data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence defined by $a_k =\\dfrac{1}{k^2+k}$ for $k\\geq 1$. Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=\\dfrac{1}{29}$, for positive integers $m$ and $n$ with $m<n$, find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 7100,
"solution": "Note that $a_1 + a_2 + \\cdots + a_i = \\dfrac{i}{i+1}$. This can be proven by induction. Thus, $\\sum\\limits_{i=m}^{n-1} a_i = \\sum\\limits_{i=1}^{n-1} a_i - \\sum\\limits_{i=1}^{m-1} a_i = \\dfrac{n-1}{n} - \\dfrac{m-1}{m} = \\dfrac{n-m}{mn} = 1/29$. Cross-multiplying yields $29n -... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence defined recursively by $u_1 = a > 0$ and\n\\[u_{n + 1} = -\\frac{1}{u_n + 1}\\]for $n \\ge 1.$ Express $u_{16}$ in terms of $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "a"
} | {
"index": 7101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{n/2}$ when $n$ is even and by $t_n=\\frac{1}{t_{n-1}}$ when $n$ is odd. Given that $t_n=\\frac{19}{87}$, find $n.$"
}
] | math | {
"style": "rule",
"ground_truth": "1905"
} | {
"index": 7102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$ . The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$ ? (The median of a sequence of six real numbers is the average... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 7103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7104,
"solution": "\nConsider the sequence given by \\( x_1 = 19, x_2 = 95, x_{n+2} = \\text{lcm}(x_{n+1}, x_n) + x_n \\) for \\( n > 1 \\). We are tasked with finding the greatest common divisor (GCD) of \\( x_{1995} \\) and \\( x_{1996} \\).\n\nFirst, observe that the relationship between consecutive t... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $$ \\mathcal{S}=\\{(a, b, c, d, e): 0<a<b<c<d<e<100\\} $$ where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\\mathcal{S}$ , find the largest integer less than... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 7105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $E = \\{5, 6, 7, 8, 9\\}$ . For any partition ${A, B}$ of $E$ , with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$ . Let $N$ be the largest prime number amonh the... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 7106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $M=\\{1,2,3,...,2020\\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$"
}
] | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"index": 7108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $S$ of permutations of $1, 2, \\dots, 2022$ such that for all numbers $k$ in the\npermutation, the number of numbers less than $k$ that follow $k$ is even.\n\nFor example, for $n=4; S = \\{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\\}$ If $|S| = (a!... | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 7109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $\\{0.34,0.304,0.034,0.43\\}$. Find the sum of the smallest and largest numbers in the set."
}
] | math | {
"style": "rule",
"ground_truth": "0.464"
} | {
"index": 7110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $\\{0.34,0.304,0.034,0.43\\}$. The sum of the smallest and largest numbers in the set is"
}
] | math | {
"style": "rule",
"ground_truth": "0.464"
} | {
"index": 7111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $\\{2, 7, 12, 17, 22, 27, 32\\}$. Calculate the number of different integers that can be expressed as the sum of three distinct members of this set."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $\\{45, 52, 87, 90, 112, 143, 154\\}$. How many subsets containing three different numbers can be selected from this set so that the sum of the three numbers is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set $\\{8, -7, 2, -4, 20\\}$. Find the smallest sum that can be achieved by adding three different numbers from this set."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 7114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set \\( S = \\{1, 2, 3, \\cdots, 2010, 2011\\} \\). A subset \\( T \\) of \\( S \\) is said to be a \\( k \\)-element RP-subset if \\( T \\) has exactly \\( k \\) elements and every pair of elements of \\( T \\) are relatively prime. Find the smallest positive inte... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set \\(S\\) of all complex numbers \\(z\\) with nonnegative real and imaginary part such that \\(\\left|z^{2}+2\\right| \\leq|z|\\). Across all \\(z \\in S\\), compute the minimum possible value of \\(\\tan \\theta\\), where \\(\\theta\\) is the angle formed betwee... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 7116,
"solution": "Let \\(z=a+bi\\). Then, \\(z^{2}+2=(a^{2}-b^{2}+2)^{2}+2ab \\cdot i\\). Recall the identity \\((a^{2}-b^{2})^{2}+(2ab)^{2}=(a^{2}+b^{2})^{2}\\), so we have \\(\\left|z^{2}+2\\right|^{2}=(a^{2}+b^{2})^{2}+4(a^{2}-b^{2})+4\\). Thus, \\(z \\in S\\) if and only if \\((a^{2}+b^{2})^{2}+4(a^... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\\in \\{-2,-1,0,1,2\\}$ and $b\\in \\{-3,-2,-1,1,2,3\\}$. No three of these parabolas have a c... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"index": 7117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of 5-tuples of positive integers at most 5. We say the tuple $\\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\\right)$ is perfect if for any distinct indices $i, j, k$, the three numbers $a_{i}, a_{j}, a_{k}$ do not form an arithmetic progression (in any order). Find ... | math | {
"style": "rule",
"ground_truth": "780"
} | {
"index": 7118,
"solution": "There are two situations. 1. The multiset is aabbc; the only condition here is $c \\neq \\frac{1}{2}(a+b)$, for $\\left(\\binom{5}{3}-|S|\\right) \\cdot\\binom{3}{1}=18$ such triples, where $S$ is the set of unordered triples $(a, b, c)$ which do not satisfy the condition, and $S=\\{(1... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} < r < 3"
} | {
"index": 7119,
"solution": "To find the largest positive real number $r$ that satisfies at least one equation of the form $x^3 + a_2x^2 + a_1x + a_0 = 0$ with $|a_i| < 2$ for $i = 0,1,2$, we need to consider how to maximize $x$ under these constraints.\n\n1. **Expression Rearrangement**: \n We start by rearrang... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all fractions $\\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\\%$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7120,
"solution": "1. **Formulate the equation**: Given that increasing both the numerator and denominator by 1 increases the fraction by 10%, we can write:\n \\[\n \\frac{x+1}{y+1} = 1.1 \\cdot \\frac{x}{y}\n \\]\n Simplifying the right side, we get:\n \\[\n \\frac{x+1}{y+1} = \\frac{11x}{10... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is exactly twice the other. If we were to plot all such points, into how many regions would the resulting graph split up the plane?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"index": 7122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + 3y = 2070$. Find the number of such distinct triangles whose area is a positive integer that is also a ... | math | {
"style": "rule",
"ground_truth": "676"
} | {
"index": 7123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + y = 1953$. Find the number of such distinct triangles whose area is a positive integer and where $x_1 \... | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 7124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 7125,
"solution": "Let the two points $P$ and $Q$ be defined with coordinates; $P=(x_1,y_1)$ and $Q=(x_2,y_2)$\n We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).\n$\\det \\left(\\begin{array}{c} P \\\\ Q\\end{array}\... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $51x + y = 2048$. Find the number of such distinct triangles whose area is a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 7126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of numbers $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7127,
"solution": "To solve the problem, we need to find the ratio of the largest element in the set $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$ to the sum of all other elements in the set. The largest element in this set is $10^{10}$.\n\n1. **Calculate the sum of the other elements:**\n The sum of the... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\\frac{m + n\\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 7128,
"solution": "[asy] size(220); import three; currentprojection = perspective(5,4,3); defaultpen(linetype(\"8 8\")+linewidth(0.6)); draw(box((0,-.1,0),(0.4,0.6,0.3))); draw(box((-.1,0,0),(0.5,0.5,0.3))); draw(box((0,0,-.1),(0.4,0.5,0.4))); draw(box((0,0,0),(0.4,0.5,0.3)),linewidth(1.2)+linetype(\"1\"... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\\displaystyle {{m+n\\pi}\\over\np}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are re... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"index": 7129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the set of points that are inside or within two units of a rectangular parallelepiped that measures 2 by 3 by 6 units. Calculate the total volume of this set, expressing your answer in the form $\\frac{m+n\\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n... | math | {
"style": "rule",
"ground_truth": "\\frac{540 + 164\\pi}{3}"
} | {
"index": 7130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the statements:\n$\\textbf{(1)}\\ p\\wedge \\sim q\\wedge r \\qquad\\textbf{(2)}\\ \\sim p\\wedge \\sim q\\wedge r \\qquad\\textbf{(3)}\\ p\\wedge \\sim q\\wedge \\sim r \\qquad\\textbf{(4)}\\ \\sim p\\wedge q\\wedge r$\nwhere $p,q$, and $r$ are propositions. How many ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7131,
"solution": "To solve this problem, we need to evaluate each statement to see if it implies the truth of $(p \\rightarrow q) \\rightarrow r$. We start by understanding the implication $p \\rightarrow q$, which is logically equivalent to $\\sim p \\vee q$. Then, we evaluate $(p \\rightarrow q) \\rig... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the statements:\n(1) p and q are both true\n(2) p is true and q is false\n(3) p is false and q is true\n(4) p is false and q is false.\nHow many of these imply the negative of the statement \"p and q are both true?\""
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7132,
"solution": "To solve this problem, we need to determine how many of the given statements imply the negation of the statement \"p and q are both true.\"\n\nThe statement \"p and q are both true\" can be represented as $p \\land q$. The negation of this statement, according to De Morgan's Laws, is $... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sum\n\\[\n S_n = \\sum_{k = 1}^n \\frac{1}{\\sqrt{2k-1}} \\, .\n\\]\nDetermine $\\lfloor S_{4901} \\rfloor$ . Recall that if $x$ is a real number, then $\\lfloor x \\rfloor$ (the *floor* of $x$ ) is the greatest integer that is less than or equal to $x$ ."... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 7133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sum $$ S =\\sum^{2021}_{j=1} \\left|\\sin \\frac{2\\pi j}{2021}\\right|. $$ The value of $S$ can be written as $\\tan \\left( \\frac{c\\pi}{d} \\right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ .... | math | {
"style": "rule",
"ground_truth": "3031"
} | {
"index": 7134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the system \\begin{align*}x + y &= z + u,\\\\2xy & = zu.\\end{align*} Find the greatest value of the real constant $m$ such that $m \\leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \\geq y$."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 7135,
"solution": "\nTo solve this problem, we need to analyze the given system of equations:\n\n\\[\n\\begin{align*}\n1) \\quad & x + y = z + u,\\\\\n2) \\quad & 2xy = zu.\n\\end{align*}\n\\]\n\nOur goal is to find the greatest value of the real constant \\( m \\) such that \\( m \\leq \\frac{x}{y} \\) ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the system of equations\n\n\\[\n8x - 6y = c,\n\\]\n\\[\n12y - 18x = d.\n\\]\n\nIf this system has a solution \\((x, y)\\) where both \\(x\\) and \\(y\\) are nonzero, find the value of \\(\\frac{c}{d}\\), assuming \\(d\\) is nonzero."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{9}"
} | {
"index": 7136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the system of equations:\n\n\\[\n8x - 6y = a,\n\\]\n\\[\n12y - 18x = b.\n\\]\nIf there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\\frac{a}{b}$, assuming $b$ is nonzero."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{9}"
} | {
"index": 7137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the system of equations:\n\\begin{align*}\n8x - 5y &= a, \\\\\n10y - 15x &= b.\n\\end{align*}\nIf this system has a solution \\((x, y)\\) where both \\(x\\) and \\(y\\) are nonzero, calculate \\(\\frac{a}{b}\\), assuming \\(b\\) is nonzero."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"index": 7138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand ex... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 7139,
"solution": "143 We can look at the twelve-hour cycle beginning at midnight and ending just before noon, since during this time, the clock goes through each possible position exactly once. The minute hand has twelve times the angular velocity of the hour hand, so if the hour hand has made $t$ revol... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the two points \\(A(4,1)\\) and \\(B(2,5)\\). For each point \\(C\\) with positive integer coordinates, we define \\(d_C\\) to be the shortest distance needed to travel from \\(A\\) to \\(C\\) to \\(B\\) moving only horizontally and/or vertically. The positive integer ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider this histogram of the scores for $81$ students taking a test:\n\nThe median is in the interval labeled"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 7141,
"solution": "1. **Identify the position of the median**: \n Since there are $81$ students, the median will be the score of the middle student when all scores are arranged in order. The middle position is calculated as $\\frac{81+1}{2} = 41$. Thus, the median is the score of the $41^\\text{st}$ st... |
DeepScaleR | [
{
"role": "user",
"content": "Consider those functions $ f: \\mathbb{N} \\mapsto \\mathbb{N}$ which satisfy the condition\n\\[ f(m \\plus{} n) \\geq f(m) \\plus{} f(f(n)) \\minus{} 1\n\\]\nfor all $ m,n \\in \\mathbb{N}.$ Find all possible values of $ f(2007).$\n\n[i]Author: Nikolai Nikolov, Bulgaria[/i]"
... | math | {
"style": "rule",
"ground_truth": "1, 2, \\ldots, 2008"
} | {
"index": 7142,
"solution": "\nTo solve the problem, we need to determine all possible values of \\( f(2007) \\) for functions \\( f: \\mathbb{N} \\to \\mathbb{N} \\) that satisfy the given functional inequality:\n\\[ \nf(m + n) \\geq f(m) + f(f(n)) - 1 \n\\]\nfor all \\( m, n \\in \\mathbb{N} \\).\n\nFirstly, let... |
DeepScaleR | [
{
"role": "user",
"content": "Consider those functions $f$ that satisfy $f(x+4)+f(x-4) = f(x)$ for all real $x$. Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 7143,
"solution": "1. **Identify the functional equation and its implications**: We are given that $f(x+4) + f(x-4) = f(x)$ for all real $x$. This equation suggests a relationship between function values at intervals of 4 units.\n\n2. **Derive additional relationships**: Substitute $x+4$ for $x$ in the o... |
DeepScaleR | [
{
"role": "user",
"content": "Consider those functions $f$ that satisfy $f(x+5)+f(x-5) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 7144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider three coins where two are fair and a third coin lands on heads with a probability of $\\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ a... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"index": 7146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider three squares: $PQRS$, $TUVW$, and $WXYZ$, where each side of the squares has length $s=1$. $S$ is the midpoint of $WY$, and $R$ is the midpoint of $WU$. Calculate the ratio of the area of the shaded quadrilateral $PQSR$ to the sum of the areas of the three squares.\nA... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 7147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider triangle $A B C$ where $B C=7, C A=8$, and $A B=9$. $D$ and $E$ are the midpoints of $B C$ and $C A$, respectively, and $A D$ and $B E$ meet at $G$. The reflection of $G$ across $D$ is $G^{\\prime}$, and $G^{\\prime} E$ meets $C G$ at $P$. Find the length $P G$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{145}}{9}"
} | {
"index": 7148,
"solution": "Observe that since $G^{\\prime}$ is a reflection and $G D=\\frac{1}{2} A G$, we have $A G=G G^{\\prime}$ and therefore, $P$ is the centroid of triangle $A C G^{\\prime}$. Thus, extending $C G$ to hit $A B$ at $F, P G=\\frac{1}{3} C G=\\frac{2}{9} C F=\\frac{2}{9} \\sqrt{\\frac{2\\left(... |
DeepScaleR | [
{
"role": "user",
"content": "Consider triangle $A B C$ with $\\angle A=2 \\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\\overline{A B}$ at $E$. If $\\frac{D E}{D C}=\\frac{1}{3}$, compute $\\frac{A B}{A C}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 7149,
"solution": "Let $A E=x$ and $B E=y$. Using angle-bisector theorem on $\\triangle A C E$ we have $x: D E=A C: D C$, so $A C=3 x$. Using some angle chasing, it is simple to see that $\\angle A D E=\\angle A E D$, so $A D=A E=x$. Then, note that $\\triangle C D A \\sim \\triangle C E B$, so $y:(D C+D... |
DeepScaleR | [
{
"role": "user",
"content": "Consider triangle $A B C$ with side lengths $A B=4, B C=7$, and $A C=8$. Let $M$ be the midpoint of segment $A B$, and let $N$ be the point on the interior of segment $A C$ that also lies on the circumcircle of triangle $M B C$. Compute $B N$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{210}}{4}"
} | {
"index": 7150,
"solution": "Let $\\angle B A C=\\theta$. Then, $\\cos \\theta=\\frac{4^{2}+8^{2}-7^{2}}{2 \\cdot 4 \\cdot 8}$. Since $A M=\\frac{4}{2}=2$, and power of a point gives $A M \\cdot A B=A N \\cdot A C$, we have $A N=\\frac{2 \\cdot 4}{8}=1$, so $N C=8-1=7$. Law of cosines on triangle $B A N$ gives $$B... |
DeepScaleR | [
{
"role": "user",
"content": "Consider triangle \\(ABC\\) where \\(BC = 7\\), \\(CA = 8\\), and \\(AB = 9\\). \\(D\\) and \\(E\\) are the midpoints of \\(BC\\) and \\(CA\\), respectively, and \\(AD\\) and \\(BE\\) meet at \\(G\\). The reflection of \\(G\\) across \\(D\\) is \\(G'\\), and \\(G'E\\) meets \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{145}}{9}"
} | {
"index": 7151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two circles defined by the equations $x^2 + y^2 - 6x + 4y - 20 = 0$ and $x^2 + y^2 - 2x - 6y + 10 = 0$. Find the slope of the line that passes through their intersection points."
}
] | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 7152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume a... | math | {
"style": "rule",
"ground_truth": "281216000"
} | {
"index": 7153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two lines $p$ and $q$ in a coordinate plane with equations $y = -3x + 9$ and $y = -6x + 9$, respectively. Determine the probability that a point randomly selected in the first quadrant and below line $p$ will fall between $p$ and $q$."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 7154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two lines: line $l$ parametrized as\n\\begin{align*} \nx &= 1 + 4t,\\\\\ny &= 4 + 3t\n\\end{align*}and the line $m$ parametrized as\n\\begin{align*} \nx &=-5 + 4s\\\\\ny &= 6 + 3s.\n\\end{align*}Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix}-6 \\\\ 8 \\end{pmatrix}"
} | {
"index": 7155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two positive even integers less than $15$ (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 7156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two right-angled triangles, ABC and DEF. Triangle ABC has a right angle at C with AB = 10 cm and BC = 7 cm. Triangle DEF has a right angle at F with DE = 3 cm and EF = 4 cm. If these two triangles are arranged such that BC and DE are on the same line segment and point ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 7157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two sets of consecutive integers. Let $A$ be the least common multiple (LCM) of the integers from $15$ to $25$ inclusive. Let $B$ be the least common multiple of $A$ and the integers $26$ to $45$. Compute the value of $\\frac{B}{A}$.\nA) 4536\nB) 18426\nC) 3 * 37 * 41 ... | math | {
"style": "rule",
"ground_truth": "3 \\cdot 37 \\cdot 41 \\cdot 43"
} | {
"index": 7158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider two solid spherical balls, one centered at $\\left( 0, 0, \\frac{21}{2} \\right),$ with radius 6, and the other centered at $(0,0,1)$ with radius $\\frac{9}{2}.$ How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls?"
}
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a circle with center \\( S \\) and radius \\( 3 \\text{ cm} \\). Construct two mutually perpendicular diameters \\( AC \\) and \\( BD \\) of this circle. Construct isosceles triangles \\( ABK, BCL, CDM, DAN \\) such that:\n\n- The base of each triangle is a side of th... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 7160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 7161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a new sequence $\\{a_n\\}$ by extracting every 2nd, 4th, 8th, ..., $2^n$th, ... term from the sequence $\\{3n+\\log_2 n\\}$ in the original order. Find the general term $a_n$ for the new sequence $\\{a_n\\}$ and calculate the sum of the first five terms, $S_5$."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 7163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a new shape by adding an eighth unit cube to the previously described configuration of seven cubes. Place this new cube adjacent to one of the six outlying cubes from the central cube. What is the ratio of the volume in cubic units to the surface area in square units ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{33}"
} | {
"index": 7164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a square \\( A B C D \\) with a side length of \\( 6 \\text{ cm} \\) and label the intersection of its diagonals as \\( S \\). Construct a point \\( K \\) such that together with points \\( S, B, \\) and \\( C \\), they form a square \\( B K C S \\). Construct a point... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a square \\(A B C D\\) with side length \\(6 \\text{ cm}\\). Construct a line \\(p\\) parallel to the diagonal \\(A C\\) passing through point \\(D\\). Construct a rectangle \\(A C E F\\) such that vertices \\(E\\) and \\(F\\) lie on the line \\(p\\).\n\nUsing the giv... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 7167,
"solution": "1. **Identify the shapes and their sides**: We are given an equilateral triangle (3 sides), a square (4 sides), a regular pentagon (5 sides), a regular hexagon (6 sides), a regular heptagon (7 sides), and a regular octagon (8 sides).\n\n2. **Determine the adjacency of the shapes**: \n ... |
DeepScaleR | [
{
"role": "user",
"content": "Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!"
}
] | math | {
"style": "rule",
"ground_truth": "8/5"
} | {
"index": 7168,
"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?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Construct the cross-section of a triangular pyramid \\( A B C D \\) with a plane passing through the midpoints \\( M \\) and \\( N \\) of edges \\( A C \\) and \\( B D \\) and the point \\( K \\) on edge \\( C D \\), for which \\( C K: K D = 1: 2 \\). In what ratio does this pl... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 7170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Container A holds 4 red balls and 6 green balls; container B holds 8 red balls and 6 green balls; container C holds 8 red balls and 6 green balls; container D holds 3 red balls and 7 green balls. A container is selected at random and then a ball is randomly selected from that c... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{28}"
} | {
"index": 7171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Container A holds 4 red balls and 6 green balls; containers B and C each hold 6 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"index": 7172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Container I holds 8 red balls and 4 green balls; containers II and III each hold 2 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 7173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \\in \\mathbb{Z}$, she ascends to he... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{22}"
} | {
"index": 7174,
"solution": "Let $P(m, n)$ be the probability that she ascends to heaven from point $(m, n)$. Then $P(6 m, 6 n)=1$ and $P(6 m+3,6 n+3)=0$ for all integers $m, n \\in \\mathbb{Z}$. At all other points, $$\\begin{equation*} 4 P(m, n)=P(m-1, n)+P(m+1, n)+P(m, n-1)+P(m, n+1) \\tag{1} \\end{equation*}$$... |
DeepScaleR | [
{
"role": "user",
"content": "Continue the problem statement \"Petya and Kolya went to the forest to pick mushrooms. In the first hour, Petya collected twice as many mushrooms as Kolya...\" so that its solution is determined by the expression $30: 3+4$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Contractor Steve agreed to complete a job in 30 days. After 6 days he found that the 8 people assigned to the work had already done $\\frac{1}{3}$ of the job. If everyone works at the same rate, what is the least number of people he must keep on the job to ensure that the job w... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $10101_3$ to a base 10 integer."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 7177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $1011001_2$ to base 4."
}
] | math | {
"style": "rule",
"ground_truth": "1121_4"
} | {
"index": 7178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $102012_{(3)}$ to base 10."
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"index": 7179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $110_{(5)}$ to binary."
}
] | math | {
"style": "rule",
"ground_truth": "11110_{(2)}"
} | {
"index": 7180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $115_{10}$ to base 11. Represent $10$ as $A$, if necessary."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{A5}_{11}"
} | {
"index": 7181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $12012_3$ to a base 10 integer."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 7182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $1357_{10}$ to base 5."
}
] | math | {
"style": "rule",
"ground_truth": "20412_5"
} | {
"index": 7183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $135_7$ to a base 10 integer."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 7184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $1729_{10}$ to base 6."
}
] | math | {
"style": "rule",
"ground_truth": "12001_6"
} | {
"index": 7185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $199_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 7186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $2014_{10}$ to base 9."
}
] | math | {
"style": "rule",
"ground_truth": "2677_9"
} | {
"index": 7187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $314_{10}$ to base 6."
}
] | math | {
"style": "rule",
"ground_truth": "1242_6"
} | {
"index": 7188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $3206_7$ to a base 10 integer."
}
] | math | {
"style": "rule",
"ground_truth": "1133"
} | {
"index": 7189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $427_8$ to base 5."
}
] | math | {
"style": "rule",
"ground_truth": "2104_5"
} | {
"index": 7190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $5214_8$ to a base 10 integer."
}
] | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"index": 7191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $6351_8$ to base 7."
}
] | math | {
"style": "rule",
"ground_truth": "12431_7"
} | {
"index": 7192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $645_{10}$ to base 5."
}
] | math | {
"style": "rule",
"ground_truth": "10400_5"
} | {
"index": 7193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $6532_8$ to base 5."
}
] | math | {
"style": "rule",
"ground_truth": "102313_5"
} | {
"index": 7194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $813_9$ to base 3."
}
] | math | {
"style": "rule",
"ground_truth": "220110_3"
} | {
"index": 7195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $BD4_{16}$ to base 4."
}
] | math | {
"style": "rule",
"ground_truth": "233110_4"
} | {
"index": 7196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $\\rm{A}03_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 in order."
}
] | math | {
"style": "rule",
"ground_truth": "2563"
} | {
"index": 7197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $\\sqrt{2} e^{11 \\pi i/4}$ to rectangular form."
}
] | math | {
"style": "rule",
"ground_truth": "-1 + i"
} | {
"index": 7198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Convert $e^{11 \\pi i/2}$ to rectangular form."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"index": 7199,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.