data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The measures of angles $X$ and $Y$ are both positive, integer numbers of degrees. The measure of angle $X$ is a multiple of the measure of angle $Y$, and angles $X$ and $Y$ are supplementary angles. How many measures are possible for angle $X$? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 28749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^3y^5$ in the expansion of $\\left(\\frac{2}{3}x - \\frac{y}{3}\\right)^8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{448}{6561}"
} | {
"split": "train",
"index": 28750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30!}{(10!)^3}"
} | {
"split": "train",
"index": 28751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add the square of the smallest area from squares of size $1 \\times 1, 2 \\times 2,$ and $3 \\times 3,$ such that the number of squares of each size is the same. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 28752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ is magical if $\\lfloor\\sqrt{\\lceil\\sqrt{n}\\rceil}\\rfloor=\\lceil\\sqrt{\\lfloor\\sqrt{n}\\rfloor}\\rceil$ where $\\lfloor\\cdot\\rfloor$ and $\\lceil\\cdot\\rceil$ represent the floor and ceiling function respectively. Find the number of magical int... | math | {
"style": "rule",
"ground_truth": "1330"
} | {
"split": "train",
"index": 28753,
"solution": "First of all, we have $\\lfloor\\sqrt{n}\\rfloor=\\lceil\\sqrt{n}\\rceil$ when $n$ is a perfect square and $\\lfloor\\sqrt{n}\\rfloor=\\lceil\\sqrt{n}\\rceil-1$ otherwise. Therefore, in the first case, the original equation holds if and only if $\\sqrt{n}$ is a per... |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first n terms of the sequence {a_n} is S_n, and a_{n+1}+a_n=2^n, find the value of S_{10}. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "682"
} | {
"split": "train",
"index": 28754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One focus of the ellipse $\\frac{x^2}{2} + y^2 = 1$ is at $F = (1,0).$ There exists a point $P = (p,0),$ where $p > 0,$ such that for any chord $\\overline{AB}$ that passes through $F,$ angles $\\angle APF$ and $\\angle BPF$ are equal. Find $p.$\n\n[asy]\nunitsize(2 cm);\n\np... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 28755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, a line $l_{1}$ is given by the equation $y = \\tan \\alpha \\cdot x \\ (0 \\leqslant \\alpha < \\pi, \\alpha \\neq \\frac{\\pi}{2})$, and a parabola $C$ is given by the parametric equations $\\begin{cases} x = t^{2} \\\\ y = -2t \\e... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $x$ , we let $\\lfloor x \\rfloor$ be the unique integer $n$ such that $n \\leq x < n+1$ . For example. $\\lfloor 31.415 \\rfloor = 31$ . Compute \\[2020^{2021} - \\left\\lfloor\\frac{2020^{2021}}{2021} \\right \\rfloor (2021).\\]\n\n\n*2021 CCA Math... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 28757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:\n\n\n\n- All ... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 28758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a}, \\mathbf{b},$ and $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 2,$ $\\|\\mathbf{b}\\| = 3,$ and $\\|\\mathbf{c}\\| = 6,$ and\n\\[\\mathbf{a} + 2\\mathbf{b} + \\mathbf{c} = \\mathbf{0}.\\]\nCompute $\\mathbf{a} \\cdot \\mathbf{b} + \\mathbf{a} \\cdot... | math | {
"style": "rule",
"ground_truth": "-19"
} | {
"split": "train",
"index": 28759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin(-60^\\circ)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 28760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P_1,P_2,\\dots,P_n$ be $n$ distinct points over a line in the plane ($n\\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\\leq{i,j}\\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.\n\nFor each positive i... | math | {
"style": "rule",
"ground_truth": "n \\geq 2^k + 1"
} | {
"split": "train",
"index": 28761,
"solution": "Consider \\( n \\) distinct points \\( P_1, P_2, \\ldots, P_n \\) arranged on a line in the plane, and we define circumferences using these points as diameters \\( P_iP_j \\) for \\( 1 \\leq i < j \\leq n \\). Each circumference is colored using one of \\( k \\) co... |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$ in the equation $\\frac{4}{7} \\cdot \\frac{1}{9} \\cdot x = 14$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220.5"
} | {
"split": "train",
"index": 28762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane Cartesian coordinate system $xOy$, let two unit vectors in the directions of the $x$-axis and $y$-axis be $\\overrightarrow{i}$ and $\\overrightarrow{j}$, respectively. Given that $\\overrightarrow{OA}=2\\overrightarrow{i}+\\overrightarrow{j}$ and $\\overrightarrow... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 28763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be a positive integer, and let $a_0, a_1,\\ldots,a_m$ be a sequence of reals such that $a_0 = 37, a_1 = 72, a_m = 0,$ and $a_{k+1} = a_{k-1} - \\frac 3{a_k}$ for $k = 1,2,\\ldots, m-1.$ Find $m.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "889"
} | {
"split": "train",
"index": 28764,
"solution": "For $0 < k < m$, we have\n$a_{k}a_{k+1} = a_{k-1}a_{k} - 3$.\n Thus the product $a_{k}a_{k+1}$ is a monovariant: it decreases by 3 each time $k$ increases by 1. For $k = 0$ we have $a_{k}a_{k+1} = 37\\cdot 72$, so when $k = \\frac{37 \\cdot 72}{3} = 888$, $a_{k}a_{... |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 28765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to construct this dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the oth... | math | {
"style": "rule",
"ground_truth": "7983360"
} | {
"split": "train",
"index": 28766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(x-1)(x+3)(x-4)(x-8)+m$ is a perfect square, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 28767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 28768,
"solution": "The problem is equivalent to finding the number of ways to partition 10 into a sum of three (unordered) positive integers. These can be computed by hand to be $(1,1,8),(1,2,7),(1,3,6),(1,4,5),(2,2,6),(2,3,5),(2,4,4),(3,3,4)$"
} |
deepscaler | [
{
"role": "user",
"content": "A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes a... | math | {
"style": "rule",
"ground_truth": "4466\\pi"
} | {
"split": "train",
"index": 28769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of eight-digit positive integers that are multiples of 9 and have all distinct digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "181440"
} | {
"split": "train",
"index": 28770,
"solution": "Note that $0+1+\\cdots+9=45$. Consider the two unused digits, which must then add up to 9. If it's 0 and 9, there are $8 \\cdot 7!$ ways to finish; otherwise, each of the other four pairs gives $7 \\cdot 7!$ ways to finish, since 0 cannot be the first digit. This g... |
deepscaler | [
{
"role": "user",
"content": "In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:\n\n$$\n0,2,3,4,4,7,8,8,9\n$$\n\nWhat is this number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 28772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with sides of length $4$ and $2$ is rolled into the lateral surface of a cylinder. The volume of the cylinder is $\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\pi}"
} | {
"split": "train",
"index": 28773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 28774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $e^{i \\theta} = \\frac{3 + i \\sqrt{8}}{4},$ then find $\\sin 6 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{855 \\sqrt{2}}{1024}"
} | {
"split": "train",
"index": 28775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$, $d$ are the thousands, hundreds, tens, and units digits of a four-digit number, respectively, and the digits in lower positions are not less than those in higher positions. When $|a-b|+|b-c|+|c-d|+|d-a|$ reaches its maximum value, the minimum value of ... | math | {
"style": "rule",
"ground_truth": "1119"
} | {
"split": "train",
"index": 28776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 28777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\frac{\\cos A}{\\cos C}=\\frac{a}{2b-c}$, find:<br/>\n$(1)$ The measure of angle $A$;<br/>\n$(2)$ If $a=\\sqrt{7}$, $c=3$, and $D$ is the midpoint of... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{19}}{2}"
} | {
"split": "train",
"index": 28778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the area of the parallelogram formed by the vectors \\( a \\) and \\( b \\).\n\n$$\n\\begin{aligned}\n& a = p - 4q \\\\\n& b = 3p + q \\\\\n& |p| = 1 \\\\\n& |q| = 2 \\\\\n& \\angle(p, q) = \\frac{\\pi}{6}\n\\end{aligned}\n$$ Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 28779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BC$ at points $D$ and $E$, respectively. The length of $AE$ is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 28780,
"solution": "1. **Identify the Geometry of the Problem:**\n - Given that $AB$ is a diameter of a circle with radius $1$, the length of $AB$ is $2$ (since the diameter is twice the radius).\n - $AB$ is also a side of an equilateral triangle $ABC$. Therefore, all sides of tri... |
deepscaler | [
{
"role": "user",
"content": "Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 28781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, three lines meet at the points \\( A, B \\), and \\( C \\). If \\( \\angle ABC = 50^\\circ \\) and \\( \\angle ACB = 30^\\circ \\), the value of \\( x \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? Let's thi... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 28783,
"solution": "1. **Understanding the Offer**: The store offers one free window for every four purchased. This means for every five windows considered (four purchased + one free), the cost is for four windows only.\n\n2. **Calculating Individual Savings**:\n - **Dave's Purchase... |
deepscaler | [
{
"role": "user",
"content": "Quantities \\(r\\) and \\( s \\) vary inversely. When \\( r \\) is \\( 1500 \\), \\( s \\) is \\( 0.4 \\). Alongside, quantity \\( t \\) also varies inversely with \\( r \\) and when \\( r \\) is \\( 1500 \\), \\( t \\) is \\( 2.5 \\). What is the value of \\( s \\) and \\( t \... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 28784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers $a, b$ and $c$ is 60. If we decrease $a$ by 7, we get the value $N$. If we increase $b$ by 7, we get the value $N$. If we multiply $c$ by 7, we also get the value $N$. What is the value of $N$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 28785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\\{ 2, 3, 5, 7 \\}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1932"
} | {
"split": "train",
"index": 28786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn the trapezoid \\( ABCD \\) with bases \\( AD \\) and \\( BC \\), the side \\( AB \\) is equal to 2. The angle bisector of \\( \\angle BAD \\) intersects the line \\( BC \\) at point \\( E \\). A circle is inscribed in triangle \\( ABE \\), touching side \\( AB \\) at point... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 28787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)$ be the number of times you have to hit the $\\sqrt{ }$ key on a calculator to get a number less than 2 starting from $n$. For instance, $f(2)=1, f(5)=2$. For how many $1<m<2008$ is $f(m)$ odd? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "242"
} | {
"split": "train",
"index": 28788,
"solution": "This is $[2^{1}, 2^{2}) \\cup [2^{4}, 2^{8}) \\cup [2^{16}, 2^{32}) \\ldots$, and $2^{8}<2008<2^{16}$ so we have exactly the first two intervals."
} |
deepscaler | [
{
"role": "user",
"content": "If the program flowchart on the right is executed, determine the value of the output S. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 28789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $EFC$ given that $[EFC]=\\left(\\frac{5}{6}\\right)[AEC]=\\left(\\frac{5}{6}\\right)\\left(\\frac{4}{5}\\right)[ADC]=\\left(\\frac{5}{6}\\right)\\left(\\frac{4}{5}\\right)\\left(\\frac{2}{3}\\right)[ABC]$ and $[ABC]=20\\sqrt{3}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{80\\sqrt{3}}{9}"
} | {
"split": "train",
"index": 28790,
"solution": "By shared bases, we know that $$[EFC]=\\left(\\frac{5}{6}\\right)[AEC]=\\left(\\frac{5}{6}\\right)\\left(\\frac{4}{5}\\right)[ADC]=\\left(\\frac{5}{6}\\right)\\left(\\frac{4}{5}\\right)\\left(\\frac{2}{3}\\right)[ABC]$$ By Heron's formula, we find that $[ABC]=\\sqr... |
deepscaler | [
{
"role": "user",
"content": "Let $\\star (x)$ be the sum of the digits of a positive integer $x$. $\\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\\mathcal{S}$, we have that $\\star (n)=12$ and $0\\le n< 10^{7}$. If $m$ is the number of elements in $\\mathcal{S}$, compute ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 28791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ satisfies $$a_{n+1}=n\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor+1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\\lfloor x\\rfloor$ denotes t... | math | {
"style": "rule",
"ground_truth": "274"
} | {
"split": "train",
"index": 28792,
"solution": "It is straightforward to show that if $a_{1}=1$, then $a_{n}=n$ for all $n$. Since $a_{n+1}$ is an increasing function in $a_{n}$, it follows that the set of possible $a_{1}$ is of the form $\\{1,2, \\ldots, m\\}$ for some $m$, which will be the answer to the probl... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\frac{x-2}{x^2+4x+1} = \\frac{x-5}{x^2-10x}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{39}{11}"
} | {
"split": "train",
"index": 28793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the largest triangle is 8 centimeters. What is the number of centimeters in the length of the longer leg of the smallest triangle? Express your ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 28794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 28795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $\\left\\{a_{n}\\right\\}$ such that $a_{n}$ is the last digit of the sum $1 + 2 + \\cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\left\\{a_{n}\\right\\}$. Find $S_{2016}$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "7066"
} | {
"split": "train",
"index": 28796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 28797,
"solution": "1. **Identify the problem and apply the Principle of Inclusion-Exclusion (PIE):**\n We need to find the number of days Mrs. Sanders does not receive a call from any of her grandchildren. Each grandchild calls her every 3, 4, and 5 days respectively. We use PIE to... |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac {x^{2}}{4}-y^{2}$=1, a line $l$ with a slope angle of $\\frac {π}{4}$ passes through the right focus $F\\_2$ and intersects the right branch of the hyperbola at points $M$ and $N$. The midpoint of the line segment $MN$ is $P$. Determine the vertical ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt {5}}{3}"
} | {
"split": "train",
"index": 28798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the prime factors of 91? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers have a square less than 10,000,000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3162"
} | {
"split": "train",
"index": 28800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pick a random integer between 0 and 4095, inclusive. Write it in base 2 (without any leading zeroes). What is the expected number of consecutive digits that are not the same (that is, the expected number of occurrences of either 01 or 10 in the base 2 representation)? Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{20481}{4096}"
} | {
"split": "train",
"index": 28801,
"solution": "Note that every number in the range can be written as a 12-digit binary string. For $i=1,2, \\ldots 11$, let $R_{i}$ be a random variable which is 1 if the $i$ th and $(i+1)$ st digits differ in a randomly chosen number in the range. By linearity of expectation, $E... |
deepscaler | [
{
"role": "user",
"content": "Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"split": "train",
"index": 28802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of five-digit palindromes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 28803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points \\((2,3)\\) and \\((3, 7)\\) lie on a circle whose center is on the \\(x\\)-axis. What is the radius of the circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{1717}}{2}"
} | {
"split": "train",
"index": 28804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O=9$, and $B C=15$, let $S$ be the sum of all possible values for the area of triangle $A G O$. Compute $S^{2}$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 28805,
"solution": "Note that we know that $O, H$, and $G$ are collinear and that $H G=2 O G$. Thus, let $O G=x$ and $H G=2 x$. We also have $\\sin A=\\frac{B C}{2 R}=\\frac{5}{6}$, so $\\cos A=\\frac{\\sqrt{11}}{6}$. Then, if $A G \\perp O G$, then we have $x^{2}+A G^{2}=O G^{2}+A G^... |
deepscaler | [
{
"role": "user",
"content": "How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,\n\na) if each digit can be used only once?\n\nb) if each digit can be used multiple times? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 28806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\\\\nIf all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\\\\nInstead, if we exchange the questions score... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28807,
"solution": "\nLet \\( x \\) be the number of hard questions and \\( 9x \\) be the number of easy questions in the test. Let the total number of questions be \\( n = x + 9x = 10x \\). \n\nGiven:\n- Easy questions are worth 3 points each.\n- Hard questions are worth \\( D \\) po... |
deepscaler | [
{
"role": "user",
"content": "A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is: Let's think step ... | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 28808,
"solution": "1. **Define Variables:**\n Let $d$ be the depth of the well in feet, $t_1$ be the time in seconds it takes for the stone to fall to the bottom of the well, and $t_2$ be the time in seconds it takes for the sound to travel back up to the top of the well. We know t... |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": ".08"
} | {
"split": "train",
"index": 28809,
"solution": "1. **Define the probabilities:** Given that if $n \\leq 50$, the probability of choosing $n$ is $p$, and if $n > 50$, the probability of choosing $n$ is $3p$. \n\n2. **Total probability must sum to 1:** Since there are 50 numbers in each category ($n \\leq 50$ and ... |
deepscaler | [
{
"role": "user",
"content": "In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\\alpha$ through $AM$ parallel to $BC$, and denote $\\angle PAM$ as $\\theta_1$. The acute d... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 28810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all real numbers \\(x\\) satisfying\n\\[\n(x^2 - 5x + 3)^{x^2 - 6x + 3} = 1.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$ , $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$ . Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$ . Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "3007"
} | {
"split": "train",
"index": 28812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given complex numbers ${z_1}=2i$, ${z_2}=1-i$, where $i$ is the imaginary unit,\n\n(1) Find the conjugate of the complex number $\\frac{z_1}{z_2}$;\n\n(2) In the complex plane, let points $Z_1$, $Z_2$ correspond to ${z_1}$, ${z_2}$ respectively, and $O$ be the origin. Form a pa... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 28813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum value of $w$ if $w=3x^2 + 4y^2 - 12x + 8y + 15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n$ is a positive integer, find the minimum value of $|n-1| + |n-2| + \\cdots + |n-100|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 28815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 28816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Between 1000 and 9999, the number of four-digit integers with distinct digits where the absolute difference between the first and last digit is 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 28817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $x$ for which $x^{2}+x+41$ is not a prime? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 28818,
"solution": "40."
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x)=\\sin \\omega x+\\sin (\\omega x- \\frac {\\pi}{2})$ ($\\omega > 0$) is symmetric about the point $\\left( \\frac {\\pi}{8},0\\right)$, and there is a zero point within $\\left(- \\frac {\\pi}{4},0\\right)$, determine the minimum value of $\\o... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \\(a\\) and \\(b\\) are obtained. What is the probability that the two-digit number \\(ab\\) (where \\(a\\) and \\(b\\) are digits) and both \\(a\\) and \\(b\\) are divisible by 4? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 28820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a deck of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$. Find the number of posit... | math | {
"style": "rule",
"ground_truth": "958"
} | {
"split": "train",
"index": 28822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(1 \\times 2 \\times 3 \\times \\ldots \\times 2018\\) be represented as \\(2018!\\). Divide \\(2018!\\) by 3 to obtain a quotient. Continue dividing the quotient by 3 until it can no longer be divided evenly by 3. How many times is 2018! divided by 3 in this process? Let... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 28823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $x$ that satisfies the equation:\n\\[\\log_2 (x + 2) + \\log_{4} (x^2 - 2) + \\log_{\\frac{1}{2}} (x + 2) = 5.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{1026}"
} | {
"split": "train",
"index": 28824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say that two natural numbers form a perfect pair when the sum and the product of these two numbers are perfect squares. For example, 5 and 20 form a perfect pair because $5+20=25=5^{2}$ and $5 \\times 20=100=10^{2}$. Does 122 form a perfect pair with any other natural number... | math | {
"style": "rule",
"ground_truth": "122 \\times 121"
} | {
"split": "train",
"index": 28825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence ${a_{n}}$, $a_{n+1}=\\begin{cases} 2a_{n}\\left(a_{n} < \\frac{1}{2}\\right) \\\\ 2a_{n}-1\\left(a_{n}\\geqslant \\frac{1}{2}\\right) \\end{cases}$, if $a_{1}=\\frac{4}{5}$, then the value of $a_{20}$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 28826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$, and $z$ be distinct real numbers that sum to 0. Find the maximum possible value of $$\\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1/2"
} | {
"split": "train",
"index": 28827,
"solution": "Note that $0=(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$. Rearranging, we get that $x y+y z+z x=-\\frac{1}{2}\\left(x^{2}+y^{2}+z^{2}\\right)$, so that in fact the quantity is always equal to $-1 / 2$."
} |
deepscaler | [
{
"role": "user",
"content": "Two different natural numbers are chosen from the set $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$. What is the probability that their greatest common factor is one? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 28828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the prime numbers between 10 and 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 28829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Machiavelli added together $1+3i$ and $2-4i$. What number did he get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3-i"
} | {
"split": "train",
"index": 28830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs $(A, B)$ such that the following conditions hold: $A$ and $B$ are disjoint subsets of $\\{1,2, \\ldots, 50\\}$, $|A|=|B|=25$, and the median of $B$ is 1 more than the median of $A$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\binom{24}{12}^{2}"
} | {
"split": "train",
"index": 28831,
"solution": "The median of both sets, which we will call $a$ and $b$ respectively, are more than exactly 12 of the members in their own set. Since $a$ and $b$ are consecutive, they must also be higher than the lower half of the other set and lower than the higher half of the ot... |
deepscaler | [
{
"role": "user",
"content": "From 6 students, 4 are to be selected to undertake four different tasks labeled A, B, C, and D. If two of the students, named A and B, cannot be assigned to task A, calculate the total number of different assignment plans. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 28832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the mean value of the measures of the three interior angles of any triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 28833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, let $F_1$ and $F_2$ be the left and right foci, and let $P$ and $Q$ be two points on the right branch. If $\\overrightarrow{PF_2} = 2\\overrightarrow{F_2Q}$ and $\\overrightarrow{F_1Q} \\cdot \\overrightarrow{PQ} = ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{3}"
} | {
"split": "train",
"index": 28834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be angles such that\n\\begin{align*}\n\\cos x &= \\tan y, \\\\\n\\cos y &= \\tan z, \\\\\n\\cos z &= \\tan x.\n\\end{align*}Find the largest possible value of $\\sin x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"split": "train",
"index": 28835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation\n\\[\\frac{1}{x} + \\frac{1}{x + 2} - \\frac{1}{x + 4} - \\frac{1}{x + 6} - \\frac{1}{x + 8} - \\frac{1}{x + 10} + \\frac{1}{x + 12} + \\frac{1}{x + 14} = 0\\]has four roots of the form $-a \\pm \\sqrt{b \\pm c \\sqrt{d}},$ where $a,$ $b,$ $c,$ $d$ are positive int... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 28836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest ... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"split": "train",
"index": 28837,
"solution": "Let $A,B,C$ be the fraction of bananas taken by the first, second, and third monkeys respectively. Then we have the system of equations \\[\\frac{3}{4}A+\\frac{3}{8}B+\\frac{11}{24}C=\\frac{1}{2}\\] \\[\\frac{1}{8}A+\\frac{1}{4}B+\\frac{11}{24}C=\\frac{1}{3}\\] \\[... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=x^{2}-2bx+3$, where $b\\in R$. \n$(1)$ Find the solution set of the inequality $f\\left(x\\right) \\lt 4-b^{2}$. \n$(2)$ When $x\\in \\left[-1,2\\right]$, the function $y=f\\left(x\\right)$ has a minimum value of $1$. Find the maximum val... | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 28838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What integer \\( n \\) satisfies \\( 0 \\leq n < 151 \\) and \n$$150n \\equiv 93 \\pmod{151}~?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 28839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_3\\frac{1}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 28841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\\$ $. How much did the ball cost? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55440"
} | {
"split": "train",
"index": 28843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 28844,
"solution": "We have\n$(x-a)\\cdot (x-b)\\cdot (x-c)=x^3-(a+b+c)x+(ab+ac+bc)x-abc$ \n As a result, we have\n$a+b+c=0$\n$ab+bc+ac=-2011$\n$abc=-m$\n So, $a=-b-c$\n As a result, $ab+bc+ac=(-b-c)b+(-b-c)c+bc=-b^2-c^2-bc=-2011$\n Solve $b=\\frac {-c+\\sqrt{c^2-4(c^2-2011)}}{2}$ and... |
deepscaler | [
{
"role": "user",
"content": "In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative while the sum of any 11 consecutive terms is positive. What is the maximum number of terms in such a sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{4q+1}"
} | {
"split": "train",
"index": 28846,
"solution": "1. **Identify the roots of the quadratic equation**: Given the quadratic equation $x^2 + px + q = 0$, the roots can be found using the quadratic formula:\n \\[\n x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\n \\]\n Here, $a = 1$, $b = p$, and $c = q$. Thus, th... |
deepscaler | [
{
"role": "user",
"content": "Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each o... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 28847,
"solution": "We simplify this problem by using complementary counting and fixing one knight in place. Then, either a knight can sit two spaces apart from the fixed knight, or a knight can sit more than two spaces apart from the fixed knight. The probability is then $\\frac{24\\... |
deepscaler | [
{
"role": "user",
"content": "Given the circle $(x+1)^2+(y-2)^2=1$ and the origin O, find the minimum value of the distance |PM| if the tangent line from point P to the circle has a point of tangency M such that |PM|=|PO|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {5}}{5}"
} | {
"split": "train",
"index": 28848,
"solution": ""
} |
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