data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Three real numbers $x, y, z$ are chosen randomly, and independently of each other, between 0 and 1, inclusive. What is the probability that each of $x-y$ and $x-z$ is greater than $-\\frac{1}{2}$ and less than $\\frac{1}{2}$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 20149,
"solution": "Consider a $1 \\times 1 \\times 1$ cube. We associate a triple $(x, y, z)$ of real numbers with $0 \\leq x \\leq 1$ and $0 \\leq y \\leq 1$ and $0 \\leq z \\leq 1$ with a point inside this cube by letting $x$ be the perpendicular distance of a point from the left f... |
deepscaler | [
{
"role": "user",
"content": "A point M on the parabola $y=-4x^2$ is at a distance of 1 from the focus. What is the y-coordinate of point M? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{16}"
} | {
"split": "train",
"index": 20150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 20151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 20152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 20153,
"solution": "To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choos... |
deepscaler | [
{
"role": "user",
"content": "The high-speed train \"Sapsan,\" approaching a railway station at a speed of \\( v = 216 \\) km/h, emits a warning sound signal lasting \\( \\Delta t = 5 \\) seconds when it is half a kilometer away from the station. What will be the duration of the signal \\( \\Delta t_{1} \\)... | math | {
"style": "rule",
"ground_truth": "4.12"
} | {
"split": "train",
"index": 20154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a number is divided by 7, the remainder is 2. What is the remainder when three times the number minus 7 is divided by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f: x \\rightarrow \\sqrt{x}$ is a function from set $A$ to set $B$.\n1. If $A=[0,9]$, then the range of the function $f(x)$ is ________.\n2. If $B={1,2}$, then $A \\cap B =$ ________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "{1}"
} | {
"split": "train",
"index": 20156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is equivalent to $\\sqrt{\\frac{x}{1-\\frac{x-1}{x}}}$ when $x < 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x"
} | {
"split": "train",
"index": 20157,
"solution": "To solve the problem, we first simplify the expression inside the square root:\n\n\\[\n\\sqrt{\\frac{x}{1-\\frac{x-1}{x}}}\n\\]\n\n1. Simplify the denominator:\n \\[\n 1 - \\frac{x-1}{x} = 1 - \\left(\\frac{x}{x} - \\frac{1}{x}\\right) = 1 - 1 + \\frac{1}{x} = ... |
deepscaler | [
{
"role": "user",
"content": "Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed i... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 20158,
"solution": "1. **Identify the initial and final odometer readings**: Barney's initial odometer reading is $1441$ and after riding for some time, the final reading is $1661$.\n\n2. **Calculate the total distance traveled**: \n \\[\n \\text{Distance} = \\text{Final odometer ... |
deepscaler | [
{
"role": "user",
"content": "Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 20159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 20160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{3, 3, 4, 4, 4, 7, 8\\}$ if no digit may be used more times than it appears in the given set of available digits? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 20161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\\frac{\\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region encl... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 20162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the leading coefficient in the polynomial $-3(x^4 - x^3 + x) + 7(x^4 + 2) - 4(2x^4 + 2x^2 + 1)$ after it is simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 20163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 20164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegat... | math | {
"style": "rule",
"ground_truth": "n\\geq4"
} | {
"split": "train",
"index": 20165,
"solution": "\nTo solve this problem, we need to determine for which natural numbers \\( n \\) there exists a set \\( S \\) of special triples, with \\( |S| = n \\), such that any special triple is bettered by at least one element of \\( S \\).\n\n### Understanding the Definiti... |
deepscaler | [
{
"role": "user",
"content": "If $0.8= \\frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x=4"
} | {
"split": "train",
"index": 20166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be real numbers such that\n\\[x^3 + y^3 + z^3 - 3xyz = 1.\\]Find the minimum value of $x^2 + y^2 + z^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 20167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $3x^2+4x-9$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{70}{9}"
} | {
"split": "train",
"index": 20168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a podcast series that lasts for 837 minutes needs to be stored on CDs and each CD can hold up to 75 minutes of audio, determine the number of minutes of audio that each CD will contain. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69.75"
} | {
"split": "train",
"index": 20169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a student multiplied the number $66$ by the repeating decimal, \\(1.\\overline{ab}\\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 20170,
"solution": "1. **Understanding the Problem:**\n We are given a repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\\overline{ab}$. The error in the calculation resulted in an answer that was ... |
deepscaler | [
{
"role": "user",
"content": "Given $m= \\int_{0}^{\\frac{\\pi}{2}} 9\\cos xdx$, then the constant term in the expansion of $\\left( \\frac{1}{\\sqrt{x}}-x\\right)^{m}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-84"
} | {
"split": "train",
"index": 20171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any number $y$, define the operations $\\&y = 2(7-y)$ and $\\&y = 2(y-7)$. What is the value of $\\&(-13\\&)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 20172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. What is {{1 1 0} {0 1} 0}? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"split": "train",
"index": 20173,
"solution": "1. **Calculate $\\{1, 1, 0\\}$**: \n The average of $1$, $1$, and $0$ is calculated as follows:\n \\[\n \\{1, 1, 0\\} = \\frac{1 + 1 + 0}{3} = \\frac{2}{3}\n \\]\n\n2. **Calculate $[0, 1]$**: \n The average of $0$ and $1$ is calculated as follows:\n \... |
deepscaler | [
{
"role": "user",
"content": "If $|x-\\log y|=x+\\log y$ where $x$ and $\\log y$ are real, then Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x(y-1)=0"
} | {
"split": "train",
"index": 20174,
"solution": "Given the equation $|x-\\log y|=x+\\log y$, we need to consider the properties of the absolute value function and the possible values of $x$ and $\\log y$.\n\n1. **Understanding the absolute value equation**:\n The absolute value equation $|a| = b$ holds if and o... |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{v} \\times \\mathbf{w} = \\begin{pmatrix} 5 \\\\ -2 \\\\ 4 \\end{pmatrix},$ then find $(\\mathbf{v} + \\mathbf{w}) \\times (\\mathbf{v} + \\mathbf{w}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 20175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sequences of four letters can be made from the letters in EQUALS if each sequence must begin with L, end with Q, and no letter can appear in a sequence more than once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a\\in R$, $b\\in R$, if the set $A=\\{a, \\frac{b}{a}, 1\\}$, $B=\\{a^{2}, a+b, 0\\}$, $A\\subseteq B$, and $B\\subseteq A$, then the value of $a^{2023}+b^{2023}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers between 30 and 50, inclusive, are not possible total scores if a multiple choice test has 10 questions, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20178,
"solution": "If 10 of 10 questions are answered correctly, the total score is $10 \\times 5=50$ points. If 9 of 10 questions are answered correctly, the score is either $9 \\times 5=45$ or $46$ points. If 8 of 10 questions are answered correctly, the score is $40,41,42$ points.... |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be positive integers such that $\\gcd(a,b) = 130$ and $\\gcd(a,c) = 770$. What is the smallest possible value of $\\gcd(b,c)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy: $|\\overrightarrow{a}| = 1$, $| \\overrightarrow{b}| = 6$, and $\\overrightarrow{a} \\cdot (\\overrightarrow{b} - \\overrightarrow{a}) = 2$, find the angle between vectors $\\overrightarrow{a}$ and $\\o... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{3}"
} | {
"split": "train",
"index": 20180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $(a-3+\\frac{1}{a-1})÷\\frac{{a}^{2}-4}{{a}^{2}+2a}⋅\\frac{1}{a-2}$, then choose a suitable number from $-2$, $-1$, $0$, $1$, $2$ to substitute and evaluate. 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": 20181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 15 bahs are equal to 24 rahs, and 9 rahs are equal in value to 15 yahs, how many bahs are equal in value to 1000 yahs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 20183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(\\frac{2}{m-3}+1\\right) \\div \\frac{2m-2}{m^2-6m+9}$, and then choose a suitable number from $1$, $2$, $3$, $4$ to substitute and evaluate. 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": 20184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin\\alpha = \\frac{3}{5}$, and $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi \\right)$. \n(1) Find the value of $\\tan\\left(\\alpha+\\frac{\\pi}{4}\\right)$; \n(2) If $\\beta \\in (0, \\frac{\\pi}{2})$, and $\\cos(\\alpha-\\beta) = \\frac{1}{3}$, find the value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2} - 4}{15}"
} | {
"split": "train",
"index": 20185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following system of equations: \\begin{align*}\n3x-5y&=-11,\\\\\n7x+2y&=-12.\n\\end{align*}Express your answer as an ordered pair $(x,y).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,1)"
} | {
"split": "train",
"index": 20186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, whose left and right foci are $F_{1}$ and $F_{2}$ respectively, and the top vertex is $B$. If the perimeter of $\\triangle BF_{1}F_{2}$ is $6$, and the distance from point $F_{1}$ to the line $BF... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 20187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $5! \\cdot 3! = n!$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r(\\theta) = \\frac{1}{1-\\theta}$. What is $r(r(r(r(r(r(30))))))$ (where $r$ is applied $6$ times)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 20189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The constant term in the expansion of $(x^2+ \\frac{1}{x^2}-2)^3$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 20190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[A=111111\\]and \\[B=142857\\]Find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $AB$ modulo 1,000,000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 20191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line in the plane is called strange if it passes through \\((a, 0)\\) and \\((0, 10-a)\\) for some \\(a\\) in the interval \\([0,10]\\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set... | math | {
"style": "rule",
"ground_truth": "50/3"
} | {
"split": "train",
"index": 20192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\\sin A = \\frac{{\\sqrt{3}}}{2}a$. Find:<br/>\n$(Ⅰ)$ The measure of angle $B$;<br/>\n$(Ⅱ)$ If triangle $\\triangle ABC$ is an acute ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 20193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $∆ABC$, the angles $A$, $B$, $C$ correspond to the sides $a$, $b$, $c$ respectively. The vector $\\overrightarrow{m}\\left(a, \\sqrt{3b}\\right)$ is parallel to $\\overrightarrow{n}=\\left(\\cos A,\\sin B\\right)$.\n\n(1) Find $A$;\n\n(2) If $a= \\sqrt{7},b=2$, find... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3 \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 20194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four contestants \\\\(A\\\\), \\\\(B\\\\), \\\\(C\\\\), and \\\\(D\\\\) participate in three competitions: shooting, ball throwing, and walking on a balance beam. Each contestant has an equal chance of passing or failing each competition. At the end of the competitions, the jud... | math | {
"style": "rule",
"ground_truth": "\\dfrac {2}{3}"
} | {
"split": "train",
"index": 20195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Figure $ABCD$ is a trapezoid with $AB \\parallel DC$, $AB=5$, $BC=3\\sqrt{2}$, $\\angle BCD=45^\\circ$, and $\\angle CDA=60^\\circ$. The length of $DC$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 + \\sqrt{3}"
} | {
"split": "train",
"index": 20196,
"solution": "1. **Identify the Trapezoid and Drop Perpendiculars**: Given that $ABCD$ is a trapezoid with $AB \\parallel DC$, drop perpendiculars from $A$ and $B$ to line $CD$. Let these perpendiculars meet $CD$ at points $F$ and $E$, respectively.\n\n2. **Analyze the Triangles... |
deepscaler | [
{
"role": "user",
"content": "Fix a sequence $a_1,a_2,a_3\\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ . Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 20197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical CDs regularly cost a total of $\\$28.$ What is the cost in dollars of five of these CDs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 20198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ and $β$ are acute angles, and $\\cos(α+β)=\\frac{3}{5}$, $\\sin α=\\frac{5}{13}$, find the value of $\\cos β$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 20199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 20200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, sides $\\overline{AB}$ and $\\overline{CD}$ are parallel, $\\angle A = 2\\angle D$, and $\\angle C = 3\\angle B$. Find $\\angle A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 20201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest two-digit multiple of 13? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 20202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-me... | math | {
"style": "rule",
"ground_truth": "\\binom{2k+1}{3} - k(2k+1)"
} | {
"split": "train",
"index": 20203,
"solution": "\nLet \\( A \\) be the number of three-member subsets such that the three distinct pairs among them use different languages. We aim to find the maximum possible value of \\( A \\).\n\nGiven that the social club has \\( 2k+1 \\) members, each fluent in \\( k \\) lan... |
deepscaler | [
{
"role": "user",
"content": "Points \\( M \\) and \\( N \\) are located on side \\( BC \\) of triangle \\( ABC \\), and point \\( K \\) is on side \\( AC \\), with \\( BM : MN : NC = 1 : 1 : 2 \\) and \\( CK : AK = 1 : 4 \\). Given that the area of triangle \\( ABC \\) is 1, find the area of quadrilateral ... | math | {
"style": "rule",
"ground_truth": "13/20"
} | {
"split": "train",
"index": 20204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there exist constants $A$ , $B$ , $C$ , and $D$ such that \\[n^4=A\\binom n4+B\\binom n3+C\\binom n2 + D\\binom n1\\] holds true for all positive integers $n\\geq 4$ . What is $A+B+C+D$ ?\n\n*Proposed by David Altizio* Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 20205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the flux of the vector field \\(\\mathbf{a} = y^2 \\mathbf{j} + z \\mathbf{k}\\) through the part of the surface \\(z = x^2 + y^2\\), cut off by the plane \\(z=2\\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid. Let's think ... | math | {
"style": "rule",
"ground_truth": "-2\\pi"
} | {
"split": "train",
"index": 20206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20207,
"solution": "1. **Assumption of Equal Height**: We assume that all pipes have the same height, which allows us to focus on comparing their cross-sectional areas to determine their water carrying capacities.\n\n2. **Cross-Sectional Area of the Larger Pipe**:\n - The diameter o... |
deepscaler | [
{
"role": "user",
"content": "At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but caref... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 20208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 700, inclusive, contain the digit 3 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "214"
} | {
"split": "train",
"index": 20210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, the line $l$ passes through the origin with an inclination angle of $\\frac{\\pi}{4}$; the parametric equations of the curve $C\\_1$ are $\\begin{cases} x = \\frac{\\sqrt{3}}{3} \\cos \\alpha \\\\ y = \\sin \\alpha \\end{cases}$ (where $\\a... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 20211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\\lambda$ intersects the $VA$ , $VB$ , $VC$ and $VD$ at $M$ , $N$ , $P$ , $Q$ respectively. Find $VQ : QD$ , if $VM : MA = 2 : 1$ , $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$ . Let... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 20212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the infinite series $$1+2\\left(\\frac{1}{1998}\\right)+3\\left(\\frac{1}{1998}\\right)^{2}+4\\left(\\frac{1}{1998}\\right)^{3}+\\ldots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1998}{1997}\\right)^{2} \\text{ or } \\frac{3992004}{3988009}"
} | {
"split": "train",
"index": 20213,
"solution": "We can rewrite the sum as \\(\\left(1+\\frac{1}{1998}+\\left(\\frac{1}{1998}\\right)^{2}+\\ldots\\right)+\\left(\\frac{1}{1998}+\\left(\\frac{1}{1998}\\right)^{2}+\\left(\\frac{1}{1998}\\right)^{3}+\\ldots\\right)+\\left(\\left(\\frac{1}{1998}\\right)^{2}+\\left(\\... |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that\n\n$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 1, 4, 1) \\text{ and } (2, 1, 1, 4)"
} | {
"split": "train",
"index": 20214,
"solution": "\nTo find all positive integers \\( a \\), \\( b \\), \\( c \\), and \\( p \\), where \\( p \\) is a prime number, satisfying the equation:\n\n\\[\n73p^2 + 6 = 9a^2 + 17b^2 + 17c^2,\n\\]\n\nwe proceed as follows:\n\n### Step 1: Investigate the Equation\n\nThe equat... |
deepscaler | [
{
"role": "user",
"content": "\\(ABCD\\) is a convex quadrilateral where \\(AB = 7\\), \\(BC = 4\\), and \\(AD = DC\\). Also, \\(\\angle ABD = \\angle DBC\\). Point \\(E\\) is on segment \\(AB\\) such that \\(\\angle DEB = 90^\\circ\\). Find the length of segment \\(AE\\). Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 20215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\\begin{cases} x=- \\dfrac { \\sqrt {3}}{2}t \\\\ y=1+ \\dfrac {1}{2}t\\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, t... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3+ \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 20216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x.$\n\n[asy]\nunitsize(0.7 cm);\n\npair A, B, C, D, O;\n\nO = (0,0);\nA = 4*dir(160);\nB = 5*dir(160 + 180);\nC = 8*dir(20);\nD = 4*dir(20 + 180);\n\ndraw(A--B);\ndraw(C--D);\ndraw(A--C);\ndraw(B--D);\n\nlabel(\"$4$\", (A + O)/2, SW);\nlabel(\"$10$\", (C + O)/2, SE);\nlab... | math | {
"style": "rule",
"ground_truth": "9 \\sqrt{2}"
} | {
"split": "train",
"index": 20217,
"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": 20218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The limit of the sum of an infinite number of terms in a geometric progression is $\\frac {a}{1 - r}$ where $a$ denotes the first term and $- 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac {a^2}{1 - r^2}"
} | {
"split": "train",
"index": 20219,
"solution": "1. **Identify the Original Series**: \n The original geometric series is given by the terms $a, ar, ar^2, ar^3, \\ldots$. Here, $a$ is the first term and $r$ is the common ratio.\n\n2. **Form the Series of Squares**: \n Squaring each term of the original seri... |
deepscaler | [
{
"role": "user",
"content": "If five geometric means are inserted between $8$ and $5832$, the fifth term in the geometric series: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 20220,
"solution": "1. **Identify the sequence and terms**: We are given that five geometric means are inserted between $8$ and $5832$. This means that the sequence starts at $8$ and ends at $5832$, with five terms in between, making a total of seven terms in the geometric sequence.\n... |
deepscaler | [
{
"role": "user",
"content": "A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many five-digit numbers are there, not divisible by 1000, in which the first, third, and last digits are even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9960"
} | {
"split": "train",
"index": 20222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \\cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one o... | math | {
"style": "rule",
"ground_truth": "(2n-1)!!"
} | {
"split": "train",
"index": 20223,
"solution": "\nConsider an integer \\( n > 0 \\) and a balance with \\( n \\) weights of weights \\( 2^0, 2^1, \\ldots, 2^{n-1} \\). Our task is to place each of these weights on the balance, one by one, so that the right pan is never heavier than the left pan. We aim to determ... |
deepscaler | [
{
"role": "user",
"content": "The solution to the inequality\n\\[y = -x^2 + ax + b \\le 0\\]is $(-\\infty,-3] \\cup [5,\\infty).$ Find the vertex of the parabola $y = -x^2 + ax + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,16)"
} | {
"split": "train",
"index": 20224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $19$ and their difference is $5$. What is their product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 20225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a unit cube \\( ABCD-A_1B_1C_1D_1 \\), let \\( O \\) be the center of the square \\( ABCD \\). Points \\( M \\) and \\( N \\) are located on edges \\( A_1D_1 \\) and \\( CC_1 \\) respectively, with \\( A_1M = \\frac{1}{2} \\) and \\( CN = \\frac{2}{3} \\). Find the volume of... | math | {
"style": "rule",
"ground_truth": "11/72"
} | {
"split": "train",
"index": 20226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability that exactly one red ball is drawn;<br/>$(2)$ Let the random variable $X$ represent the number of red b... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 20227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest real number $\\theta$ less than $\\pi$ (i.e. $\\theta<\\pi$ ) such that $\\prod_{k=0}^{10} \\cos \\left(2^{k} \\theta\\right) \\neq 0$ and $\\prod_{k=0}^{10}\\left(1+\\frac{1}{\\cos \\left(2^{k} \\theta\\right)}\\right)=1 ? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{2046 \\pi}{2047}"
} | {
"split": "train",
"index": 20228,
"solution": "For equality to hold, note that $\\theta$ cannot be an integer multiple of $\\pi$ (or else $\\sin =0$ and $\\cos = \\pm 1$ ). Let $z=e^{i \\theta / 2} \\neq \\pm 1$. Then in terms of complex numbers, we want $\\prod_{k=0}^{10}\\left(1+\\frac{2}{z^{2^{k+1}}+z^{-2^{k... |
deepscaler | [
{
"role": "user",
"content": "The circular region of the sign (below, left) has an area of 154 square inches. Vanessa would like to place a tiny ribbon (shaded) around the circle's edge. To be sure she has enough ribbon, she decides to buy 2 inches more of the ribbon than the original circle's circumference... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 20229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere with a radius of $1$ is placed inside a cone and touches the base of the cone. The minimum volume of the cone is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{8\\pi}{3}"
} | {
"split": "train",
"index": 20230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of edges of a given square pyramid determine a plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 20231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a six-digit decimal number $\\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \\leqslant i \\leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does no... | math | {
"style": "rule",
"ground_truth": "13056"
} | {
"split": "train",
"index": 20232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{100}{x - 100} = x.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 20233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $ \\frac {\\pi}{2}\\leqslant \\beta\\leqslant \\alpha\\leqslant \\frac {3\\pi}{4} $, $ \\cos (\\alpha-\\beta) = \\frac {12}{13} $, $ \\sin (\\alpha+\\beta) = -\\frac {3}{5} $, find the values of $ \\sin 2\\alpha $ and $ \\cos 2\\beta $. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-\\frac{63}{65}"
} | {
"split": "train",
"index": 20234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 hands... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the equation $x^{2}+(k^{2}-4)x+k-1=0$ has two roots that are opposite numbers, solve for $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 20236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In digital communication, signals are sequences composed of the digits \"$0$\" and \"$1$\". Signals are transmitted continuously $n$ times, with each transmission of \"$0$\" and \"$1$\" being equally likely. Let $X$ denote the number of times the signal \"$1$\" is transmitted.<... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 20237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$, and $S_n$ as the sum of its first $n$ terms, if $a_2$, $a_3$, and $a_6$ form a geometric sequence and $a_{10} = -17$, determine the minimum value of $\\frac{S_n}{2^n}$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 20238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of triangle \\(ABC\\) is 1. Circle \\(\\omega\\) is tangent to side \\(BC\\), the extension of side \\(AB\\) at point \\(P\\), and the extension of side \\(AC\\) at point \\(Q\\). A line passing through the midpoints of \\(AB\\) and \\(AC\\) intersects the circumc... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 20239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 20240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 20241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the base-10 logarithms of the divisors of $10^n$ is $792$. What is $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 20242,
"solution": "\nTo solve the problem, we need to find the value of $n$ such that the sum of the base-$10$ logarithms of the divisors of $10^n$ equals $792$.\n\n#### Step 1: Understanding the divisors of $10^n$\n$10^n = 2^n \\cdot 5^n$. Any divisor of $10^n$ can be expressed in t... |
deepscaler | [
{
"role": "user",
"content": "A sign has 31 spaces on a single line. The word RHOMBUS is written from left to right in 7 consecutive spaces. There is an equal number of empty spaces on each side of the word. Counting from the left, in what space number should the letter $R$ be put? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 20243,
"solution": "Since the letters of RHOMBUS take up 7 of the 31 spaces on the line, there are $31-7=24$ spaces that are empty. Since the numbers of empty spaces on each side of RHOMBUS are the same, there are $24 \\div 2=12$ empty spaces on each side. Therefore, the letter R is p... |
deepscaler | [
{
"role": "user",
"content": "Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 20244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos(2x+ \\frac {\\pi}{3})$, determine the horizontal shift required to obtain this function from the graph of $y=\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 20245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lucy surveyed a group of people about their knowledge of mosquitoes. To the nearest tenth of a percent, she found that $75.3\\%$ of the people surveyed thought mosquitoes transmitted malaria. Of the people who thought mosquitoes transmitted malaria, $52.8\\%$ believed that mosq... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 20246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be largest number such that \\[ \\frac{2014^{100!}-2011^{100!}}{3^n} \\] is still an integer. Compute the remainder when $3^n$ is divided by $1000$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 20247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\\left(a_{n-1}^{2}+a_{n}^{2}\\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer less than $a_{8}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 20248,
"solution": "Let $t_{n}$ be the larger real such that $a_{n}=t_{n}+\\frac{1}{t_{n}}$. Then $t_{1}=\\frac{3+\\sqrt{5}}{2}$. We claim that $t_{n}=2 t_{n-1}$. Writing the recurrence as a quadratic polynomial in $a_{n}$, we have: $$4 a_{n}^{2}-10 a_{n-1} a_{n}+4 a_{n-1}^{2}+9=0$$ U... |
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