data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a square and $M$ and $N$ are the midpoints of $\\overline{BC}$ and $\\overline{CD},$ respectively. Find $\\sin \\theta.$\n\n[asy]\nunitsize(1.5 cm);\n\ndraw((0,0)--(2,0)--(2,2)--(0,2)--cycle);\ndraw((0,0)--(2,1));\ndraw((0,0)--(1,2));\nlabel(\"$A$\", (0,0), SW);\nlab... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 19849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\) be a set containing only positive integers, and for any elements \\( x \\) and \\( y \\) in \\( A \\), \\(|x-y| \\geq \\frac{x y}{30}\\). Determine at most how many elements \\( A \\) may contain. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ryan is learning number theory. He reads about the *Möbius function* $\\mu : \\mathbb N \\to \\mathbb Z$ , defined by $\\mu(1)=1$ and\n\\[ \\mu(n) = -\\sum_{\\substack{d\\mid n d \\neq n}} \\mu(d) \\]\nfor $n>1$ (here $\\mathbb N$ is the set of positive integers).\nHowe... | math | {
"style": "rule",
"ground_truth": "14013"
} | {
"split": "train",
"index": 19851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 19852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The U.S. produces about 8 million tons of apples each year. Initially, $30\\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\\%$ for each additional million tons. Of the remai... | math | {
"style": "rule",
"ground_truth": "2.24"
} | {
"split": "train",
"index": 19853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = |x-1| + |x+1|$, where $x \\in \\mathbb{R}$.\n\n(Ⅰ) Solve the inequality $f(x) \\leq 4$;\n\n(Ⅱ) If there exists a non-zero real number $b$ such that the inequality $f(x) \\geq \\frac{|2b+1| + |1-b|}{|b|}$ holds, find the maximum value of $x$ when $x$ is a negative nu... | math | {
"style": "rule",
"ground_truth": "-1.5"
} | {
"split": "train",
"index": 19854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first n terms of the sequence {a_n} is S_n, with a_1 = 2 and S_{n+1} = 2S_n - 1 for all positive integers n, find the value of a_{10}. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 19855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ \\triangle ABC $, the sides opposite angles A, B, C are respectively $ a, b, c $, with $ A = \\frac{\\pi}{4} $, $ \\sin A + \\sin(B - C) = 2\\sqrt{2}\\sin 2C $ and the area of $ \\triangle ABC $ is 1. Find the length of side $ BC $. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 19857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has side lengths 7, 8, and 9. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\\theta$ be the acute angle between these two lines. Find $\\tan \\theta.$\n\n[asy]\nunitsize(0.5 cm);\n\npair A, B, C, P, Q, R, S, X;... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{5} + 2 \\sqrt{10}"
} | {
"split": "train",
"index": 19858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which values of \\( x \\) and \\( y \\) the number \\(\\overline{x x y y}\\) is a square of a natural number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"split": "train",
"index": 19859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $$α∈(0， \\frac {π}{3})$$ and vectors $$a=( \\sqrt {6}sinα， \\sqrt {2})$$, $$b=(1，cosα- \\frac { \\sqrt {6}}{2})$$ are orthogonal,\n\n(1) Find the value of $$tan(α+ \\frac {π}{6})$$;\n\n(2) Find the value of $$cos(2α+ \\frac {7π}{12})$$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}- \\sqrt {30}}{8}"
} | {
"split": "train",
"index": 19860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ 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": "f(x) = 0"
} | {
"split": "train",
"index": 19861,
"solution": "\nTo find all functions \\( f:\\mathbb{R}\\rightarrow\\mathbb{R} \\) that satisfy the functional equation\n\n\\[\nf(x^2y) = f(xy) + yf(f(x) + y)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), let's proceed as follows:\n\n1. **Initial Substitution and Simplif... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = e^x \\cos x - x$.\n\n(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;\n\n(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \\frac{\\pi}{2}]$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 19862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 19863,
"solution": "1. **Identify the powers of $(1+i)$ and $(1-i)$:**\n We start by calculating $(1+i)^2$ and $(1-i)^2$:\n \\[\n (1+i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i\n \\]\n \\[\n (1-i)^2 = 1^2 - 2i + i^2 = 1 - 2i - 1 = -2i\n \\]\n\n2. **Raise both expressions to ... |
deepscaler | [
{
"role": "user",
"content": "Suppose you have an unlimited number of pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 19864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a set \\( \\mathcal{T} \\) of distinct positive integers such that for every integer \\( y \\) in \\( \\mathcal{T}, \\) the geometric mean of the values obtained by omitting \\( y \\) from \\( \\mathcal{T} \\) remains a positive integer. In addition, assume that 1 is a m... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line $2mx+ny-4=0$ passes through the point of intersection of the function $y=\\log _{a}(x-1)+2$ where $a>0$ and $a\\neq 1$, find the minimum value of $\\frac{1}{m}+\\frac{4}{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3+2\\sqrt{2}"
} | {
"split": "train",
"index": 19866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19867,
"solution": "1. **Understanding the problem**: Cindy was supposed to subtract 3 from a number $x$ and then divide by 9. Instead, she subtracted 9 and divided by 3, resulting in 43.\n\n2. **Setting up the equation for Cindy's incorrect operation**: \n \\[\n \\frac{x - 9}{3} ... |
deepscaler | [
{
"role": "user",
"content": "Let $f_1(x) = \\frac23 - \\frac3{3x+1}$, and for $n \\ge 2$, define $f_n(x) = f_1(f_{n-1}(x))$. The value of $x$ that satisfies $f_{1001}(x) = x-3$ can be expressed in the form $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 19868,
"solution": "After evaluating the first few values of $f_k (x)$, we obtain $f_4(x) = f_1(x) = \\frac{2}{3} - \\frac{3}{3x+1} = \\frac{6x-7}{9x+3}$. Since $1001 \\equiv 2 \\mod 3$, $f_{1001}(x) = f_2(x) = \\frac{3x+7}{6-9x}$. We set this equal to $x-3$, i.e.\n$\\frac{3x+7}{6-9x}... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\\frac{3}{5}$ and a minor axis length of $8$,\n\n(1) Find the standard equation of the ellipse $C$;\n\n(2) Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 19869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 19870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$=（sinx，cosx）, $\\overrightarrow {b}$=（1，$\\sqrt {3}$）.\n(1) If $\\overrightarrow {a}$$∥ \\overrightarrow {b}$, find the value of tanx;\n(2) Let f(x) = $\\overrightarrow {a}$$$\\cdot \\overrightarrow {b}$, stretch the horizontal coordinates of... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 19871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these 4 numbers? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19872,
"solution": "Two opposite vertices are relatively prime, but they both share a factor with their common neighbor. So that common neighbor must have two prime factors. So each of the 4 numbers has two prime factors, which are not shared with the opposite vertex. Moreover, it suf... |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\\angle C K L = \\angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ acc... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 19874,
"solution": "At each point of reflection, we pretend instead that the light continues to travel straight. [asy] pathpen = linewidth(0.7); size(250); real alpha = 28, beta = 36; pair B = MP(\"B\",(0,0),NW), C = MP(\"C\",D((1,0))), A = MP(\"A\",expi(alpha * pi/180),N); path r = C... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\\overrightarrow{m}=(\\sqrt{3}b-c,\\cos C)$, $\\overrightarrow{n}=(a,\\cos A)$. Given that $\\overrightarrow{m} \\parallel \\overrightarrow{n}$, determine the value o... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 19875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Let $\\frac{m}{n}$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le\\left|v+w\\right|$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "582"
} | {
"split": "train",
"index": 19876,
"solution": "Since $z^{1997}=1$, the roots will have magnitude $1$. Thus, the roots can be written as $\\cos(\\theta)+i\\sin(\\theta)$ and $\\cos(\\omega)+i\\sin(\\omega)$ for some angles $\\theta$ and $\\omega$. We rewrite the requirement as $\\sqrt{2+\\sqrt3}\\le|\\cos(\\thet... |
deepscaler | [
{
"role": "user",
"content": "Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 19877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let positive integers $a$, $b$, $c$ satisfy $ab + bc = 518$ and $ab - ac = 360$. The maximum value of $abc$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 19878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 19879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{12} \\times 5^8 $. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that sha... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 19881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two mutually perpendicular chords \\( AB \\) and \\( CD \\) are drawn in a circle. Determine the distance between the midpoint of segment \\( AD \\) and the line \\( BC \\), given that \\( BD = 6 \\), \\( AC = 12 \\), and \\( BC = 10 \\). If necessary, round your answer to two ... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 19882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \\ge 0$ lines. How many different values of $k$ are possible? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19883,
"solution": "To determine the number of different values of $k$, the number of lines simultaneously tangent to both circles, we analyze the possible configurations of the two circles:\n\n1. **Concentric Circles**: If the circles are concentric (i.e., they have the same center),... |
deepscaler | [
{
"role": "user",
"content": "Find the numerical value of\n\\[\\frac{\\sin 18^\\circ \\cos 12^\\circ + \\cos 162^\\circ \\cos 102^\\circ}{\\sin 22^\\circ \\cos 8^\\circ + \\cos 158^\\circ \\cos 98^\\circ}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 19884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and it satisfies $g(3i)=g(3+i)=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 19885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\\frac{m}{n},$ where $m$ and $... | math | {
"style": "rule",
"ground_truth": "051"
} | {
"split": "train",
"index": 19886,
"solution": "Denote by $N_{i,j}$ the optimal expected number of cards that Alice guesses correctly, where the number of cards are $i$ and $j \\ge i.$\n If $i = 0$ then Alice guesses correctly all cards, so $N_{0,j} = j.$\n If $j = i$ then Alice guesses next card with probabilit... |
deepscaler | [
{
"role": "user",
"content": "Mafia is a game where there are two sides: The village and the Mafia. Every night, the Mafia kills a person who is sided with the village. Every day, the village tries to hunt down the Mafia through communication, and at the end of every day, they vote on who they think the maf... | math | {
"style": "rule",
"ground_truth": "319/512"
} | {
"split": "train",
"index": 19887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the two-dimensional rectangular coordinate system, given the vector $\\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\\sin θ,t)(0≤θ≤\\frac {π}{2})$.\n(1) If $\\overrightarrow{AB} \\perp \\overrightarrow{a}$, and $|\\overrightarrow{AB}|= \\sqrt {5}|\\overright... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 19888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $ a,b,c,d$ are rational numbers with $ a>0$ , find the minimal value of $ a$ such that the number $ an^{3} + bn^{2} + cn + d$ is an integer for all integers $ n \\ge 0$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 19889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$14 N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $N$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "35964"
} | {
"split": "train",
"index": 19890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. If $A_{10}^{m} =10×9×…×5$, then $m=$ ______.\n2. The number of ways for A, B, C, and D to take turns reading the same book, with A reading first, is ______.\n3. If five boys and two girls are to be arranged in a row for a photo, with boy A required to stand in the middle and... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 19891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\in \\mathbf{Z}_{+} \\). When \\( n > 100 \\), the first two digits of the decimal part of \\( \\sqrt{n^{2}+3n+1} \\) are ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 19892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let triangle $A B C$ have $A B=5, B C=6$, and $A C=7$, with circumcenter $O$. Extend ray $A B$ to point $D$ such that $B D=5$, and extend ray $B C$ to point $E$ such that $O D=O E$. Find $C E$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}-3"
} | {
"split": "train",
"index": 19893,
"solution": "Because $O D=O E, D$ and $E$ have equal power with respect to the circle, so $(E C)(E B)=(D B)(D A)=50$. Letting $E C=x$, we have $x(x+6)=50$, and taking the positive root gives $x=\\sqrt{59}-3$."
} |
deepscaler | [
{
"role": "user",
"content": "The equations of the asymptotes of a hyperbola are $y = 2x+5$ and $y = -2x+1.$ Given that the hyperbola passes through the point $(0, 7),$ the standard form for the equation of the hyperbola is \\[\\frac{(y-k)^2}{a^2} - \\frac{(x-h)^2}{b^2} = 1,\\]where $a,$ $b$, $h,$ and $k$ a... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}-1"
} | {
"split": "train",
"index": 19894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $n \\times m$ maze is an $n \\times m$ grid in which each cell is one of two things: a wall, or a blank. A maze is solvable if there exists a sequence of adjacent blank cells from the top left cell to the bottom right cell going through no walls. (In particular, the top left... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19895,
"solution": "We must have both top-left and bottom-right cells blank, and we cannot have both top-right and bottom-left cells with walls. As long as those conditions are satisfied, the maze is solvable, so the answer is 3."
} |
deepscaler | [
{
"role": "user",
"content": "Given the complex numbers $z\\_1=a^2-2-3ai$ and $z\\_2=a+(a^2+2)i$, if $z\\_1+z\\_2$ is a purely imaginary number, determine the value of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 19897,
"solution": "To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.\n\n1. **Analyze the discriminants:**\n - For $x^2 + ... |
deepscaler | [
{
"role": "user",
"content": "On bookshelf A, there are 4 English books and 2 Chinese books, while on bookshelf B, there are 2 English books and 3 Chinese books. \n$(Ⅰ)$ Without replacement, 2 books are taken from bookshelf A, one at a time. Find the probability of getting an English book on the first draw... | math | {
"style": "rule",
"ground_truth": "\\frac{93}{280}"
} | {
"split": "train",
"index": 19898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sherry starts at the number 1. Whenever she's at 1, she moves one step up (to 2). Whenever she's at a number strictly between 1 and 10, she moves one step up or one step down, each with probability $\\frac{1}{2}$ . When she reaches 10, she stops. What is the expected numbe... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 19899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of solutions to\n\\[2\\sin^3 x - 5 \\sin^2 x + 2 \\sin x = 0\\]in the range $0 \\le x \\le 2 \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height of trapezoid $ABCD$ is 5, and the bases $BC$ and $AD$ are 3 and 5 respectively. Point $E$ is on side $BC$ such that $BE=2$. Point $F$ is the midpoint of side $CD$, and $M$ is the intersection point of segments $AE$ and $BF$. Find the area of quadrilateral $AMFD$. Let... | math | {
"style": "rule",
"ground_truth": "12.25"
} | {
"split": "train",
"index": 19901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$\\frac{4x-12}{x^2+2x-15}=x+2.$$If $p > q$, what is the value of $p - q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\\overline{AC}$ with $V$ on $\\overline{AW}$, points $X$ and $Y$ are on $\\overline{BC}$ with $X$ on $\\overline{CY}$, and points $Z$ and $U$ are on $\\overline{AB}$ with $Z$ on $\\overline{BU}$.... | math | {
"style": "rule",
"ground_truth": "318"
} | {
"split": "train",
"index": 19903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=4\\cos x\\sin \\left(x- \\frac{\\pi}{3}\\right)+a$ has a maximum value of $2$.\n\n$(1)$ Find the value of $a$ and the smallest positive period of the function $f(x)$;\n\n$(2)$ In $\\triangle ABC$, if $A < B$, and $f(A)=f(B)=1$, find the value of $\\frac... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 19904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exists a positive number $m$ such that the positive roots of the equation $\\sqrt{3} \\sin x - \\cos x = m$ form an arithmetic sequence in ascending order. If the point $A(1, m)$ lies on the line $ax + by - 2 = 0 (a > 0, b > 0)$, find the minimum value of $\\frac{1}{a} + ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 19905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n ($1$) for each $k,$ the math... | math | {
"style": "rule",
"ground_truth": "332"
} | {
"split": "train",
"index": 19906,
"solution": "It is a well-known fact that the set $0, a, 2a, ... (n-1)a$ forms a complete set of residues if and only if $a$ is relatively prime to $n$.\n Thus, we have $a$ is relatively prime to $n$. In addition, for any seats $p$ and $q$, we must have $ap - aq$ not be equival... |
deepscaler | [
{
"role": "user",
"content": "Find $\\tan A$ in the right triangle shown below.\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (40,0);\nC = (0,15);\ndraw(A--B--C--A);\ndraw(rightanglemark(B,A,C,20));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);\nlabel(\"$41$\", (B+C)/2,NE);\nlabel(\"$40$\", B/2,... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{40}"
} | {
"split": "train",
"index": 19907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $(a+1)x - 1 - \\ln x \\leqslant 0$ holds for any $x \\in [\\frac{1}{2}, 2]$, find the maximum value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 2\\ln 2"
} | {
"split": "train",
"index": 19908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four spheres, each with a radius of 1, are placed on a horizontal table with each sphere tangential to its neighboring spheres (the centers of the spheres form a square). There is a cube whose bottom face is in contact with the table, and each vertex of the top face of the cube... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 19909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 19910,
"solution": "Write $x+y=a / 60+b / 70=(7 a+6 b) / 420$. Since $a$ is relatively prime to 60 and $b$ is relatively prime to 70 , it follows that none of the primes $2,3,7$ can divide $7 a+6 b$, so we won't be able to cancel any of these factors in the denominator. Thus, after re... |
deepscaler | [
{
"role": "user",
"content": "Robert reads 90 pages per hour. How many 270-page books can he read in six hours? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight spheres of radius 2 are each tangent to the faces of an inner cube centered at the origin, with each sphere located in one of the octants, and the side length of the cube is 6. Find the radius of the smallest sphere, centered at the origin, that encloses these eight spher... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} + 2"
} | {
"split": "train",
"index": 19912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a round-robin chess tournament, 30 players are participating. To achieve the 4th category norm, a player needs to score 60% of the possible points. What is the maximum number of players who can achieve the category norm by the end of the tournament? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 19913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\), \\( A \\cup B = \\{a_1, a_2, a_3\\} \\). When \\( A \\neq B \\), pairs \\((A, B)\\) and \\((B, A)\\) are considered different. How many such pairs \\((A, B)\\) are there? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the first eight prime numbers that have a units digit of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "394"
} | {
"split": "train",
"index": 19915,
"solution": ""
} |
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": 19916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There were initially 2013 empty boxes. Into one of them, 13 new boxes (not nested into each other) were placed. As a result, there were 2026 boxes. Then, into another box, 13 new boxes (not nested into each other) were placed, and so on. After several such operations, there wer... | math | {
"style": "rule",
"ground_truth": "28182"
} | {
"split": "train",
"index": 19917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1<x_2<x_3$ be the three real roots of the equation $\\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19918,
"solution": "Let $x =\\frac{y}{\\sqrt{2014}}.$ The original equation simplifies to $\\frac{y^3}{2014} -\\frac{4029y^2}{2014}+2 = 0 \\implies y^3 - 4029y^2 + 4028=0.$ Here we clearly see that $y=1$ is a root. Dividing $y-1$ from the sum we find that $(y-1)(y^2-4028y-4028)=0.$ Fr... |
deepscaler | [
{
"role": "user",
"content": "Define $x \\star y=\\frac{\\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute $$((\\cdots((2007 \\star 2006) \\star 2005) \\star \\cdots) \\star 1)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{9}"
} | {
"split": "train",
"index": 19919,
"solution": "Note that $x \\star 2=\\frac{\\sqrt{x^{2}+6 x+4-2 x-4+4}}{2 x+4}=\\frac{\\sqrt{(x+2)^{2}}}{2(x+2)}=\\frac{1}{2}$ for $x>-2$. Because $x \\star y>0$ if $x, y>0$, we need only compute $\\frac{1}{2} \\star 1=\\frac{\\sqrt{\\frac{1}{4}+\\frac{3}{2}+1-3+4}}{\\frac{1}{2}... |
deepscaler | [
{
"role": "user",
"content": "One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\\%$ more than the number of seniors, and the mean score of the seniors was $50\\%$ higher than that of the non-senio... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 19920,
"solution": "1. **Define Variables:**\n Let $s$ be the number of seniors and $n$ be the number of non-seniors. According to the problem, the number of non-seniors is $50\\%$ more than the number of seniors, so we can write:\n \\[\n n = s + 0.5s = 1.5s\n \\]\n\n2. **Tota... |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent to the curve $y=x^{3}-4x$ at the point $(1,-3)$ is calculated in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}\\pi"
} | {
"split": "train",
"index": 19921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 19922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a paper, a $4 \\times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.\n\nObserve that the diagonal $AB$ intersects the grid at 9 points.\n\nIf the grid were of size $12 \\times 17$, at how many points would the diagonal $AB$ intersect the grid? Let's thi... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 19923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the residue of $-811\\pmod{24}$. Your answer should be an integer in the range $0,1,2,\\ldots,22,23$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set \\{1,2, \\ldots, 2012\\}. A perfectutation is a bijective function $h$ from $S$ to itself such that there exists an $a \\in S$ such that $h(a) \\neq a$, and that for any pair of integers $a \\in S$ and $b \\in S$ such that $h(a) \\neq a, h(b) \\neq b$, there ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19925,
"solution": "Note that both $f$ and $g$, when written in cycle notation, must contain exactly one cycle that contains more than 1 element. Assume $f$ has $k$ fixed points, and that the other $2012-k$ elements form a cycle, (of which there are (2011 - $k$ )! ways). Then note tha... |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5625"
} | {
"split": "train",
"index": 19926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following co... | math | {
"style": "rule",
"ground_truth": "3030"
} | {
"split": "train",
"index": 19927,
"solution": "\nTo address this problem, we need to determine the smallest number of beams that can be placed inside a \\(2020 \\times 2020 \\times 2020\\) cube such that they satisfy the given conditions: they must be \\(1 \\times 1 \\times 2020\\) and can only touch the faces ... |
deepscaler | [
{
"role": "user",
"content": "Find the area enclosed by the graph \\( x^{2}+y^{2}=|x|+|y| \\) on the \\( xy \\)-plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi + 2"
} | {
"split": "train",
"index": 19928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The \"Hua Luogeng\" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first \"Hua Cup\" is: \\( A_1 = 1 + 9 + 8 + 6 = 24 \\). The ... | math | {
"style": "rule",
"ground_truth": "629"
} | {
"split": "train",
"index": 19929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plan... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 19930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "18.4"
} | {
"split": "train",
"index": 19931,
"solution": "Since $P Q R S$ is rectangular, then $\\angle S R Q=\\angle S P Q=90^{\\circ}$. Also, $S R=P Q=20$ and $S P=Q R=15$. By the Pythagorean Theorem in $\\triangle S P Q$, since $Q S>0$, we have $Q S=\\sqrt{S P^{2}+P Q^{2}}=\\sqrt{15^{2}+20^{2}}=\\sqrt{225+400}=\\sqrt{6... |
deepscaler | [
{
"role": "user",
"content": "Define a modified Ackermann function \\( A(m, n) \\) with the same recursive relationships as the original problem: \n\\[ A(m,n) = \\left\\{\n\\begin{aligned} \n&n+1& \\text{ if } m = 0 \\\\\n&A(m-1, 1) & \\text{ if } m > 0 \\text{ and } n = 0 \\\\\n&A(m-1, A(m, n-1))&\\text{ i... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 19932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \\( \\frac{48}{\\sqrt{K}} \\). What is \\( K \\)? L... | math | {
"style": "rule",
"ground_truth": "\\frac{36}{\\pi^3}"
} | {
"split": "train",
"index": 19933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square \\(PQRS\\) with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc \\(TRU\\) has center \\(P\\); arc \\(VPW\\) has center \\(R\\); arc \\(UV\\) has center \\(S\\); and arc \\(WT\\) has center \\(Q\\). What is the length of t... | math | {
"style": "rule",
"ground_truth": "(2\\sqrt{2} - 1)\\pi"
} | {
"split": "train",
"index": 19934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When \\( n \\) is a positive integer, the function \\( f \\) satisfies \\( f(n+3)=\\frac{f(n)-1}{f(n)+1} \\), with \\( f(1) \\neq 0 \\) and \\( f(1) \\neq \\pm 1 \\). Find the value of \\( f(8) \\cdot f(2018) \\). Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three cars start simultaneously from City A, heading towards City B along the same highway. The second car travels 4 kilometers per hour less than the first car and 6 kilometers per hour more than the third car. The second car arrives at City B 3 minutes later than the first ca... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 19936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ (x_1,x_2,\\cdots)$ be a sequence of positive numbers such that $ (8x_2 \\minus{} 7x_1)x_1^7 \\equal{} 8$ and\r\n\\[ x_{k \\plus{} 1}x_{k \\minus{} 1} \\minus{} x_k^2 \\equal{} \\frac {x_{k \\minus{} 1}^8 \\minus{} x_k^8}{x_k^7x_{k \\minus{} 1}^7} \\text{ for }k \\equa... | math | {
"style": "rule",
"ground_truth": "8^{1/8}"
} | {
"split": "train",
"index": 19937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 un... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 19938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning. Le... | math | {
"style": "rule",
"ground_truth": "258"
} | {
"split": "train",
"index": 19939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $4a\\sin B = \\sqrt{7}b$. \n$(1)$ If $a = 6$ and $b+c = 8$, find the area of $\\triangle ABC$. \n$(2)$ Find the value of $\\sin (2A+\\frac{2\\pi}{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}-3\\sqrt{7}}{16}"
} | {
"split": "train",
"index": 19940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $5-7\\left(8-3^2\\right)4.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 19941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A recent report about the amount of plastic created in the last 65 years stated that the 8.3 billion tonnes produced is as heavy as 25000 Empire State Buildings in New York or a billion elephants. On that basis, how many elephants have the same total weight as the Empire State ... | math | {
"style": "rule",
"ground_truth": "40000"
} | {
"split": "train",
"index": 19942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten students are taking both algebra and drafting. There are 24 students taking algebra. There are 11 students who are taking drafting only. How many students are taking algebra or drafting but not both? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 19943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find: $\\frac{2}{7}+\\frac{8}{10}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{38}{35}"
} | {
"split": "train",
"index": 19944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ points $D$ and $E$ lie on $\\overline{BC}$ and $\\overline{AC}$, respectively. If $\\overline{AD}$ and $\\overline{BE}$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$?\n\n[asy]\npair A,B,C,D,I,T;\nA=(0,0);\nB=(6,8);\nC=(11,0);\nD=(9.33,2.... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 19945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon a... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 19946,
"solution": "Alternatively, use basic geometry. First, scale everything down by dividing everything by 6. Let $a/6=p$. Then, the dimensions of the central rectangle in the hexagon is p x 4, and the original rectangle is 6 x p. By Pythagorean theorem and splitting the end triang... |
deepscaler | [
{
"role": "user",
"content": "Express $537_8 + 1C2E_{16}$ as a base 10 integer, where $C$ and $E$ denote the hexadecimal digits with values 12 and 14 respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7565"
} | {
"split": "train",
"index": 19947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 19948,
"solution": "To find the fractional part of the students who walk home, we need to subtract the fractions of students who use other means of transportation from the whole.\n\n1. **Identify the fractions for each mode of transportation:**\n - School bus: $\\frac{1}{2}$\n - A... |
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