^3(5x^4-2)"
} | {
"split": "train",
"index": 38074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a circular table, there are 5 people seated: Arnaldo, Bernaldo, Cernaldo, Dernaldo, and Ernaldo, each in a chair. Analyzing clockwise, we have:\n\nI. There is 1 empty chair between Arnaldo and Bernaldo;\n\nII. There are 5 chairs between Bernaldo and Cernaldo;\n\nIII. There a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 3x_2 + 5x_3 = 100.$ Find the smallest possible value of\n\\[ x_1^2 + x_2^2 + x_3^2. \\]\n\n--- Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2000}{7}"
} | {
"split": "train",
"index": 38076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Defined on $\\mathbf{R}$, the function $f$ satisfies\n$$\nf(1+x)=f(9-x)=f(9+x).\n$$\n\nGiven $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 38077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $x = k$ intersects the graph of the parabola $x = -2y^2 - 3y + 5$ at exactly one point. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{8}"
} | {
"split": "train",
"index": 38078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 10x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 38079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sets of 8 positive odd integers sum to 20 ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38080,
"solution": "This is the same as the number of ways 8 nonnegative even integers sum to 12 (we subtract 1 from each integer in the above sum). All 11 possibilities are (leaving out 0s): $12,10+2,8+4,8+2+2,6+6,6+4+2,6+2+2+2+2,4+4+4,4+4+2+2$, $4+2+2+2+2,2+2+2+2+2+2$."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ is an integer that satisfies the following congruences:\n\\[\n4 + x \\equiv 3^2 \\pmod{2^3}, \\\\\n6 + x \\equiv 2^3 \\pmod{3^3}, \\\\\n8 + x \\equiv 7^2 \\pmod{5^3}.\n\\]\nWhat is the remainder when $x$ is divided by $30$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38081,
"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, and it is given that $\\cos \\frac{A}{2}= \\frac{2 \\sqrt{5}}{5}, \\overrightarrow{AB} \\cdot \\overrightarrow{AC}=15$. \n$(1)$ Find the area of $\\triangle ABC$;... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"split": "train",
"index": 38082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a $\\spadesuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"split": "train",
"index": 38083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance betwee... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 38084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( t = \\frac{1}{1 - \\sqrt[4]{2}} \\), simplify the expression for \\( t \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-(1 + \\sqrt[4]{2})(1 + \\sqrt{2})"
} | {
"split": "train",
"index": 38085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height $BL$ of the rhombus $ABCD$, dropped perpendicular to the side $AD$, intersects the diagonal $AC$ at point $E$. Find $AE$ if $BL = 8$ and $AL:LD = 3:2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"split": "train",
"index": 38086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the positive integer divisors of 23? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a rectangular field that measures $20\\text{m}$ by $15\\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\\text{m}$ sides such that he is $6\\text{m}$ from a corner. He flies in a straight line to another point on the per... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive real numbers satisfying $a + 2b = 1$, find the minimum value of $a^2 + 4b^2 + \\frac{1}{ab}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"split": "train",
"index": 38089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $ab= \\frac{1}{4}$, $a$, $b \\in (0,1)$, find the minimum value of $\\frac{1}{1-a}+ \\frac{2}{1-b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4+ \\frac{4 \\sqrt{2}}{3}"
} | {
"split": "train",
"index": 38090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, $c$ are positive integers such that $a+b+c=23$ and $\\gcd(a,b)+\\gcd(b,c)+\\gcd(c,a)=9.$ What is the sum of all possible distinct values of $a^2+b^2+c^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "438"
} | {
"split": "train",
"index": 38091,
"solution": "We are given that $a+b+c=23$ and $\\gcd(a,b)+\\gcd(b,c)+\\gcd(c,a)=9$, where $a$, $b$, and $c$ are positive integers. We need to find the sum of all possible distinct values of $a^2+b^2+c^2$.\n\n#### Case Analysis:\nSince $a+b+c=23$ is odd, the integers $a$, $b$, a... |
deepscaler | [
{
"role": "user",
"content": "Point A is the intersection of the unit circle and the positive half of the x-axis, and point B is in the second quadrant. Let θ be the angle formed by the rays OA and OB. Given sin(θ) = 4/5, calculate the value of sin(π + θ) + 2sin(π/2 - θ) divided by 2tan(π - θ). Let's think ... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 38092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the line integral\n\n$$\n\\int_{L} \\frac{y}{3} d x - 3 x d y + x d z\n$$\n\nalong the curve \\( L \\), which is given parametrically by\n\n$$\n\\begin{cases}\nx = 2 \\cos t \\\\\ny = 2 \\sin t \\\\\nz = 1 - 2 \\cos t - 2 \\sin t\n\\end{cases}\n\\quad \\text{for} \\qu... | math | {
"style": "rule",
"ground_truth": "2 - \\frac{13\\pi}{3}"
} | {
"split": "train",
"index": 38093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b$ and $c$ be real numbers, and define the polynomial $P(x)=x^{2}+b x+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \\neq P(2)$. Find $P(0)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 38094,
"solution": "Since $P(P(1))=P(P(2))=0$, but $P(1) \\neq P(2)$, it follows that $P(1)=1+b+c$ and $P(2)=4+2 b+c$ are the distinct roots of the polynomial $P(x)$. Thus, $P(x)$ factors: $$P(x) =x^{2}+b x+c =(x-(1+b+c))(x-(4+2 b+c)) =x^{2}-(5+3 b+2 c) x+(1+b+c)(4+2 b+c)$$ It follows... |
deepscaler | [
{
"role": "user",
"content": "Find the measure of the angle \n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2907^{\\circ}+\\sin 2908^{\\circ}+\\cdots+\\sin 6507^{\\circ}\\right)^{\\cos 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}}\\right)\n$$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 38095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \\max\\{f(x),g(x),h(x)\\},$$$$k(x) = \\min\\{f(x),g(x),h(x)\\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greate... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 38096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\\_1$, $F\\_2$, $b = 4$, and an eccentricity of $\\frac{3}{5}$. A line passing through $F\\_1$ intersects the ellipse at points $A$ and $B$. Find the perimeter of $\\trian... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is formed using the digits \\(1, 2, \\ldots, 9\\). Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number?... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 38098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Say that a polynomial with real coefficients in two variables, $x,y$, is \\emph{balanced} if\nthe average value of the polynomial on each circle centered at the origin is $0$.\nThe balanced polynomials of degree at most $2009$ form a vector space $V$ over $\\mathbb{R}$.\nFind t... | math | {
"style": "rule",
"ground_truth": "2020050"
} | {
"split": "train",
"index": 38099,
"solution": "Any polynomial $P(x,y)$ of degree at most $2009$ can be written uniquely\nas a sum $\\sum_{i=0}^{2009} P_i(x,y)$ in which $P_i(x,y)$ is a homogeneous\npolynomial of degree $i$.\nFor $r>0$, let $C_r$ be the path $(r\\cos \\theta, r\\sin \\theta)$\nfor $0 \\leq \\the... |
deepscaler | [
{
"role": "user",
"content": "Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 38100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the planar vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, with magnitudes $|\\overrightarrow{a}| = 1$ and $|\\overrightarrow{b}| = 2$, and their dot product $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 1$. If $\\overrightarrow{e}$ is a unit vector in the pl... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 38101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{81}"
} | {
"split": "train",
"index": 38102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \\(S, T, U,\\) and \\(V\\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \\... | math | {
"style": "rule",
"ground_truth": "29.3"
} | {
"split": "train",
"index": 38103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has vertices at the corners and edges joining them similarly to a cube. The prism dimensions differ along each axis; therefore, no two adjoining sides are of the same length. If one side has a length ratio of 2:3 with another, and there are three dimensions ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], determine the range of the function g(x) on the interval [-2020,2020]. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "[-4043,4041]"
} | {
"split": "train",
"index": 38105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the magical forest of Santa Claus, cedars grow one and a half times taller than firs and grow for 9 hours. Firs grow for 2 hours. Santa Claus planted cedar seeds at 12 o'clock and fir seeds at 2 o'clock in the afternoon. What time was it when the trees were of the same heigh... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right-angled triangle has an area of \\( 36 \\mathrm{~m}^2 \\). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.\n\nDetermine the area of this square. L... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_n\\}$, it is given that $a_{13}=1$ and $a_{12} > a_{13}$. Find the largest integer $n$ for which $(a_1-\\frac{1}{a_1})+(a_2-\\frac{1}{a_2})+(a_3-\\frac{1}{a_3})+\\cdots+(a_n-\\frac{1}{a_n}) > 0$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 38109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan 2\\alpha= \\frac {3}{4}$, $\\alpha\\in(- \\frac {\\pi}{2}, \\frac {\\pi}{2})$, $f(x)=\\sin (x+\\alpha)+\\sin (\\alpha-x)-2\\sin \\alpha$, and for any $x\\in\\mathbb{R}$, it always holds that $f(x)\\geqslant 0$, find the value of $\\sin (\\alpha- \\frac {\\pi}{4})$.... | math | {
"style": "rule",
"ground_truth": "- \\frac {2 \\sqrt {5}}{5}"
} | {
"split": "train",
"index": 38110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, 3, \\cdots, 1995\\} \\) and \\( A \\subseteq M \\), with the constraint that if \\( x \\in A \\), then \\( 19x \\notin A \\). Find the maximum value of \\( |A| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1890"
} | {
"split": "train",
"index": 38111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $6 - 5\\left[7 - (\\sqrt{16} + 2)^2\\right] \\cdot 3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-429"
} | {
"split": "train",
"index": 38112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA \\( 5 \\mathrm{~cm} \\) by \\( 5 \\mathrm{~cm} \\) pegboard and a \\( 10 \\mathrm{~cm} \\) by \\( 10 \\mathrm{~cm} \\) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \\( 1 \\mathrm{~cm} \\) intervals from each edge. Peg... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $2$ and $100$ inclusive *cannot* be written as $m \\cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$ ? Note that there are $25$ primes less than $100.$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 38114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the base of the rectangular prism $A B C D-A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$ be a rhombus with an area of $2 \\sqrt{3}$ and $\\angle ABC = 60^\\circ$. Points $E$ and $F$ lie on edges $CC'$ and $BB'$, respectively, such that $EC = BC = 2FB$. What is the volume... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 38115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle HAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 38116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$.... | math | {
"style": "rule",
"ground_truth": "-0.25"
} | {
"split": "train",
"index": 38117,
"solution": "We are given a special key operation on a calculator that transforms the number $x$ displayed into $\\frac{1}{1-x}$. We need to determine the result after applying this operation 100 times starting with the number 5.\n\n1. **First Press:**\n \\[\n x_1 = \\frac{... |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{15})$ be a permutation of $(1,2,3,\\ldots,15)$ for which\n$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14} < a_{15}.$\nFind the number of such permutations. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 38118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38119,
"solution": "1. **Calculate the number of potatoes peeled by Homer alone**: Homer peels potatoes at a rate of 3 potatoes per minute. He peels alone for 4 minutes. Therefore, the number of potatoes peeled by Homer in these 4 minutes is:\n \\[\n 4 \\text{ minutes} \\times 3 \... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (\\cos \\frac{3x}{2}, \\sin \\frac{3x}{2})$ and $\\overrightarrow{b} = (\\cos \\frac{x}{2}, -\\sin \\frac{x}{2})$, with $x \\in \\left[-\\frac{\\pi}{3}, \\frac{\\pi}{4}\\right]$,\n(Ⅰ) Find $\\overrightarrow{a} \\cdot \\overrightarrow{b}$ and... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 38120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and t... | math | {
"style": "rule",
"ground_truth": "6084000"
} | {
"split": "train",
"index": 38121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The stem and leaf plot shows the heights, in inches, of the players on the Westvale High School boys' basketball team. Calculate the mean height of the players on the team. (Note: $6|2$ represents 62 inches.)\n\nHeight of the Players on the Basketball Team (inches)\n\n$5|7$\n\n... | math | {
"style": "rule",
"ground_truth": "68.6"
} | {
"split": "train",
"index": 38122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\\sqrt{17}$.\n(1) Find the equation of circle C;\n(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 38123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\frac{1}{2}\\cos^{2}x-\\frac{1}{2}\\sin^{2}x+1-\\sqrt{3}\\sin x \\cos x$. \n$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing. \n$(2)$ Find the minimum value of $f(x)$ on $[0,\\frac{\\pi}{2}]$ and the corresponding set of independent... | math | {
"style": "rule",
"ground_truth": "\\left\\{\\frac{\\pi}{3}\\right\\}"
} | {
"split": "train",
"index": 38124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \\(2^{10}\\). How many numbers will he have to w... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 38125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (piec... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 38126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first four terms of an arithmetic sequence is $10$. If the fifth term is $5$, what is the sixth term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 38127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $|(4\\sqrt{2}-4i)(\\sqrt{3}+3i)|$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 38128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\[\\cot(\\cot^{-1}3+\\cot^{-1}7+\\cot^{-1}13+\\cot^{-1}21).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 38129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-3 + 6 - 12 + 24 - \\cdots - 768$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "514"
} | {
"split": "train",
"index": 38130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 38131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 300 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 38132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the function $f(x)$ on $\\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\\left(\\frac{x}{5}\\right)=\\frac{1}{2}f(x)$, and for $0 \\leq x_1 < x_2 \\leq 1$, $f(x_1) \\leq f(x_2)$. Find the value of $f\\left(\\frac{1}{2007}\\right)$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"split": "train",
"index": 38133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $WXYZ$ is a trapezoid where $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. If $YZ = 20$, $\\tan Z = 2$, and $\\tan X = 2.5$, then what is the length of $XY$? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{116}"
} | {
"split": "train",
"index": 38134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. A... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 38135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ 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": "\\frac{1}{8}"
} | {
"split": "train",
"index": 38136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $x^2+kx+6=0$ has one root as $2$, find the other root and the value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 38137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 20x^3 - 8x^2 + 35x + 12$ and $x = -2$, apply Horner's method to calculate the value of $v_4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 38138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular parallelepiped with dimensions AB = 4, BC = 2, and CG = 5, point M is the midpoint of EF, calculate the volume of the rectangular pyramid with base BDFE and apex M. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 38139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 38140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangular prism $S-ABC$, where the base is an isosceles right triangle with $AB$ as the hypotenuse, $SA = SB = SC = 2$, and $AB = 2$, and points $S$, $A$, $B$, and $C$ all lie on a sphere centered at point $O$, find the distance from point $O$ to the plane $ABC$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 38141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 0, 1, 2, 3, 4, and 5, form six-digit numbers without repeating any digit. \n(1) How many such six-digit odd numbers are there? \n(2) How many such six-digit numbers are there where the digit 5 is not in the unit place? \n(3) How many such six-digit numbers a... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"split": "train",
"index": 38142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction t... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 38143,
"solution": "Disclaimer: This is for the people who may not understand calculus, and is also how I did it. First, we assume several things. First, we assume the speeds must be multiples of 15 to maximize cars, because any less will be a waste. Second, we start with one car in f... |
deepscaler | [
{
"role": "user",
"content": "Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}.$ What is the value of $a^{4} + a^{ - 4}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 38144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 20$, calculate the probability that 3 numbers randomly selected form an arithmetic sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{38}"
} | {
"split": "train",
"index": 38145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = 4$, and $\\vec{a}\\cdot \\vec{b}=2$, determine the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi }{3}"
} | {
"split": "train",
"index": 38146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $t(x) = 3x-8$ and $s(t(x)) = x^2 + 3x - 2$. Find $s(1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all values of $x$ such that the set $\\{107, 122,127, 137, 152,x\\}$ has a mean that is equal to its median. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "234"
} | {
"split": "train",
"index": 38148,
"solution": ""
} |
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