data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ACB$ is 30 degrees. If ray $CA$ is rotated 510 degrees about point $C$ clockwise, what will be the positive measure of the 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": 38749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 99 positive integers, and their sum is 101101. Find the greatest possible value of the greatest common divisor of these 99 positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 38750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that both $m$ and $n$ are non-negative integers, find the number of \"simple\" ordered pairs $(m, n)$ with a value of 2019. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 38751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly $12$ positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 38752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 38753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Exploring with a counter) Write any multiple of 3, cube each of its digits and add them together, then cube each digit of the new number obtained and add them together to get another new number, and keep repeating this process... \n(1) Write out the calculation formulas; \n(... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 38754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180^{\\circ}"
} | {
"split": "train",
"index": 38755,
"solution": "1. **Understanding the Problem**: We are given a quadrilateral inscribed in a circle. We need to find the sum of the angles inscribed in the four arcs cut off by the sides of the quadrilateral.\n\n2. **Total Arc in a Circle**: The total measure of the arcs in a cir... |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(1.90 \\frac{1}{1-\\sqrt[4]{3}}+\\frac{1}{1+\\sqrt[4]{3}}+\\frac{2}{1+\\sqrt{3}}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 38756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\sec \\frac{5 \\pi}{3}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 38757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ms. Johnson awards bonus points to students in her class whose test scores are above the median. The class consists of 81 students. What is the maximum number of students who could receive bonus points? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 38758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $Q(1,-1), R(-1,0)$ and $S(0,1)$ are three vertices of a parallelogram. What could be the coordinates of the fourth vertex of the parallelogram? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,2)"
} | {
"split": "train",
"index": 38759,
"solution": "We plot the first three vertices on a graph. We see that one possible location for the fourth vertex, $V$, is in the second quadrant. If $V S Q R$ is a parallelogram, then $S V$ is parallel and equal to $Q R$. To get from $Q$ to $R$, we go left 2 units and up 1 uni... |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of the positive integer \\(N\\) is three times the sum of the digits of \\(N+1\\). What is the smallest possible sum of the digits of \\(N\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the common foci of the ellipse $C_1: \\frac{x^2}{4} + y^2 = 1$ and the hyperbola $C_2$, and $A, B$ are the common points of $C_1$ and $C_2$ in the second and fourth quadrants, respectively. If the quadrilateral $AF_1BF_2$ is a rectangle, determine... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 38761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be positive integers such that\\[\\frac{3}{5} < \\frac{p}{q} < \\frac{2}{3}\\]and $q$ is as small as possible. What is $q - p$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discov... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{45}"
} | {
"split": "train",
"index": 38763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the product of all positive integer values of \\( c \\) such that \\( 9x^2 + 24x + c = 0 \\) has real roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20922789888000"
} | {
"split": "train",
"index": 38764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $(100110_2 + 1001_2) \\times 110_2 \\div 11_2$. Express your answer in base 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1011110_2"
} | {
"split": "train",
"index": 38765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the probability \\( P(B^c | A^c) \\) given the probabilities:\n\n\\[ P(A \\cap B) = 0.72, \\quad P(A \\cap \\bar{B}) = 0.18 \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.90"
} | {
"split": "train",
"index": 38766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two prime numbers $p$ so that $5 p$ can be expressed in the form $\\left\\lfloor\\frac{n^{2}}{5}\\right\\rfloor$ for some positive integer $n$. What is the sum of these two prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 38767,
"solution": "Note that the remainder when $n^{2}$ is divided by 5 must be 0,1 , or 4 . Then we have that $25 p=n^{2}$ or $25 p=n^{2}-1$ or $25 p=n^{2}-4$. In the first case there are no solutions. In the second case, if $25 p=(n-1)(n+1)$, then we must have $n-1=25$ or $n+1=25$ ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\dfrac{\\sqrt[6]{5}}{\\sqrt[4]{5}}$. What power of 5 does this expression represent? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{12}"
} | {
"split": "train",
"index": 38768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 38769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( A = \\{1, 2, 3, 4, 5, 6\\} \\) and the mapping \\( f: A \\rightarrow A \\). If the triple composition \\( f \\cdot f \\cdot f \\) is an identity mapping, how many such functions \\( f \\) are there? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 38770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For certain ordered pairs $(a,b)\\,$ of real numbers, the system of equations\n$ax+by=1\\,$\n$x^2+y^2=50\\,$\n has at least one solution, and each solution is an ordered pair $(x,y)\\,$ of integers. How many such ordered pairs $(a,b)\\,$ are there? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 38771,
"solution": "$x^2+y^2=50$ is the equation of a circle of radius $\\sqrt{50}$, centered at the origin. The lattice points on this circle are $(\\pm1,\\pm7)$, $(\\pm5,\\pm5)$, and $(\\pm7,\\pm1)$.\n$ax+by=1$ is the equation of a line that does not pass through the origin. (Since ... |
deepscaler | [
{
"role": "user",
"content": "The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has sides $EF=86$, $FG=60$, $GH=26$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find $EQ$, expressed as a fraction $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and co... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 38773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 38774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given function $f(x)$ defined on $\\mathbb{R}$ satisfies: (1) The graph of $y = f(x - 1)$ is symmetric about the point $(1, 0)$; (2) For all $x \\in \\mathbb{R}$, $$f\\left( \\frac {3}{4}-x\\right) = f\\left( \\frac {3}{4}+x\\right)$$ holds; (3) When $x \\in \\left(- \\frac {3}... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 38775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is a positive integer, then let $f(n)$ be the sum of the digits of $\\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $A$ is an angle such that $\\tan A + \\sec A = 2,$ enter all possible values of $\\cos A,$ separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 38777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=2\\sin (\\omega x+\\varphi )\\left( \\omega \\gt 0,\\left| \\varphi \\right|\\lt \\frac{\\pi }{2} \\right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\\left( \\frac{\\pi }{18},\\frac{\\pi }{3} \\right)... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant and $\\cos(α+π)=\\frac{4}{5}$, find the value of $\\tan 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"split": "train",
"index": 38779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A stock investment increased by 30% in 2006. Starting at this new value, what percentage decrease is needed in 2007 to return the stock to its original price at the beginning of 2006? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23.077\\%"
} | {
"split": "train",
"index": 38780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child gave Carlson 111 candies. They ate some of them right away, 45% of the remaining candies went to Carlson for lunch, and a third of the candies left after lunch were found by Freken Bok during cleaning. How many candies did she find? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + f(y)) = x + y\\]for all real numbers $x$ and $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\\equal{}f(x), g_{n\\plus{}1}(x)\\equal{}f(g_n(x))$ , for all $ n \\in N$ . Let $ r_n$ be the average of $ g_n(x)$ 's roots. If $ r_{19}\\equal{}99$ , find $ r_{99}... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 38783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty percent less than 60 is one-third more than what number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38784,
"solution": "1. **Calculate 20% less than 60**: \n To find 20% less than a number, we subtract 20% of the number from the number itself. \n \\[\n 60 - 0.20 \\times 60 = 60 - 12 = 48\n \\]\n Alternatively, we can calculate this as 80% of 60:\n \\[\n 0.80 \\times 60... |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\\overrightarrow{PF_{1}} \\cdot (\\overrightarrow{OF_{1}} + \\overrightarrow{OP}) = 0$, if $|\\overrightarrow{... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6} - \\sqrt{3}"
} | {
"split": "train",
"index": 38785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a 6 by 6 grid of points, what fraction of the larger square's area is inside the new shaded square? Place the bottom-left vertex of the square at grid point (3,3) and the square rotates 45 degrees (square's sides are diagonals of the smaller grid cells).\n```\n[asy]\nsize(6c... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"split": "train",
"index": 38786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 15 cards, each with 3 different Chinese characters. No two cards have the exact same set of Chinese characters, and in any set of 6 cards, there are always at least 2 cards that share a common Chinese character. What is the maximum number of different Chinese characte... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 38787,
"solution": ""
} |
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 $(p+q+r+s)^2 + (t+u+v+w)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^{x-3}=4^2$, find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of 81 consecutive integers is $9^5$. What is their median? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 38790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16:3"
} | {
"split": "train",
"index": 38791,
"solution": "1. **Convert the given ratios to equations:**\n - The ratio of $w$ to $x$ is given as $4:3$. This can be written as:\n \\[\n \\frac{w}{x} = \\frac{4}{3}\n \\]\n - The ratio of $y$ to $z$ is given as $3:2$. This can be written as:\n \\[\n \\f... |
deepscaler | [
{
"role": "user",
"content": "\nNeznaika does not know about multiplication and exponentiation operations. However, he is good at addition, subtraction, division, and square root extraction, and he knows how to use parentheses. While practicing, Neznaika chose three numbers 20, 2, and 2, and formed the expr... | math | {
"style": "rule",
"ground_truth": "20 + 10\\sqrt{2}"
} | {
"split": "train",
"index": 38792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \\(\\sqrt{6}\\). Calculate the length of the other side. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 38793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r,$ $s,$ $t$ be the roots of $2x^3 - 7x^2 - 6 = 0.$ Find $rst.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 38794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equations $z^2=4+4\\sqrt{15}i$ and $z^2=2+2\\sqrt 3i,$ where $i=\\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\\sqrt q-r\\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positiv... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38795,
"solution": "We solve each equation separately and find the vertices of the parallelogram formed by the solutions in the complex plane.\n\n#### Solving $z^2 = 4 + 4\\sqrt{15}i$:\nLet $z = r(\\cos\\theta + i\\sin\\theta) = r\\operatorname{cis}\\theta$, where $r$ is the magnitude... |
deepscaler | [
{
"role": "user",
"content": "The base of the quadrilateral pyramid \\( S A B C D \\) is a rhombus \\( A B C D \\) with an acute angle at vertex \\( A \\). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \\( S \\) onto the plane of the ba... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 38796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overset{ .}{a}=(\\sin x, \\frac{1}{2})$, $\\overset{ .}{b}=( \\sqrt {3}\\cos x+\\sin x,-1)$, and the function $f(x)= \\overset{ .}{a}\\cdot \\overset{ .}{b}$:\n\n(1) Find the smallest positive period of the function $f(x)$;\n(2) Find the maximum and minimum val... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d,$ $e$ be positive real numbers such that $a^2 + b^2 + c^2 + d^2 + e^2 = 100.$ Let $N$ be the maximum value of\n\\[ac + 3bc + 4cd + 8ce,\\]and let $a_N,$ $b_N$, $c_N,$ $d_N,$ $e_N$ be the values of $a,$ $b,$ $c,$ $d,$ $e,$ respectively, that produce the ma... | math | {
"style": "rule",
"ground_truth": "16 + 150\\sqrt{10} + 5\\sqrt{2}"
} | {
"split": "train",
"index": 38798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x) = \\frac{x}{5} + 3$ and $g(x) = 4-x$. If $f(g(a)) = 5$, find $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 38799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A new pyramid is added on one of the pentagonal faces of a pentagonal prism. Calculate the total number of exterior faces, vertices, and edges of the composite shape formed by the fusion of the pentagonal prism and the pyramid. What is the maximum value of this sum? Let's think... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Without using a calculator, find the largest prime factor of $15^4+2\\times15^2+1-14^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "211"
} | {
"split": "train",
"index": 38801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x_{k+1} = x_k + \\frac{1}{2}$ for $k=1, 2, \\dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \\dots + x_n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{n^2+3n}{4}"
} | {
"split": "train",
"index": 38802,
"solution": "1. **Identify the sequence type**: Given the recurrence relation $x_{k+1} = x_k + \\frac12$, we observe that each term increases by $\\frac12$ from the previous term. This indicates that the sequence is an arithmetic sequence.\n\n2. **Determine the first term and c... |
deepscaler | [
{
"role": "user",
"content": "While moving down an escalator, Walker counted 50 steps. Trotman, who moved three times faster than Walker, counted 75 steps. If the escalator were to stop, how many steps could be counted on its visible part? It is assumed that both individuals moved at a constant speed and th... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every card in a deck displays one of three symbols - star, circle, or square, each filled with one of three colors - red, yellow, or blue, and each color is shaded in one of three intensities - light, normal, or dark. The deck contains 27 unique cards, each representing a diffe... | math | {
"style": "rule",
"ground_truth": "702"
} | {
"split": "train",
"index": 38804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible length of an arithmetic progression formed of positive primes less than $1,000,000$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{n = 1}^{15} \\frac{n + 4}{n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11628"
} | {
"split": "train",
"index": 38806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{3}\\cos x\\sin x - \\frac{1}{2}\\cos 2x$.\n\n(1) Find the smallest positive period of $f(x)$.\n\n(2) Find the maximum and minimum values of $f(x)$ on the interval $\\left[0, \\frac{\\pi}{2}\\right]$ and the corresponding values of $x$. Let's th... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 38807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the sum $-1 + 2 - 3 + 4 - 5 + 6 - 7 +\\dots+ 10,\\!000$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"split": "train",
"index": 38808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(3\\sqrt{5},d+3)$ is $3d$ units away from the origin. What is the smallest possible value of $d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 38809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). What is the total number of dots NOT visible in this view? [asy]\n/* AMC8 2000 #8 Problem */\ndraw((0,0)--(1,0)--(1.5,0.... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 38810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area in the plane contained by the graph of\n\\[|2x + 3y| + |2x - 3y| \\le 12.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. T... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 38812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jia and his four friends each have a private car. The last digit of Jia's license plate is 0, and the last digits of his four friends' license plates are 0, 2, 1, 5, respectively. To comply with the local traffic restrictions from April 1st to 5th (cars with odd-numbered last d... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 38813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac {16!}{2^8}"
} | {
"split": "train",
"index": 38814,
"solution": "\nTo solve this problem, we need to determine the number of ways the spider can put on its socks and shoes such that each sock is put on before its corresponding shoe on each leg.\n\n#### Step-by-step Analysis:\n\n1. **Total Items to Wear:** The spider has 8 legs, ... |
deepscaler | [
{
"role": "user",
"content": "In the multiplication shown, $P, Q,$ and $R$ are all different digits such that\n\n$$\n\\begin{array}{r}\nP P Q \\\\\n\\times \\quad Q \\\\\n\\hline R Q 5 Q\n\\end{array}\n$$\n\nWhat is the value of $P + Q + R$? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 552 weights with masses of 1g, 2g, 3g, ..., 552g. Divide them into three equal weight piles. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50876"
} | {
"split": "train",
"index": 38816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $(3^{12} \\times 9^{-3})^2$? Write your answer as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "531441"
} | {
"split": "train",
"index": 38817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability of getting a sum of 30 when rolling a fair six-sided die 10 times, where the probability of each face (1, 2, 3, 4, 5, or 6) is $\\frac{1}{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0485"
} | {
"split": "train",
"index": 38818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your... | math | {
"style": "rule",
"ground_truth": "30.8"
} | {
"split": "train",
"index": 38819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure shown, segment $AB$ is parallel to segment $YZ$. If $AZ = 42$ units, $BQ = 12$ units, and $QY = 24$ units, what is the length of segment $QZ$? [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\npair Y = (0,0), Z = (16,0), A = (0,8), B... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 38820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2x - 5)(x + 7) - (x + 5)(2x - 1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 38821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 38822,
"solution": "First of all, draw a good diagram! This is always the key to solving any geometry problem. Once you draw it, realize that $EF=2, FA=16$. Why? Because since the hexagon is equiangular, we can put an equilateral triangle around it, with side length $6+8+10=24$. Then,... |
deepscaler | [
{
"role": "user",
"content": "Find the inverse of the matrix\n\\[\\begin{pmatrix} 6 & -4 \\\\ -3 & 2 \\end{pmatrix}.\\]If the inverse does not exist, then enter the zero matrix. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 38823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 38824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of integers \\(n \\in\\{1,2, \\ldots, 300\\}\\) such that \\(n\\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 38825,
"solution": "Let \\(n=p \\cdot q\\) for primes \\(p<q\\). If \\(n\\) is the second largest side of a right triangle there exist integers \\(c, a\\) such that \\(a<p q\\) and \\((p q)^{2}=c^{2}-a^{2}=(c-a)(c+a)\\). Since \\(c-a<c+a\\) there are three cases for the values of \\(c... |
deepscaler | [
{
"role": "user",
"content": "In the side face $A A^{\\prime} B^{\\prime} B$ of a unit cube $A B C D - A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\\prime} C^{\\prime}$ are equal. What is the minimum distance from a point on t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 38826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nathan will roll two six-sided dice. What is the probability that he will roll a number less than three on the first die and a number greater than three on the second die? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 38827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangle is inscribed in a circle with a diameter of $\\sqrt{12}$. Another regular triangle is constructed on its height as a side, and a new circle is inscribed in this triangle. Find the radius of this circle. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 38828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 38829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(4,3)$, respectively. What is its area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37.5\\sqrt{3}"
} | {
"split": "train",
"index": 38830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 12$, $BC = 35$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\overline{AI}$ ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle that satisfies $A B=13, B C=14, A C=15$. Given a point $P$ in the plane, let $P_{A}, P_{B}, P_{C}$ be the reflections of $A, B, C$ across $P$. Call $P$ good if the circumcircle of $P_{A} P_{B} P_{C}$ intersects the circumcircle of $A B C$ at exactly 1 ... | math | {
"style": "rule",
"ground_truth": "\\frac{4225}{64} \\pi"
} | {
"split": "train",
"index": 38832,
"solution": "By the properties of reflection, the circumradius of $P_{A} P_{B} P_{C}$ equals the circumradius of $A B C$. Therefore, the circumcircle of $P_{A} P_{B} P_{C}$ must be externally tangent to the circumcircle of $A B C$. Now it's easy to see that the midpoint of the ... |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, $c=2a$, $B={120}^{\\circ}$, and the area of $\\Delta ABC$ is $\\frac{\\sqrt{3}}{2}$.\n\n(I) Find the value of $b$;\n\n(II) Find the value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{5}"
} | {
"split": "train",
"index": 38833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the equation $x^3 - 6x^2 + 11x - 6 = 0$ have three real roots $a$, $b$, and $c$. Find $\\frac{1}{a^3} + \\frac{1}{b^3} + \\frac{1}{c^3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{251}{216}"
} | {
"split": "train",
"index": 38834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given: $\\sqrt{23.6}=4.858$, $\\sqrt{2.36}=1.536$, then calculate the value of $\\sqrt{0.00236}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.04858"
} | {
"split": "train",
"index": 38835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the ... | math | {
"style": "rule",
"ground_truth": "4(\\pi - \\sqrt{3})"
} | {
"split": "train",
"index": 38836,
"solution": "1. **Calculate the area of the circle:**\n The formula for the area of a circle is $\\pi r^2$. Given the radius $r = 2$, the area of the circle is:\n \\[\n \\text{Area of the circle} = \\pi \\times 2^2 = 4\\pi\n \\]\n\n2. **Calculate the area of the equilat... |
deepscaler | [
{
"role": "user",
"content": "If the solution set of the inequality $tx^2-6x+t^2<0$ with respect to $x$ is $(-\\infty, a) \\cup (1, +\\infty)$, then the value of $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 38837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two identical, uniform tetrahedral dice, each with the numbers $1$, $2$, $3$, $5$ on their faces, are rolled simultaneously and $x$ represents the sum of the numbers on the two faces facing down, determine the probability that $x$ is not less than $6$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube. Find $\\cos \\angle GAC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 38839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $8^x = 32$, then $x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 38840,
"solution": "1. **Express 8 and 32 as powers of 2**: \n We know that $8 = 2^3$ and $32 = 2^5$. Therefore, the equation $8^x = 32$ can be rewritten using the base 2:\n \\[\n (2^3)^x = 2^5\n \\]\n\n2. **Simplify the left-hand side**:\n Using the power of a power propert... |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided by $(x-13)(x-17)$? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2x-20"
} | {
"split": "train",
"index": 38841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit number $2\\Box\\Box5$ is divisible by $45$. How many such four-digit numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38843,
"solution": "1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:\n - Multiplies by $3$: $3x$\n - Adds $11$: $3x + 11$\n - Switches the digits of the result.... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence ${a_n}$, where $a_1=1$, and $P(a_n,a_{n+1})(n∈N^{+})$ is on the line $x-y+1=0$. If the function $f(n)= \\frac {1}{n+a_{1}}+ \\frac {1}{n+a_{2}}+ \\frac {1}{n+a_{3}}+…+ \\frac {1}{n+a_{n}}(n∈N^{\\*})$, and $n\\geqslant 2$, find the minimum value of the functio... | math | {
"style": "rule",
"ground_truth": "\\frac {7}{12}"
} | {
"split": "train",
"index": 38844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $X$ follows a normal distribution $N(2,\\sigma^{2})$, and its normal distribution density curve is the graph of the function $f(x)$, and $\\int_{0}^{2} f(x)dx=\\dfrac{1}{3}$, calculate $P(x > 4)$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 38845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The extension of the altitude \\( BH \\) of triangle \\( ABC \\) intersects the circumcircle at point \\( D \\) (points \\( B \\) and \\( D \\) lie on opposite sides of line \\( AC \\)). The measures of arcs \\( AD \\) and \\( CD \\) that do not contain point \\( B \\) are \\( ... | math | {
"style": "rule",
"ground_truth": "1: \\sqrt{3}"
} | {
"split": "train",
"index": 38846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point $P$ on the curve $y= \\frac{1}{2}e^{x}$ and a point $Q$ on the curve $y=\\ln (2x)$, determine the minimum value of $|PQ|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}(1-\\ln 2)"
} | {
"split": "train",
"index": 38847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an 11x11 grid making up a square, there are 121 uniformly spaced grid points including those on the edges. The point P is located in the very center of the square. A point Q is randomly chosen from the other 120 points. What is the probability that the line PQ is a line of s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 38848,
"solution": ""
} |
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