data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be ... | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 35149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, 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": "2"
} | {
"split": "train",
"index": 35150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\frac{1}{x-3} + \\frac{1}{x-5} = \\frac{5}{x-4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 - \\frac{\\sqrt{15}}{3}"
} | {
"split": "train",
"index": 35151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.\n\n$$\ny = \\arcsin x, \\quad y = \\arccos x, \\quad y = 0\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 35152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many obtuse interior angles are in an obtuse triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 35153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A magician writes the numbers 1 to 16 on 16 positions of a spinning wheel. Four audience members, A, B, C, and D, participate in the magic show. The magician closes his eyes, and then A selects a number from the wheel. B, C, and D, in that order, each choose the next number in ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 35154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call a natural number \"remarkable\" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit \"remarkable\" numbers exist? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 35155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chessboard has its squares labeled according to the rule $\\frac{1}{c_i + r_j}$, where $c_i$ is the column number and $r_j$ is the row number. Eight squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 35156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $\\angle BAC = 90^\\circ$. A circle is tangent to the sides $AB$ and $AC$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $BC$. Given that $AB = 6$, find the area of the port... | math | {
"style": "rule",
"ground_truth": "\\pi - 2"
} | {
"split": "train",
"index": 35157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\frac{(N-1)!(N)}{(N+1)!}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{N+1}"
} | {
"split": "train",
"index": 35158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second t... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P_{1}: y=x^{2}+\\frac{101}{100}$ and $P_{2}: x=y^{2}+\\frac{45}{4}$ be two parabolas in the Cartesian plane. Let $\\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\\mathcal{L}$ is written in the form $ax+by=c$ for positive inte... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 35160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are la... | math | {
"style": "rule",
"ground_truth": "650"
} | {
"split": "train",
"index": 35161,
"solution": "For each $i$th switch (designated by $x_{i},y_{i},z_{i}$), it advances itself only one time at the $i$th step; thereafter, only a switch with larger $x_{j},y_{j},z_{j}$ values will advance the $i$th switch by one step provided $d_{i}= 2^{x_{i}}3^{y_{i}}5^{z_{i}}$ d... |
deepscaler | [
{
"role": "user",
"content": "How many ways can you tile the white squares of the following \\(2 \\times 24\\) grid with dominoes? (A domino covers two adjacent squares, and a tiling is a non-overlapping arrangement of dominoes that covers every white square and does not intersect any black square.) Let's t... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 35162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \\frac{1}{3}x^3 + \\frac{1}{2}ax^2 + bx$ has an extreme value. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{36}"
} | {
"split": "train",
"index": 35163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the determinant operation $\\begin{vmatrix} a_{1} & a_{2} \\\\ b_{1} & b_{2}\\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \\begin{vmatrix} \\sqrt {3} & \\sin x \\\\ 1 & \\cos x\\end{vmatrix}$. If the graph of this function is translated to the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"split": "train",
"index": 35164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 4-3x$ and $g(x) = x^2 +1$, find $f(g(\\sqrt{2}))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 35165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), a circle is constructed with diameter \\(AC\\), which intersects side \\(AB\\) at point \\(M\\) and side \\(BC\\) at point \\(N\\). Given that \\(AC = 2\\), \\(AB = 3\\), and \\(\\frac{AM}{MB} = \\frac{2}{3}\\), find \\(AN\\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{\\sqrt{145}}"
} | {
"split": "train",
"index": 35166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\\sin \\angle GAC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt{14}}"
} | {
"split": "train",
"index": 35167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vertices of a pentagon at coordinates $(1, 1)$, $(4, 1)$, $(5, 3)$, $(3, 5)$, and $(1, 4)$, calculate the area of this pentagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 35168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A motorcyclist left point A for point B, and at the same time, a pedestrian left point B for point A. When they met, the motorcyclist took the pedestrian on his motorcycle to point A and then immediately went back to point B. As a result, the pedestrian reached point A 4 times ... | math | {
"style": "rule",
"ground_truth": "2.75"
} | {
"split": "train",
"index": 35169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed t... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 35170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\frac {1}{3} \\leq a \\leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$. \n(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$); \n(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be ... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 35171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When ethane is mixed with chlorine gas under lighting conditions, determine the maximum number of substances that can be generated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of Natalie's blueberry bushes yields eight containers of blueberries. If she can trade five containers of blueberries for two zucchinis, how many bushes does Natalie need to pick in order to have forty-eight zucchinis? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Matěj had written six different natural numbers in a row in his notebook. The second number was double the first, the third was double the second, and similarly, each subsequent number was double the previous one. Matěj copied all these numbers into the following table in rando... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 35174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $254_{9} + 627_{9} + 503_{9}$. Express your answer in base 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1485_{9}"
} | {
"split": "train",
"index": 35175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a die is thrown twice in succession, the numbers obtained are recorded as $a$ and $b$, respectively. The probability that the line $ax+by=0$ and the circle $(x-3)^2+y^2=3$ have no points in common is ______. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 35176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area enclosed by the curves \\( y = \\sin x \\) and \\( y = \\left(\\frac{4}{\\pi}\\right)^{2} \\sin \\left(\\frac{\\pi}{4}\\right) x^{2} \\) (the latter is a quadratic function). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}\\left(1 + \\frac{\\pi}{12}\\right)"
} | {
"split": "train",
"index": 35177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sum shown, each of the letters \\( D, O, G, C, A \\), and \\( T \\) represents a different digit.\n$$\n\\begin{array}{r}\nD O G \\\\\n+C A T \\\\\n\\hline 1000\n\\end{array}\n$$\n\nWhat is the value of \\( D + O + G + C + A + T \\)? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 35178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on ... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 35179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be positive real numbers such that $a + b + c + d + e + f = 7.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{4}{b} + \\frac{9}{c} + \\frac{16}{d} + \\frac{25}{e} + \\frac{36}{f}.\\] Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 35180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $\\left\\{a_{n}\\right\\}_{n \\geq 1}$ of positive reals is defined by the rule $a_{n+1} a_{n-1}^{5}=a_{n}^{4} a_{n-2}^{2}$ for integers $n>2$ together with the initial values $a_{1}=8$ and $a_{2}=64$ and $a_{3}=1024$. Compute $$\\sqrt{a_{1}+\\sqrt{a_{2}+\\sqrt{a_{3}... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 35181,
"solution": "Taking the base-2 $\\log$ of the sequence $\\left\\{a_{n}\\right\\}$ converts the multiplicative rule to a more familiar additive rule: $\\log _{2}\\left(a_{n+1}\\right)-4 \\log _{2}\\left(a_{n}\\right)+5 \\log _{2}\\left(a_{n-1}\\right)-2 \\log _{2}\\left(a_{n-2}\... |
deepscaler | [
{
"role": "user",
"content": "Call an integer \\( n > 1 \\) radical if \\( 2^n - 1 \\) is prime. What is the 20th smallest radical number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4423"
} | {
"split": "train",
"index": 35182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ben rolls four fair 20-sided dice, and each of the dice has faces numbered from 1 to 20. What is the probability that exactly two of the dice show an even number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 35183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(500) = 3$, what is the value of $f(600)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 35184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation \"中环杯是 + 最棒的 = 2013\", different Chinese characters represent different digits. What is the possible value of \"中 + 环 + 杯 + 是 + 最 + 棒 + 的\"? (If there are multiple solutions, list them all). Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "1250 + 763"
} | {
"split": "train",
"index": 35185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "430"
} | {
"split": "train",
"index": 35186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of a rectangular yard exceeds twice its width by 30 feet, and the perimeter of the yard is 700 feet. What is the area of the yard in square feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25955.56"
} | {
"split": "train",
"index": 35187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a monic polynomial of degree 6 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and $p(6) = 6.$ Find $p(7).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "727"
} | {
"split": "train",
"index": 35188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ Enter all possible values of $k,$ separated by commas. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "1,-1,i,-i"
} | {
"split": "train",
"index": 35189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quagga is an extinct chess piece whose move is like a knight's, but much longer: it can move 6 squares in any direction (up, down, left, or right) and then 5 squares in a perpendicular direction. Find the number of ways to place 51 quaggas on an $8 \\times 8$ chessboard in su... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 35190,
"solution": "Represent the 64 squares of the board as vertices of a graph, and connect two vertices by an edge if a quagga can move from one to the other. The resulting graph consists of 4 paths of length 5 and 4 paths of length 3 (given by the four rotations of the two paths s... |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression \\[\\sqrt{37-20\\sqrt3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5-2\\sqrt3"
} | {
"split": "train",
"index": 35191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $(\\overrightarrow{a}+\\overrightarrow{b})\\perp\\overrightarrow{a}$, determine the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 35192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x)=a^{x-2}-2a (a > 0, a \\neq 1)$ always passes through the fixed point $\\left(x\\_0, \\frac{1}{3}\\right)$, then the minimum value of the function $f(x)$ on $[0,3]$ is equal to \\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 35193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $M(m,n)$ is any point on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.\n\n(I) Find the maximum and minimum values of $|MQ|$;\n\n(II) Find the maximum and minimum values of $\\frac{n-3}{m+2}$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2-\\sqrt{3}"
} | {
"split": "train",
"index": 35194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ between $1$ and $50$, inclusive, is $\\frac{(n^2-1)!}{(n!)^n}$ an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 35195,
"solution": "1. **Understanding the Expression**: We start by analyzing the expression \\[\\frac{(n^2-1)!}{(n!)^n}.\\] We need to determine for how many integers $n$ between $1$ and $50$ this expression is an integer.\n\n2. **Relating to a Known Integer Expression**: We know th... |
deepscaler | [
{
"role": "user",
"content": "When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers, all of which are nonprime? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 35196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ , point $D$ lies on side $AC$ such that $\\angle ABD=\\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\\sqrt{3}$ ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers \\( n \\) satisfy \\[ (n + 9)(n - 4)(n - 13) < 0 \\]? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 35198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$ .\nFind the greatest value of $M = abc$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6171"
} | {
"split": "train",
"index": 35199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( F \\) is the right focus of the ellipse \\(\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) (\\(a > b > 0\\)), and the eccentricity of the ellipse is \\(\\frac{\\sqrt{3}}{2}\\), a line \\( l \\) passing through point \\( F \\) intersects the ellipse at points... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 35200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{8 + x} + \\sqrt{15 - x} = 6$, what is the value of $(8 + x)(15 - x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{169}{4}"
} | {
"split": "train",
"index": 35201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy and Bobbi each selected a positive integer less than 500. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{50}"
} | {
"split": "train",
"index": 35202,
"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 satisfies $2b\\sin \\left(C+ \\frac {\\pi}{6}\\right)=a+c$. \n(I) Find the magnitude of angle $B$; \n(II) If point $M$ is the midpoint of $BC$, and $AM=AC... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {7}}{7}"
} | {
"split": "train",
"index": 35203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several girls (all of different ages) were picking white mushrooms in the forest. They distributed the mushrooms as follows: the youngest received 20 mushrooms and 0.04 of the remainder. The next oldest received 21 mushrooms and 0.04 of the remainder, and so on. It turned out t... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 35204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5),$ and the other on the sphere of radius 87 with center $(12,8,-16)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 35205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 35206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two ... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 35207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ is $60^{\\circ}$, and $\\overrightarrow{a}=2\\overrightarrow{e_{1}}-\\overrightarrow{e_{2}}$, find the projection of $\\overrightarrow{a}$ in the direction of $\\overrightarrow... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"split": "train",
"index": 35208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a \\(7 \\times 7\\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the number... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 35209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange the numbers $1,2,3,4,5,6$ on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 35210,
"solution": "Label the vertices of the hexagon $a b c d e f$. The numbers that are larger than both of their neighbors can't be adjacent, so assume (by rotation) that these numbers take up slots ace. We also have that 6 and 5 cannot be smaller than both of their neighbors, so a... |
deepscaler | [
{
"role": "user",
"content": "When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2395"
} | {
"split": "train",
"index": 35211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last two digits of the following sum: $$5! + 10! + 15! + \\cdots + 100!$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence? Express your answer in simplest radical form. [asy]\nsize(150); defaultpen(linewidth(2));\nreal loc = 0;\nfor(int i = 0; i < 11; ++i) {\n\n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"split": "train",
"index": 35213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say a point is contained in a square if it is in its interior or on its boundary. Three unit squares are given in the plane such that there is a point contained in all three. Furthermore, three points $A, B, C$, are given, each contained in at least one of the squares. Find ... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} / 2"
} | {
"split": "train",
"index": 35214,
"solution": "Let $X$ be a point contained in all three squares. The distance from $X$ to any point in any of the three squares is at most $\\sqrt{2}$, the length of the diagonal of the squares. Therefore, triangle $A B C$ is contained in a circle of radius $\\sqrt{2}$, so its c... |
deepscaler | [
{
"role": "user",
"content": "O is the center of square ABCD, and M and N are the midpoints of BC and AD, respectively. Points \\( A', B', C', D' \\) are chosen on \\( \\overline{AO}, \\overline{BO}, \\overline{CO}, \\overline{DO} \\) respectively, so that \\( A' B' M C' D' N \\) is an equiangular hexagon. ... | math | {
"style": "rule",
"ground_truth": "8634"
} | {
"split": "train",
"index": 35215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos \\left(x+ \\frac {\\pi}{3}\\right)$, derive the horizontal shift of the graph of the function $y=\\sin x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5\\pi}{6}"
} | {
"split": "train",
"index": 35216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of an arithmetic sequence $\\{a\\_n\\}$ is $S\\_n$, with $a\\_2 = 4$ and $S\\_{10} = 110$, find the minimum value of $\\frac{S\\_n + 64}{a\\_n}$. 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": 35217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in the expansion of $(1-2x)^{n} (n \\in \\mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 35218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three sets of cards in red, yellow, and blue, with five cards in each set, labeled with the letters $A, B, C, D,$ and $E$. If 5 cards are drawn from these 15 cards, with the condition that all letters must be different and all three colors must be included, how many d... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 35219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \\( B \\) (see the drawing). Find the measure of angle \\( ABC \\). \n\n(A polygon is called regular if all its sides are equal and all its angles are equal.) Let... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 35220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of positive integers $a_1, a_2, a_3, \\ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \\ldots$), let $b_j = k_1 + k_2 + \\ldots + k_j$ ($j=1, 2, 3, \\ldots$), \ndefine $g(m) = b_1 + b_2 + \\ldots + b_m - 100m$ ($m=1, 2, 3, \\ldot... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 35221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the decomposition rate $v$ of a certain type of garbage satisfies the function relationship $v=a\\cdot b^{t}$ (where $a$ and $b$ are non-zero constants) with time $t$ (unit: months), and after $6$ months, the decomposition rate is $5\\%$, and after $12$ months, the d... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 35222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest whole number value of $n$ that makes the following inequality true? $$\\frac13 + \\frac{n}7 < 1$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 35223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On square $ABCD$, points $E,F,G$, and $H$ lie on sides $\\overline{AB},\\overline{BC},\\overline{CD},$ and $\\overline{DA},$ respectively, so that $\\overline{EG} \\perp \\overline{FH}$ and $EG=FH = 34$. Segments $\\overline{EG}$ and $\\overline{FH}$ intersect at a point $P$, a... | math | {
"style": "rule",
"ground_truth": "850"
} | {
"split": "train",
"index": 35224,
"solution": "Continue in the same way as solution 1 to get that $POK$ has area $3a$, and $OK = \\frac{d}{10}$. You can then find $PK$ has length $\\frac 32$.\n Then, if we drop a perpendicular from $H$ to $BC$ at $L$, We get $\\triangle HLF \\sim \\triangle OPK$.\n Thus, $LF = ... |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volum... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 35225,
"solution": "Let $P'$ be the prism similar to $P$, and let the sides of $P'$ be of length $x,y,z$, such that $x \\le y \\le z$. Then\n\\[\\frac{x}{a} = \\frac{y}{b} = \\frac zc < 1.\\]\n Note that if the ratio of similarity was equal to $1$, we would have a prism with zero volu... |
deepscaler | [
{
"role": "user",
"content": "If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\\sin(\\pi x)$ ($0<x<2$), find the smallest positive period for $y=\\tan\\left(\\frac{(a+b)x}{2}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 35226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\sqrt{3}\\cos^2{\\omega}x+\\sin{\\omega}x\\cos{\\omega}x (\\omega>0)$, if there exists a real number $x_{0}$ such that for any real number $x$, $f(x_{0})\\leq f(x)\\leq f(x_{0}+2022\\pi)$ holds, then the minimum value of $\\omega$ is ____. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4044}"
} | {
"split": "train",
"index": 35227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of a trapezoid are mutually perpendicular, and one of them is 13. Find the area of the trapezoid if its height is 12. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1014/5"
} | {
"split": "train",
"index": 35228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3932"
} | {
"split": "train",
"index": 35229,
"solution": "To determine the total number of toothpicks used in constructing a grid that is $60$ toothpicks long and $32$ toothpicks wide, we need to consider both the vertical and horizontal toothpicks.\n\n1. **Vertical Toothpicks:**\n - Each vertical line extends from the ... |
deepscaler | [
{
"role": "user",
"content": "Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 35230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest even integer that cannot be written as the sum of two odd composite numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 35231,
"solution": "Let $n$ be an integer that cannot be written as the sum of two odd composite numbers. If $n>33$, then $n-9,n-15,n-21,n-25,n-27,$ and $n-33$ must all be prime (or $n-33=1$, which yields $n=34=9+25$ which does not work). Thus $n-9,n-15,n-21,n-27,$ and $n-33$ form a p... |
deepscaler | [
{
"role": "user",
"content": "Billy's age is twice Joe's age and the sum of their ages is 45. How old is Billy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{243}{1024}"
} | {
"split": "train",
"index": 35233,
"solution": "1. **Understanding the Problem**: We start with an equilateral triangle that is entirely black. Each time a change occurs, the middle fourth of each black triangle turns white. We need to determine the fraction of the original triangle that remains black after five... |
deepscaler | [
{
"role": "user",
"content": "In a two-day math competition, Charlie and Delta both attempted a total of 600 points. Charlie scored 200 points out of 400 attempted on the first day, and 160 points out of 200 attempted on the second day. Delta, who did not attempt 400 points on the first day, scored a positi... | math | {
"style": "rule",
"ground_truth": "\\frac{479}{600}"
} | {
"split": "train",
"index": 35234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, w... | math | {
"style": "rule",
"ground_truth": "923"
} | {
"split": "train",
"index": 35235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bob's Rice ID number has six digits, each a number from 1 to 9, and any digit can be used any number of times. The ID number satisfies the following property: the first two digits is a number divisible by 2, the first three digits is a number divisible by 3, etc. so that the ID... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 35236,
"solution": "We will count the number of possibilities for each digit in Bob's ID number, then multiply them to find the total number of possibilities for Bob's ID number. There are 3 possibilities for the first digit given any last 5 digits, because the entire number must be d... |
deepscaler | [
{
"role": "user",
"content": "Let $C=\\{ z \\in \\mathbb{C} : |z|=1 \\}$ be the unit circle on the complex plane. Let $z_1, z_2, \\ldots, z_{240} \\in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:\n(1) For any open arc $\\Gamma$ of length $\\pi$ on $C$, th... | math | {
"style": "rule",
"ground_truth": "80 + 40\\sqrt{3}"
} | {
"split": "train",
"index": 35237,
"solution": "\nLet \\( C = \\{ z \\in \\mathbb{C} : |z| = 1 \\} \\) be the unit circle on the complex plane. Let \\( z_1, z_2, \\ldots, z_{240} \\in C \\) (not necessarily different) be 240 complex numbers satisfying the following two conditions:\n1. For any open arc \\(\\Gamma... |
deepscaler | [
{
"role": "user",
"content": "A shop sells two kinds of products $A$ and $B$ at the price $\\$ 2100$. Product $A$ makes a profit of $20\\%$, while product $B$ makes a loss of $20\\%$. Calculate the net profit or loss resulting from this deal. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "-175"
} | {
"split": "train",
"index": 35238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $x \\ge 0$ the smallest value of $\\frac {4x^2 + 8x + 13}{6(1 + x)}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35239,
"solution": "1. **Define the function**: Let's denote the function by $f(x) = \\frac {4x^2 + 8x + 13}{6(1 + x)}$ for $x \\geq 0$.\n\n2. **Simplify the function**:\n \\[\n f(x) = \\frac {4x^2 + 8x + 13}{6(1 + x)} = \\frac{4(x^2+2x) + 13}{6(x+1)} = \\frac{4(x^2+2x+1-1)+13}{6(... |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the x-axis as the polar axis. Given circle C: ρ = 2cosθ - 2sinθ, and the parametric equation of line l is x = t, y = -1 + 2√2t (t is the para... | math | {
"style": "rule",
"ground_truth": "\\frac{10\\sqrt{5}}{9}"
} | {
"split": "train",
"index": 35240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What integer value of $n$ will satisfy $n + 10 > 11$ and $-4n > -12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of tim... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 35242,
"solution": "Let $p$ be the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. (We will call this event \"a win\", and the opposite event will be \"a loss\".)\n Let both players roll their first die.\n W... |
deepscaler | [
{
"role": "user",
"content": "Express the sum of $0.\\overline{123}+0.\\overline{0123}+0.\\overline{000123}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{123 \\times 1000900}{999 \\times 9999 \\times 100001}"
} | {
"split": "train",
"index": 35243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $|x| + x + y = 10$ and $x + |y| - y = 12,$ find $x + y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{5}"
} | {
"split": "train",
"index": 35244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from Stockholm to Malmö on a map is 120 cm. The scale on the map is 1 cm: 20 km. If there is a stop in between at Lund, which is 30 cm away from Malmö on the same map, how far is it from Stockholm to Malmö passing through Lund, in kilometers? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 35245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 cards, each with a number written on it: 1, 2, 2, 3, 4, 5, 6. Three cards are randomly drawn from these 7 cards. Let the minimum value of the numbers on the drawn cards be $\\xi$. Find $P\\left(\\xi =2\\right)=$____ and $E\\left(\\xi \\right)=$____. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"split": "train",
"index": 35246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square has a side length of $40\\sqrt{3}$ cm. Calculate the length of the diagonal of the square and the area of a circle inscribed within it. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200\\pi"
} | {
"split": "train",
"index": 35247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $5\\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 35248,
"solution": ""
} |
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