data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1,\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. ... | math | {
"style": "rule",
"ground_truth": "(8\\sqrt3, 14, 2, 4)"
} | {
"split": "train",
"index": 29049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a workshop, each participant has a 1 in 40 chance of being late. What is the probability that out of any three participants chosen at random, exactly one will be late? Express your answer as a percent rounded to the nearest tenth. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "7.1\\%"
} | {
"split": "train",
"index": 29050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The grade received on a certain teacher's 100-point test varies in direct proportion to the amount of time a student spends preparing for the test. If a student receives 72 points on a test for which she spent 3 hours preparing, what score would she receive on the next test if ... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 29051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( AB \\parallel EF \\parallel DC \\). Given that \\( AC + BD = 250 \\), \\( BC = 100 \\), and \\( EC + ED = 150 \\), find \\( CF \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, A and B are points moving on the x-axis and y-axis, respectively. If the circle C with AB as its diameter is tangent to the line $3x+y-4=0$, then the minimum area of circle C is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac {2}{5}\\pi"
} | {
"split": "train",
"index": 29053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\sqrt{16-8\\sqrt{3}}+\\sqrt{16+8\\sqrt{3}}$.\nA) $8\\sqrt{2}$\nB) $8\\sqrt{3}$\nC) $12\\sqrt{3}$\nD) $4\\sqrt{6}$\nE) $16$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 29054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given natural numbers \\( m \\) and \\( n \\) where \\( n > m > 1 \\), the last three digits of the decimal representation of \\( 1978^m \\) and \\( 1978^n \\) are the same. Find \\( m \\) and \\( n \\) such that \\( m+n \\) is minimized. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 29055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct diagonals of a convex heptagon (7-sided polygon) can be drawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 29056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 29057,
"solution": "1. **Identify the thickness per sheet**: Given that $500$ sheets of paper have a total thickness of $5$ cm, we can calculate the thickness of one sheet of paper. This is done by dividing the total thickness by the number of sheets:\n \\[\n \\text{Thickness per ... |
deepscaler | [
{
"role": "user",
"content": "Given that $y=f\\left(x\\right)+x^{2}$ is an odd function, and $f\\left(1\\right)=1$, if $g\\left(x\\right)=f\\left(x\\right)+2$, then $g\\left(-1\\right)=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 29059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain middle school, there are 180 students in both the eighth and ninth grades. To understand the physical health of students in these two grades, a sampling survey was conducted as follows:\n$(1)$ Data Collection:\nTwenty students were randomly selected from each of th... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 29060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can we fill the cells of a $4\\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 29061,
"solution": "\nTo solve the problem, we need to fill the cells of a \\(4 \\times 4\\) grid such that each cell contains exactly one positive integer, and the product of the numbers in each row and each column is 2020. We must determine the number of ways to achieve this configu... |
deepscaler | [
{
"role": "user",
"content": "If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 29062,
"solution": "1. **Identify the values of \\(y_1\\) and \\(y_2\\) using the given quadratic equation**: \n Given the quadratic equation \\(y = ax^2 + bx + c\\), we substitute the x-values of the points into the equation:\n - For the point \\((1, y_1)\\), substituting \\(x = ... |
deepscaler | [
{
"role": "user",
"content": "Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.\n\na) Write down any five-digit palindromic number that is divisible by 5.\n\nb) How many five-digit palindromic numbers are there that are... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an acute scalene triangle with circumcircle $\\omega$. The tangents to $\\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2... | math | {
"style": "rule",
"ground_truth": "717"
} | {
"split": "train",
"index": 29064,
"solution": "Using the Claim (below) we get $\\triangle ABC \\sim \\triangle XTM \\sim \\triangle YMT.$\n Corresponding sides of similar $\\triangle XTM \\sim \\triangle YMT$ is $MT,$ so\n$\\triangle XTM = \\triangle YMT \\implies MY = XT, MX = TY \\implies XMYT$ – parallelogra... |
deepscaler | [
{
"role": "user",
"content": "1. The converse of the proposition \"If $x > 1$, then ${x}^{2} > 1$\" is ________.\n2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.\n3. In a geom... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{5}"
} | {
"split": "train",
"index": 29065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two of the roots of \\[2x^3 + 8x^2 - 120x + k = 0\\]are equal, find the value of $k,$ given that $k$ is positive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{6400}{27}"
} | {
"split": "train",
"index": 29066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt[3]{1+27} \\cdot \\sqrt[3]{1+\\sqrt[3]{27}} \\cdot \\sqrt{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\cdot \\sqrt[3]{112}"
} | {
"split": "train",
"index": 29067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\\overline{a b c d e} \\cdot 9 = \\overline{e d c b a}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10989"
} | {
"split": "train",
"index": 29068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, points $M$ and $N$ lie on $\\overline{PQ}$ and $\\overline{PR}$, respectively. If $\\overline{PM}$ and $\\overline{QN}$ intersect at point $S$ so that $PS/SM = 4$ and $QS/SN = 3$, what is $RN/NQ$? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 29069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kevin Kangaroo starts at 0 on a number line and aims to reach the point 2, but his hopping strategy alternates. On odd-numbered hops, he covers $\\frac{1}{2}$ of the distance to his goal, while on even-numbered hops, he covers $\\frac{1}{4}$ of the distance remaining to the goa... | math | {
"style": "rule",
"ground_truth": "\\frac{485}{256}"
} | {
"split": "train",
"index": 29070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{P}$ such that for any vector $\\mathbf{v},$ $\\mathbf{P} \\mathbf{v}$ is the projection of $\\mathbf{v}$ onto the vector $\\begin{pmatrix} 2 \\\\ -2 \\\\ -1 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{4}{9} & -\\frac{4}{9} & -\\frac{2}{9} \\\\ -\\frac{4}{9} & \\frac{4}{9} & \\frac{2}{9} \\\\ -\\frac{2}{9} & \\frac{2}{9} & \\frac{1}{9} \\end{pmatrix}"
} | {
"split": "train",
"index": 29071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\dfrac{12}{11}\\cdot\\dfrac{15}{28}\\cdot\\dfrac{44}{45}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"split": "train",
"index": 29072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of a circle has coordinates $(6,-5)$. The circle is reflected about the line $y=x$. What are the $x,y$ coordinates of the center of the image circle? State the $x$ coordinate first. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-5, 6)"
} | {
"split": "train",
"index": 29073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all the positive factors of integer $x$ is 24. If one of the factors is 3, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Let \\( x_i \\in \\{0,1\\} \\) (for \\( i=1,2,\\cdots,n \\)). If the value of the function \\( f=f(x_1, x_2, \\cdots, x_n) \\) only takes 0 or 1, then \\( f \\) is called an n-ary Boolean function. We denote\n \\[\n D_{n}(f)=\\left\\{(x_1, x_2, \\cdots, x_n) \\mid f(x_1,... | math | {
"style": "rule",
"ground_truth": "565"
} | {
"split": "train",
"index": 29075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 474 inclusive does the decimal representation of $\\frac{n}{475}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 29076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of h... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 29077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of $23^{23}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z \\) be complex numbers such that\n\\[\nxy + 5y = -25, \\\\\nyz + 5z = -25, \\\\\nzx + 5x = -25.\n\\]\nFind all possible values of \\( xyz \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 29079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of t... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"split": "train",
"index": 29080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose you have three children and 40 pieces of candy. How many ways are there to distribute the candy such that each child gets more than one but fewer than 20 pieces? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 29081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Football tickets cost $\\$13.50$ each. What is the maximum number of tickets Jane can buy with $\\$100.00$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$. Let's... | math | {
"style": "rule",
"ground_truth": "a = 2, b = -14"
} | {
"split": "train",
"index": 29083,
"solution": "\nGiven the polynomial \\( Q(x) = x^3 - 21x + 35 \\), which has three different real roots, we need to find real numbers \\( a \\) and \\( b \\), such that the polynomial \\( P(x) = x^2 + ax + b \\) cyclically permutes the roots of \\( Q \\). Let the roots of \\( Q... |
deepscaler | [
{
"role": "user",
"content": "A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the... | math | {
"style": "rule",
"ground_truth": "998"
} | {
"split": "train",
"index": 29084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\mathrm{ABCD}$ is a right trapezoid with $\\angle \\mathrm{DAB} = \\angle \\mathrm{ABC} = 90^\\circ$. A rectangle $\\mathrm{ADEF}$ is constructed externally along $\\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\\mathrm{BE}$ intersects $\\mathrm... | math | {
"style": "rule",
"ground_truth": "3.18"
} | {
"split": "train",
"index": 29085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rhombus \\(ABCD\\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \\(B\\), draw the altitudes \\(BE\\) and \\(BF\\). Calculate the area of the quadrilateral \\(BFDE\\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4.32"
} | {
"split": "train",
"index": 29086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \\{ 1,2,\\ldots, mn\\}$. Suppose that there exists a $2n$-element set $T$ such that\r\n(a) each element of $T$ is an $m$-element subset of $S_{m}$;\r\n(b) each pair of elements of $T$ shares at most... | math | {
"style": "rule",
"ground_truth": "2n - 1"
} | {
"split": "train",
"index": 29087,
"solution": "\nLet \\( n \\) be an integer greater than 1. For a positive integer \\( m \\), let \\( S_{m} = \\{ 1, 2, \\ldots, mn \\} \\). Suppose that there exists a \\( 2n \\)-element set \\( T \\) such that:\n(a) each element of \\( T \\) is an \\( m \\)-element subset of \... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 29088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.15625"
} | {
"split": "train",
"index": 29089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Samia set off on her bicycle to visit her friend, traveling at an average speed of $17$ kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at $5$ kilometers per hour. In all it took her $44$ minut... | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"split": "train",
"index": 29090,
"solution": "1. **Define the total distance and split it into two equal parts**: Let the total distance Samia had to travel be $2x$ kilometers. This means she biked for $x$ kilometers and walked for $x$ kilometers.\n\n2. **Calculate the time for each part of the journey**:\n ... |
deepscaler | [
{
"role": "user",
"content": "Find $\\sin \\left( -\\frac{\\pi}{2} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a four-digit number such that the square of the sum of the two-digit number formed by its first two digits and the two-digit number formed by its last two digits is exactly equal to the four-digit number itself. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"split": "train",
"index": 29092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\nHow many different paths can the particle take f... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 29093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(AB = 2\\) and \\(AD = 1\\). Let \\(P\\) be a moving point on side \\(DC\\) (including points \\(D\\) and \\(C\\)), and \\(Q\\) be a moving point on the extension of \\(CB\\) (including point \\(B\\)). The points \\(P\\) and \\(Q\\) satisfy \\(|\\over... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 29094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a positive multiple of 8 and $x^2>100$, but $x<20$, what is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum value of the expression $x^2+y^2-6x+4y+18$ for real $x$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \\sqrt {2}\\sin ωx-\\cos ωx$ $(x\\in\\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 29097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a club of 60 members, the average age is 30 years. The club consists of 25 women, 20 men, and 15 senior citizens. If the average age of the women is 28 years and the average age of the men is 35 years, what is the average age of the senior citizens? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{80}{3}"
} | {
"split": "train",
"index": 29098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\\overarc{AB}$ on that face measures $120^\\t... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 29099,
"solution": "Label the points where the plane intersects the top face of the cylinder as $C$ and $D$, and the center of the cylinder as $O$, such that $C,O,$ and $A$ are collinear. Let $T$ be the center of the bottom face, and $M$ the midpoint of $\\overline{AB}$. Then $OT=4$, ... |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ and its derivative $f'(x)$ have a domain of $R$, and $f(x+2)$ is an odd function, ${f'}(2-x)+{f'}(x)=2$, ${f'}(2)=2$, then $\\sum_{i=1}^{50}{{f'}}(i)=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 29100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral $\\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\\triangle RDE$ and $\\triangle SDE$ form a $150^{\\circ}... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 29101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Painting the surface of a large metal ball requires 2.4 kilograms of paint. If this large metal ball is melted down to make 64 identical small metal balls, without considering any loss, the amount of paint needed to coat the surfaces of these small metal balls is \\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "9.6"
} | {
"split": "train",
"index": 29102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A beaver walks from $(0,0)$ to $(4,4)$ in the plane, walking one unit in the positive $x$ direction or one unit in the positive $y$ direction at each step. Moreover, he never goes to a point $(x, y)$ with $y>x$. How many different paths can he walk? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 29103,
"solution": "$C(4)=14$."
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{2, a-1\\}$, $B=\\{a^2-7, -1\\}$, and $A\\cap B=\\{2\\}$, find the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 29104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{x}$, for real numbers $m$, $n$, $p$, it is known that $f(m+n)=f(m)+f(n)$ and $f(m+n+p)=f(m)+f(n)+f(p)$. Determine the maximum value of $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\ln2-\\ln3"
} | {
"split": "train",
"index": 29105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$ be complex numbers such that \\frac{x^{2}+y^{2}}{x+y}=4$ and \\frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \\frac{x^{6}+y^{6}}{x^{5}+y^{5}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\pm 2 \\sqrt{17}"
} | {
"split": "train",
"index": 29106,
"solution": "Let $A=\\frac{1}{x}+\\frac{1}{y}$ and let $B=\\frac{x}{y}+\\frac{y}{x}$. Then $$ \\frac{B}{A}=\\frac{x^{2}+y^{2}}{x+y}=4 $$ so $B=4 A$. Next, note that $$ B^{2}-2=\\frac{x^{4}+y^{4}}{x^{2} y^{2}} \\text { and } A B-A=\\frac{x^{3}+y^{3}}{x^{2} y^{2}} $$ so $$ \\frac... |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\dfrac{20}{21} \\cdot \\dfrac{35}{54} \\cdot \\dfrac{63}{50}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 29107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 29108,
"solution": "1. **Identify the Lines and Their Intersection**: \n The lines given are $y = x$ and $y = 3x$. Both lines pass through the origin and form an acute angle there.\n\n2. **Calculate the Slopes of the Lines**:\n - The slope of the line $y = x$ is $1$.\n - The slo... |
deepscaler | [
{
"role": "user",
"content": "Compute the number of positive integers that divide at least two of the integers in the set $\\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 29109,
"solution": "For a positive integer $n$, let \\operatorname{rad} n be the product of the distinct prime factors of $n$. Observe that if $n \\mid m^{m}$, all prime factors of $n$ must divide $m$, so \\operatorname{rad} n \\mid m. Therefore, if $n$ is such an integer, \\operatorn... |
deepscaler | [
{
"role": "user",
"content": "A sphere is cut into four congruent wedges. The circumference of the sphere is $12\\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\\pi$.\n\nNote: To measure the circumference, take the largest circle on the surface o... | math | {
"style": "rule",
"ground_truth": "72\\pi"
} | {
"split": "train",
"index": 29110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three couples dine at the same restaurant every Saturday at the same table. The table is round and the couples agreed that:\n(a) under no circumstances should husband and wife sit next to each other; and\n(b) the seating arrangement of the six people at the table must be differ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$ . Let $R_1$ , $R_2$ , $R_3$ , $R_4$ be the circumradii of triangles $APB$ , $BPC$ , $CPD$ , $DPA$ respectively. If $R_1=31$ and ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 29112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school plans to purchase volleyball and basketball in one go. The price of each basketball is $30$ yuan more expensive than a volleyball. It costs a total of $340$ yuan to buy $2$ volleyballs and $3$ basketballs.<br/>$(1)$ Find the price of each volleyball and basketball.<br/... | math | {
"style": "rule",
"ground_truth": "3660"
} | {
"split": "train",
"index": 29113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The least common multiple of two integers is 36 and 6 is their greatest common divisor. What is the product of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 29114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$ 's place. Tom counted in the eggs, but his count was off by $1$ in the $10$ 's place. Raoul counted the eggs, but his count was off by $1$ in the $100$ 's place. Sasha, Jose, Pe... | math | {
"style": "rule",
"ground_truth": "439"
} | {
"split": "train",
"index": 29115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a point inside an equilateral triangle with side length \\( d \\) whose distances from the vertices are 3, 4, and 5 units. Find the side length \\( d \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{25 + 12 \\sqrt{3}}"
} | {
"split": "train",
"index": 29116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a WeChat group, five people, namely A, B, C, D, and E, are playing a game of grabbing red packets. There are $4$ red packets in total, each person can grab at most one, and all red packets will be grabbed. Among the $4$ red packets, there are two $2$ yuan packets, one $3$ yu... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $\\left( 2 \\sqrt{3}, \\frac{2 \\pi}{3} \\right)$ in polar coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-\\sqrt{3}, 3)"
} | {
"split": "train",
"index": 29118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shooter fires at a target until the first hit, with a hit rate of 0.6 for each shot. If there are 4 bullets in total, what is the expected number of remaining bullets $\\xi$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.376"
} | {
"split": "train",
"index": 29119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.\n\n[asy]\nimport three;\ntriple A,B,C,D,EE,F,G,H;\nA = (0,0,0);\nB = (1,0,0);\nC = (1,1,0);\nD= (0,1,0);\nEE = (0,0,1);\nF = B+EE;\nG = C + EE;\nH = D + EE;\ndraw(B--C--D);\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac16"
} | {
"split": "train",
"index": 29120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bike shed, there are bicycles (two wheels), tricycles, and cars (four wheels). The number of bicycles is four times the number of cars. Several students counted the total number of wheels in the shed, but each of them obtained a different count: $235, 236, 237, 238, 239$. ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 29121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\\%$ of the group are girls. How many girls were initially in the group? Let's think... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29122,
"solution": "1. **Define Variables:**\n Let $p$ be the total number of people initially in the group. Since $40\\%$ of the group are girls, the number of girls initially is $0.4p$.\n\n2. **Change in Group Composition:**\n After two girls leave and two boys arrive, the total... |
deepscaler | [
{
"role": "user",
"content": "Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "43.86\\%"
} | {
"split": "train",
"index": 29123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "15015"
} | {
"split": "train",
"index": 29124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 29125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $2^{x-3}=4^{x+1}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 29126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\\prod_{i=1}^6 h(y_i)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "676"
} | {
"split": "train",
"index": 29127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that positive integers \\( a, b, c \\) (\\( a < b < c \\)) form a geometric sequence, and\n\\[ \\log_{2016} a + \\log_{2016} b + \\log_{2016} c = 3, \\]\nfind the maximum value of \\( a + b + c \\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4066273"
} | {
"split": "train",
"index": 29128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD,\\ BC=8,\\ CD=12,\\ AD=10,$ and $m\\angle A= m\\angle B = 60^\\circ.$ Given that $AB = p + \\sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 29129,
"solution": "Draw the perpendiculars from $C$ and $D$ to $AB$, labeling the intersection points as $E$ and $F$. This forms 2 $30-60-90$ right triangles, so $AE = 5$ and $BF = 4$. Also, if we draw the horizontal line extending from $C$ to a point $G$ on the line $DE$, we find an... |
deepscaler | [
{
"role": "user",
"content": "On a rectangular sheet of paper, a picture in the shape of a \"cross\" was drawn using two rectangles $ABCD$ and $EFGH$, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$. Let's think... | math | {
"style": "rule",
"ground_truth": "52.5"
} | {
"split": "train",
"index": 29130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the chord cut by the line $y= \\frac{1}{2}x+1$ on the ellipse $x^2+4y^2=16$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{35}"
} | {
"split": "train",
"index": 29131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\\circ.$ \n$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$ \nSimilarly, $C_{1}$ is the fo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29132,
"solution": "\nGiven an acute triangle \\( ABC \\) inscribed in a circle with radius 1 and center \\( O \\), where all angles of \\( \\triangle ABC \\) are greater than \\( 45^\\circ \\), we are tasked with finding the circumradius of triangle \\( A_3B_3C_3 \\). The constructio... |
deepscaler | [
{
"role": "user",
"content": "Ten standard 6-sided dice are rolled. What is the probability that exactly one of the dice shows a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.323"
} | {
"split": "train",
"index": 29133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a right triangle is 1, and its hypotenuse is \\(\\sqrt{5}\\). Find the cosine of the acute angle between the medians of the triangle that are drawn to its legs. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{34}}{34}"
} | {
"split": "train",
"index": 29134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Base prime representation of a natural number is defined using the exponents of its prime factorization as follows. Each place in a base prime represents a prime number, and it is occupied by the corresponding exponent of that prime, starting on the right side with the smallest... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 29135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle and a regular pentagon have the same perimeter. Let $A$ be the area of the circle circumscribed about the rectangle, with the rectangle having side lengths in the ratio 3:1, and $B$ the area of the circle circumscribed around the pentagon. Find $A/B$.\nA) $\\frac{12... | math | {
"style": "rule",
"ground_truth": "\\frac{5(5-\\sqrt{5})}{64}"
} | {
"split": "train",
"index": 29136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer $n$, greater than $1$, such that $n^{-1}\\pmod{2310}$ is defined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 29137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\\left\\{{\\begin{array}{l}{x=3-\\frac{{\\sqrt{3}}}{2}t,}\\\\{y=\\sqrt{3}-\\frac{1}{2}t}\\end{array}}\\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 29138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers, $y > x > 0,$ such that\n\\[\\frac{x}{y} + \\frac{y}{x} = 6.\\]Find the value of \\[\\frac{x + y}{x - y}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 29139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $$ \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\infty} \\cdots \\sum_{a_{7}=0}^{\\infty} \\frac{a_{1}+a_{2}+\\cdots+a_{7}}{3^{a_{1}+a_{2}+\\cdots+a_{7}}} $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15309 / 256"
} | {
"split": "train",
"index": 29140,
"solution": "Note that, since this is symmetric in $a_{1}$ through $a_{7}$, $$ \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\infty} \\cdots \\sum_{a_{7}=0}^{\\infty} \\frac{a_{1}+a_{2}+\\cdots+a_{7}}{3^{a_{1}+a_{2}+\\cdots+a_{7}}} =7 \\sum_{a_{1}=0}^{\\infty} \\sum_{a_{2}=0}^{\\... |
deepscaler | [
{
"role": "user",
"content": "On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\\overline{2011 \\mathrm{ABCD}}$. What is $\\overline{\\mathr... | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"split": "train",
"index": 29141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $((5p+1)-2p\\cdot4)(3)+(4-1\\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13p-30"
} | {
"split": "train",
"index": 29142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the equation of the circle that is tangent to the x-axis, has its center on the line $3x - y = 0$, and the chord cut by the line $x - y = 0$ has a length of 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{7}"
} | {
"split": "train",
"index": 29143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 \\angle D$ and $\\angle BAC = m \\pi$ in radians, determine the value of $m$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"split": "train",
"index": 29144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $y_1 = x^2 - 7x + 6$, $y_2 = 7x - 3$, and $y = y_1 + xy_2$, find the value of $y$ when $x = 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 29145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a quadratic polynomial such that $[p(x)]^3 - x$ is divisible by $(x - 1)(x + 1)(x - 8).$ Find $p(13).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 29146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression $+1-2+3-4+5-6+7-8+9-11+12-\\cdots-199$. What is the value of the resulting exp... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 29147,
"solution": "Group the numbers into $(1-2+3-4+\\ldots+18-19)+(21-22+\\ldots+38-39)+\\ldots+(181-182+\\ldots+198-199)$. We can easily show that each group is equal to -10, and so the answer is -100."
} |
deepscaler | [
{
"role": "user",
"content": "A $3 \\times 3$ table is initially filled with zeros. In one move, any $2 \\times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a \"pattern\" any arrangement of crosses and zeros in the table. How many di... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29148,
"solution": ""
} |
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