data_source stringclasses 1
value | prompt stringlengths 80 5.22k | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR-Qwen-base | Let $x, y$ be complex numbers such that \frac{x^{2}+y^{2}}{x+y}=4$ and \frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \frac{x^{6}+y^{6}}{x^{5}+y^{5}}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "10 \\pm 2 \\sqrt{17}",
"style": "rule"
} | {
"index": "DeepScaleR-4200"
} |
DeepScaleR-Qwen-base | A subset $S$ of the set $\{1,2, \ldots, 10\}$ is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of $S$. (By convention, the product of the elements of the empty set is 1.) Let's think step by step and output the ... | math | {
"ground_truth": "\\frac{375}{8}",
"style": "rule"
} | {
"index": "DeepScaleR-4201"
} |
DeepScaleR-Qwen-base | Find the largest real $C$ such that for all pairwise distinct positive real $a_{1}, a_{2}, \ldots, a_{2019}$ the following inequality holds $$\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2018}}{\left|a_{2019}-a_{1}\right|}+\frac{a_{2019}}{\left|a_{1}-a_{2}\right|}>C$$ Le... | math | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "DeepScaleR-4202"
} |
DeepScaleR-Qwen-base | Let $x, y$, and $N$ be real numbers, with $y$ nonzero, such that the sets $\left\{(x+y)^{2},(x-y)^{2}, x y, x / y\right\}$ and $\{4,12.8,28.8, N\}$ are equal. Compute the sum of the possible values of $N$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "85.2",
"style": "rule"
} | {
"index": "DeepScaleR-4203"
} |
DeepScaleR-Qwen-base | Let $A$ and $B$ be points in space for which $A B=1$. Let $\mathcal{R}$ be the region of points $P$ for which $A P \leq 1$ and $B P \leq 1$. Compute the largest possible side length of a cube contained within $\mathcal{R}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{\\sqrt{10}-1}{3}",
"style": "rule"
} | {
"index": "DeepScaleR-4204"
} |
DeepScaleR-Qwen-base | A sequence $\left\{a_{n}\right\}_{n \geq 1}$ of positive reals is defined by the rule $a_{n+1} a_{n-1}^{5}=a_{n}^{4} a_{n-2}^{2}$ for integers $n>2$ together with the initial values $a_{1}=8$ and $a_{2}=64$ and $a_{3}=1024$. Compute $$\sqrt{a_{1}+\sqrt{a_{2}+\sqrt{a_{3}+\cdots}}}$$ Let's think step by step and output t... | math | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "DeepScaleR-4205"
} |
DeepScaleR-Qwen-base | Let $x$ and $y$ be positive real numbers. Define $a=1+\frac{x}{y}$ and $b=1+\frac{y}{x}$. If $a^{2}+b^{2}=15$, compute $a^{3}+b^{3}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "DeepScaleR-4206"
} |
DeepScaleR-Qwen-base | Acute triangle $A B C$ has circumcenter $O$. The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\angle B A C=n^{\circ}$, where $n$ is a positive integer, compute the largest possible value of $n$. Let's think st... | math | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "DeepScaleR-4207"
} |
DeepScaleR-Qwen-base | The integers \(1,2,3,4,5,6,7,8,9,10\) are written on a blackboard. Each day, a teacher chooses one of the integers uniformly at random and decreases it by 1. Let \(X\) be the expected value of the number of days which elapse before there are no longer positive integers on the board. Estimate \(X\). An estimate of \(E\)... | math | {
"ground_truth": "120.75280458176904",
"style": "rule"
} | {
"index": "DeepScaleR-4208"
} |
DeepScaleR-Qwen-base | Let $\Omega$ be a sphere of radius 4 and $\Gamma$ be a sphere of radius 2 . Suppose that the center of $\Gamma$ lies on the surface of $\Omega$. The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumference. Let's think step by step and output the final answer within \boxed{... | math | {
"ground_truth": "\\pi \\sqrt{15}",
"style": "rule"
} | {
"index": "DeepScaleR-4209"
} |
DeepScaleR-Qwen-base | Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\sqrt[3]{n}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "DeepScaleR-4210"
} |
DeepScaleR-Qwen-base | A $10 \times 10$ table consists of 100 unit cells. A block is a $2 \times 2$ square consisting of 4 unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n-1$ blocks of $C$ cover the table. Find the largest possible value of n. Let's thi... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "DeepScaleR-4211"
} |
DeepScaleR-Qwen-base | Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}} $$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "46 / 3",
"style": "rule"
} | {
"index": "DeepScaleR-4212"
} |
DeepScaleR-Qwen-base | Consider the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$, where $\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions: (i) $f(0)=0$, (ii) $f(2n)=2f(n)$ and (iii) $f(2n+1)=n+2f(n)$ for all $n \geq 0$. (a) Determine the three sets $L:=\{n \mid f(n)<f(n+1)\}, E:=\{n \mid... | math | {
"ground_truth": "a_{k}=k2^{k-1}-2^{k}+1",
"style": "rule"
} | {
"index": "DeepScaleR-4213"
} |
DeepScaleR-Qwen-base | Find all integers $n$ satisfying $n \geq 2$ and \(\frac{\sigma(n)}{p(n)-1}=n\), in which \(\sigma(n)\) denotes the sum of all positive divisors of \(n\), and \(p(n)\) denotes the largest prime divisor of \(n\). Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "n=6",
"style": "rule"
} | {
"index": "DeepScaleR-4214"
} |
DeepScaleR-Qwen-base | Suppose $a, b, c$, and $d$ are pairwise distinct positive perfect squares such that $a^{b}=c^{d}$. Compute the smallest possible value of $a+b+c+d$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "305",
"style": "rule"
} | {
"index": "DeepScaleR-4215"
} |
DeepScaleR-Qwen-base | Fran writes the numbers \(1,2,3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (inc... | math | {
"ground_truth": "\\frac{131}{10}",
"style": "rule"
} | {
"index": "DeepScaleR-4216"
} |
DeepScaleR-Qwen-base | Compute the positive real number $x$ satisfying $x^{\left(2 x^{6}\right)}=3$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\sqrt[6]{3}",
"style": "rule"
} | {
"index": "DeepScaleR-4217"
} |
DeepScaleR-Qwen-base | Let $N$ be a positive integer whose decimal representation contains 11235 as a contiguous substring, and let $k$ be a positive integer such that $10^{k}>N$. Find the minimum possible value of $$ \frac{10^{k}-1}{\operatorname{gcd}\left(N, 10^{k}-1\right)} $$ Let's think step by step and output the final answer within \b... | math | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "DeepScaleR-4218"
} |
DeepScaleR-Qwen-base | For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ r... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "DeepScaleR-4219"
} |
DeepScaleR-Qwen-base | A standard $n$-sided die has $n$ sides labeled 1 to $n$. Luis, Luke, and Sean play a game in which they roll a fair standard 4-sided die, a fair standard 6-sided die, and a fair standard 8-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Comp... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "DeepScaleR-4220"
} |
DeepScaleR-Qwen-base | Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1,2, \ldots, a b$, putting the numbers $1,2, \ldots, b$ in the first row, $b+1, b+2, \ldots, 2 b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $i j... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "DeepScaleR-4221"
} |
DeepScaleR-Qwen-base | A Sudoku matrix is defined as a $9 \times 9$ array with entries from \{1,2, \ldots, 9\} and with the constraint that each row, each column, and each of the nine $3 \times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal... | math | {
"ground_truth": "\\frac{2}{21}",
"style": "rule"
} | {
"index": "DeepScaleR-4222"
} |
DeepScaleR-Qwen-base | Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers satisfying the following equations: $$\frac{a_{1}}{k^{2}+1}+\frac{a_{2}}{k^{2}+2}+\frac{a_{3}}{k^{2}+3}+\frac{a_{4}}{k^{2}+4}+\frac{a_{5}}{k^{2}+5}=\frac{1}{k^{2}} \text { for } k=1,2,3,4,5$$ Find the value of $\frac{a_{1}}{37}+\frac{a_{2}}{38}+\frac{a_{3}}{39}+\f... | math | {
"ground_truth": "\\frac{187465}{6744582}",
"style": "rule"
} | {
"index": "DeepScaleR-4223"
} |
DeepScaleR-Qwen-base | Let $N=2^{(2^{2})}$ and $x$ be a real number such that $N^{(N^{N})}=2^{(2^{x})}$. Find $x$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "66",
"style": "rule"
} | {
"index": "DeepScaleR-4224"
} |
DeepScaleR-Qwen-base | Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \ldots, A_{8}$, such that each is a subset (possibly empty) of \{1,2\}, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2025",
"style": "rule"
} | {
"index": "DeepScaleR-4225"
} |
DeepScaleR-Qwen-base | Let $P_{1}, P_{2}, \ldots, P_{8}$ be 8 distinct points on a circle. Determine the number of possible configurations made by drawing a set of line segments connecting pairs of these 8 points, such that: (1) each $P_{i}$ is the endpoint of at most one segment and (2) two no segments intersect. (The configuration with no ... | math | {
"ground_truth": "323",
"style": "rule"
} | {
"index": "DeepScaleR-4226"
} |
DeepScaleR-Qwen-base | Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of non-negative integers in $S$ less than 2008. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "DeepScaleR-4227"
} |
DeepScaleR-Qwen-base | Compute $$ \sum_{a_{1}=0}^{\infty} \sum_{a_{2}=0}^{\infty} \cdots \sum_{a_{7}=0}^{\infty} \frac{a_{1}+a_{2}+\cdots+a_{7}}{3^{a_{1}+a_{2}+\cdots+a_{7}}} $$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "15309 / 256",
"style": "rule"
} | {
"index": "DeepScaleR-4228"
} |
DeepScaleR-Qwen-base | Let $A B C$ be a triangle with $A B=7, B C=9$, and $C A=4$. Let $D$ be the point such that $A B \| C D$ and $C A \| B D$. Let $R$ be a point within triangle $B C D$. Lines $\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\ell$ meets $A B$ and $B C$ at $P$ and $P^{\prime}$ respectivel... | math | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "DeepScaleR-4229"
} |
DeepScaleR-Qwen-base | Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "222",
"style": "rule"
} | {
"index": "DeepScaleR-4230"
} |
DeepScaleR-Qwen-base | Let $P(n)=\left(n-1^{3}\right)\left(n-2^{3}\right) \ldots\left(n-40^{3}\right)$ for positive integers $n$. Suppose that $d$ is the largest positive integer that divides $P(n)$ for every integer $n>2023$. If $d$ is a product of $m$ (not necessarily distinct) prime numbers, compute $m$. Let's think step by step and outpu... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "DeepScaleR-4231"
} |
DeepScaleR-Qwen-base | Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "1540",
"style": "rule"
} | {
"index": "DeepScaleR-4232"
} |
DeepScaleR-Qwen-base | Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and $H M O M T$. Compute t... | math | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "DeepScaleR-4233"
} |
DeepScaleR-Qwen-base | Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered $1,2,3, \ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, the... | math | {
"ground_truth": "324",
"style": "rule"
} | {
"index": "DeepScaleR-4234"
} |
DeepScaleR-Qwen-base | We are given some similar triangles. Their areas are $1^{2}, 3^{2}, 5^{2} \ldots$, and $49^{2}$. If the smallest triangle has a perimeter of 4, what is the sum of all the triangles' perimeters? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "DeepScaleR-4235"
} |
DeepScaleR-Qwen-base | Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations: - if $n$ is even, she can replace $n$ with $\frac{n}{2}$; - if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$; and - if $n \geq 64$, she can replace ... | math | {
"ground_truth": "163",
"style": "rule"
} | {
"index": "DeepScaleR-4236"
} |
DeepScaleR-Qwen-base | Let $n$ be an integer of the form $a^{2}+b^{2}$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $a b$. Determine all such $n$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "n = 2, 5, 13",
"style": "rule"
} | {
"index": "DeepScaleR-4237"
} |
DeepScaleR-Qwen-base | A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where the bug is located, uses no edge multiple times, and uses at most two of the edges adjacent to any particular face. Find the number of healthy paths. Let's think step by step and output the final ... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "DeepScaleR-4238"
} |
DeepScaleR-Qwen-base | A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find \lfloor 2017 P(2017)\rfloor. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "DeepScaleR-4239"
} |
DeepScaleR-Qwen-base | Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "100800",
"style": "rule"
} | {
"index": "DeepScaleR-4240"
} |
DeepScaleR-Qwen-base | Triangle \(\triangle P N R\) has side lengths \(P N=20, N R=18\), and \(P R=19\). Consider a point \(A\) on \(P N\). \(\triangle N R A\) is rotated about \(R\) to \(\triangle N^{\prime} R A^{\prime}\) so that \(R, N^{\prime}\), and \(P\) lie on the same line and \(A A^{\prime}\) is perpendicular to \(P R\). Find \(\fra... | math | {
"ground_truth": "\\frac{19}{18}",
"style": "rule"
} | {
"index": "DeepScaleR-4241"
} |
DeepScaleR-Qwen-base | Compute the number of ways to tile a $3 \times 5$ rectangle with one $1 \times 1$ tile, one $1 \times 2$ tile, one $1 \times 3$ tile, one $1 \times 4$ tile, and one $1 \times 5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.) Let's think step by step and outp... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "DeepScaleR-4242"
} |
DeepScaleR-Qwen-base | Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north or one step south, and 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 Joules of energy, and hops till he falls asleep with 0 energy.... | math | {
"ground_truth": "10201",
"style": "rule"
} | {
"index": "DeepScaleR-4243"
} |
DeepScaleR-Qwen-base | Tanks has a pile of 5 blue cards and 5 red cards. Every morning, he takes a card and throws it down a well. What is the probability that the first card he throws down and the last card he throws down are the same color? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "DeepScaleR-4244"
} |
DeepScaleR-Qwen-base | Let $\omega$ be a circle of radius 1 centered at $O$. Let $B$ be a point on $\omega$, and let $l$ be the line tangent to $\omega$ at $B$. Let $A$ be on $l$ such that $\angle A O B=60^{\circ}$. Let $C$ be the foot of the perpendicular from $B$ to $O A$. Find the length of line segment $O C$. Let's think step by step and... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "DeepScaleR-4245"
} |
DeepScaleR-Qwen-base | An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2 \\sqrt{17}",
"style": "rule"
} | {
"index": "DeepScaleR-4246"
} |
DeepScaleR-Qwen-base | Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "DeepScaleR-4247"
} |
DeepScaleR-Qwen-base | Let $a_{0}, a_{1}, \ldots$ be a sequence such that $a_{0}=3, a_{1}=2$, and $a_{n+2}=a_{n+1}+a_{n}$ for all $n \geq 0$. Find $\sum_{n=0}^{8} \frac{a_{n}}{a_{n+1} a_{n+2}}$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{105}{212}",
"style": "rule"
} | {
"index": "DeepScaleR-4248"
} |
DeepScaleR-Qwen-base | Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers \underline{a} \underline{b} \underline{c}, if \underline{a} \underline{b} \underline{c} is a multiple of $x$, then the three-digit (base 10) number \underline{b} \underline{c} \underline{a} is also a multipl... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "DeepScaleR-4249"
} |
DeepScaleR-Qwen-base | Some squares of a $n \times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "3n^{2}-5n+2",
"style": "rule"
} | {
"index": "DeepScaleR-4250"
} |
DeepScaleR-Qwen-base | Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n)=n^{3}$ for all $n \in\{1,2,3,4,5\}$, compute $f(0)$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{24}{17}",
"style": "rule"
} | {
"index": "DeepScaleR-4251"
} |
DeepScaleR-Qwen-base | In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1,1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subseque... | math | {
"ground_truth": "n^{2}+c^{2}-nc-c",
"style": "rule"
} | {
"index": "DeepScaleR-4252"
} |
DeepScaleR-Qwen-base | Let $f$ be a function that takes in a triple of integers and outputs a real number. Suppose that $f$ satisfies the equations $f(a, b, c) =\frac{f(a+1, b, c)+f(a-1, b, c)}{2}$, $f(a, b, c) =\frac{f(a, b+1, c)+f(a, b-1, c)}{2}$, $f(a, b, c) =\frac{f(a, b, c+1)+f(a, b, c-1)}{2}$ for all integers $a, b, c$. What is the min... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "DeepScaleR-4253"
} |
DeepScaleR-Qwen-base | Richard starts with the string HHMMMMTT. A move consists of replacing an instance of HM with MH , replacing an instance of MT with TM, or replacing an instance of TH with HT. Compute the number of possible strings he can end up with after performing zero or more moves. Let's think step by step and output the final answ... | math | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "DeepScaleR-4254"
} |
DeepScaleR-Qwen-base | Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value ... | math | {
"ground_truth": "50308",
"style": "rule"
} | {
"index": "DeepScaleR-4255"
} |
DeepScaleR-Qwen-base | Let $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ be a polynomial whose roots are all negative integers. If $a+b+c+d=2009$, find $d$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "528",
"style": "rule"
} | {
"index": "DeepScaleR-4256"
} |
DeepScaleR-Qwen-base | Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4$$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(-\\frac{13}{96}, \\frac{13}{40})",
"style": "rule"
} | {
"index": "DeepScaleR-4257"
} |
DeepScaleR-Qwen-base | Compute the largest positive integer such that $\frac{2007!}{2007^{n}}$ is an integer. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "DeepScaleR-4258"
} |
DeepScaleR-Qwen-base | Let \(A B C\) be a triangle with \(\angle A=18^{\circ}, \angle B=36^{\circ}\). Let \(M\) be the midpoint of \(A B, D\) a point on ray \(C M\) such that \(A B=A D ; E\) a point on ray \(B C\) such that \(A B=B E\), and \(F\) a point on ray \(A C\) such that \(A B=A F\). Find \(\angle F D E\). Let's think step by step an... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "DeepScaleR-4259"
} |
DeepScaleR-Qwen-base | There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of th... | math | {
"ground_truth": "2014",
"style": "rule"
} | {
"index": "DeepScaleR-4260"
} |
DeepScaleR-Qwen-base | What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "DeepScaleR-4261"
} |
DeepScaleR-Qwen-base | Let $A, E, H, L, T$, and $V$ be chosen independently and at random from the set $\left\{0, \frac{1}{2}, 1\right\}$. Compute the probability that $\lfloor T \cdot H \cdot E\rfloor=L \cdot A \cdot V \cdot A$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{55}{81}",
"style": "rule"
} | {
"index": "DeepScaleR-4262"
} |
DeepScaleR-Qwen-base | Lily and Sarah are playing a game. They each choose a real number at random between -1 and 1. They then add the squares of their numbers together. If the result is greater than or equal to 1, Lily wins, and if the result is less than 1, Sarah wins. What is the probability that Sarah wins? Let's think step by step and o... | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "DeepScaleR-4263"
} |
DeepScaleR-Qwen-base | You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut... | math | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": "DeepScaleR-4264"
} |
DeepScaleR-Qwen-base | Compute the number of ways to color 3 cells in a $3 \times 3$ grid so that no two colored cells share an edge. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "DeepScaleR-4265"
} |
DeepScaleR-Qwen-base | Let $A_{11}$ denote the answer to problem 11. Determine the smallest prime $p$ such that the arithmetic sequence $p, p+A_{11}, p+2 A_{11}, \ldots$ begins with the largest possible number of primes. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "DeepScaleR-4266"
} |
DeepScaleR-Qwen-base | Let $S=\{1,2, \ldots, 2008\}$. For any nonempty subset $A \subset S$, define $m(A)$ to be the median of $A$ (when $A$ has an even number of elements, $m(A)$ is the average of the middle two elements). Determine the average of $m(A)$, when $A$ is taken over all nonempty subsets of $S$. Let's think step by step and outpu... | math | {
"ground_truth": "\\frac{2009}{2}",
"style": "rule"
} | {
"index": "DeepScaleR-4267"
} |
DeepScaleR-Qwen-base | Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, ... | math | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "DeepScaleR-4268"
} |
DeepScaleR-Qwen-base | Distinct prime numbers $p, q, r$ satisfy the equation $2 p q r+50 p q=7 p q r+55 p r=8 p q r+12 q r=A$ for some positive integer $A$. What is $A$ ? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "1980",
"style": "rule"
} | {
"index": "DeepScaleR-4269"
} |
DeepScaleR-Qwen-base | A student at Harvard named Kevin was counting his stones by 11. He messed up $n$ times and instead counted 9s and wound up at 2007. How many values of $n$ could make this limerick true? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "DeepScaleR-4270"
} |
DeepScaleR-Qwen-base | Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "DeepScaleR-4271"
} |
DeepScaleR-Qwen-base | For how many integer values of $b$ does there exist a polynomial function with integer coefficients such that $f(2)=2010$ and $f(b)=8$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "DeepScaleR-4272"
} |
DeepScaleR-Qwen-base | Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(-7,-7,18,-7,-7,-7,18,-7,-7,18,-7,-7,-7,18,-7,-7)",
"style": "rule"
} | {
"index": "DeepScaleR-4273"
} |
DeepScaleR-Qwen-base | Victoria wants to order at least 550 donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost \$7.49, what is the minimum amount Victoria needs to pay, in dollars? Let's think step by step and output the final answer within \boxed... | math | {
"ground_truth": "344.54",
"style": "rule"
} | {
"index": "DeepScaleR-4274"
} |
DeepScaleR-Qwen-base | There are two prime numbers $p$ so that $5 p$ can be expressed in the form $\left\lfloor\frac{n^{2}}{5}\right\rfloor$ for some positive integer $n$. What is the sum of these two prime numbers? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "DeepScaleR-4275"
} |
DeepScaleR-Qwen-base | A circle $\omega_{1}$ of radius 15 intersects a circle $\omega_{2}$ of radius 13 at points $P$ and $Q$. Point $A$ is on line $P Q$ such that $P$ is between $A$ and $Q$. $R$ and $S$ are the points of tangency from $A$ to $\omega_{1}$ and $\omega_{2}$, respectively, such that the line $A S$ does not intersect $\omega_{1}... | math | {
"ground_truth": "14+\\sqrt{97}",
"style": "rule"
} | {
"index": "DeepScaleR-4276"
} |
DeepScaleR-Qwen-base | Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\omega_{1}, \omega_{2}, \ldots$ of circles such that $\omega_{1}$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the ratio of the radius of $\omega_{i+1}$ to th... | math | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "DeepScaleR-4277"
} |
DeepScaleR-Qwen-base | A computer program is a function that takes in 4 bits, where each bit is either a 0 or a 1, and outputs TRUE or FALSE. How many computer programs are there? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "65536",
"style": "rule"
} | {
"index": "DeepScaleR-4278"
} |
DeepScaleR-Qwen-base | How many two-digit prime numbers have the property that both digits are also primes? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "DeepScaleR-4279"
} |
DeepScaleR-Qwen-base | Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\omega_{1}, \omega_{2}, \ldots$ of circles such that $\omega_{1}$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the total area contained in all the circles. Le... | math | {
"ground_truth": "\\frac{180 \\pi}{13}",
"style": "rule"
} | {
"index": "DeepScaleR-4280"
} |
DeepScaleR-Qwen-base | Find the range of $$f(A)=\frac{(\sin A)\left(3 \cos ^{2} A+\cos ^{4} A+3 \sin ^{2} A+\left(\sin ^{2} A\right)\left(\cos ^{2} A\right)\right)}{(\tan A)(\sec A-(\sin A)(\tan A))}$$ if $A \neq \frac{n \pi}{2}$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(3,4)",
"style": "rule"
} | {
"index": "DeepScaleR-4281"
} |
DeepScaleR-Qwen-base | In general, if there are $d$ doors in every room (but still only 1 correct door) and $r$ rooms, the last of which leads into Bowser's level, what is the expected number of doors through which Mario will pass before he reaches Bowser's level? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{d\\left(d^{r}-1\\right)}{d-1}",
"style": "rule"
} | {
"index": "DeepScaleR-4282"
} |
DeepScaleR-Qwen-base | Find all odd positive integers $n>1$ such that there is a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the numbers $1,2, \ldots, n$, where $n$ divides one of the numbers $a_{k}^{2}-a_{k+1}-1$ and $a_{k}^{2}-a_{k+1}+1$ for each $k, 1 \leq k \leq n$ (we assume $a_{n+1}=a_{1}$ ). Let's think step by step and output the fi... | math | {
"ground_truth": "n=3",
"style": "rule"
} | {
"index": "DeepScaleR-4283"
} |
DeepScaleR-Qwen-base | 8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "DeepScaleR-4284"
} |
DeepScaleR-Qwen-base | Let $ABC$ be a right triangle with hypotenuse $AC$. Let $B^{\prime}$ be the reflection of point $B$ across $AC$, and let $C^{\prime}$ be the reflection of $C$ across $AB^{\prime}$. Find the ratio of $[BCB^{\prime}]$ to $[BC^{\prime}B^{\prime}]$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "DeepScaleR-4285"
} |
DeepScaleR-Qwen-base | Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. Find the distance from the ball's point of first contact with a wall to the nearest vertex. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "DeepScaleR-4286"
} |
DeepScaleR-Qwen-base | A cube has side length 1. Find the product of the lengths of the diagonals of this cube (a diagonal is a line between two vertices that is not an edge). Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "DeepScaleR-4287"
} |
DeepScaleR-Qwen-base | Let $A B C D$ be a quadrilateral inscribed in a circle with diameter $\overline{A D}$. If $A B=5, A C=6$, and $B D=7$, find $C D$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\sqrt{38}",
"style": "rule"
} | {
"index": "DeepScaleR-4288"
} |
DeepScaleR-Qwen-base | Let $x_{1}, x_{2}, \ldots, x_{2022}$ be nonzero real numbers. Suppose that $x_{k}+\frac{1}{x_{k+1}}<0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_{1}$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_{n}>0$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "1010",
"style": "rule"
} | {
"index": "DeepScaleR-4289"
} |
DeepScaleR-Qwen-base | Pick a random digit in the decimal expansion of $\frac{1}{99999}$. What is the probability that it is 0? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "DeepScaleR-4290"
} |
DeepScaleR-Qwen-base | Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1,2,3,4,5\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2664",
"style": "rule"
} | {
"index": "DeepScaleR-4291"
} |
DeepScaleR-Qwen-base | Suppose that $x, y, z$ are real numbers such that $x=y+z+2$, $y=z+x+1$, and $z=x+y+4$. Compute $x+y+z$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "DeepScaleR-4292"
} |
DeepScaleR-Qwen-base | Let $f$ be a function from the nonnegative integers to the positive reals such that $f(x+y)=f(x) \cdot f(y)$ holds for all nonnegative integers $x$ and $y$. If $f(19)=524288 k$, find $f(4)$ in terms of $k$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "16 k^{4 / 19}",
"style": "rule"
} | {
"index": "DeepScaleR-4293"
} |
DeepScaleR-Qwen-base | Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, \ldots, 8,9,10\}$ that satisfy $|S|+\min (S)$. $\max (S)=0$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "335",
"style": "rule"
} | {
"index": "DeepScaleR-4294"
} |
DeepScaleR-Qwen-base | $A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\prime} B C D, A B^{\prime} C D$, $A B C^{\prime} D$, and $A B C D^{\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$? Let's think step by step and output the final an... | math | {
"ground_truth": "\\frac{125}{27}",
"style": "rule"
} | {
"index": "DeepScaleR-4295"
} |
DeepScaleR-Qwen-base | Let $S$ be a randomly chosen 6-element subset of the set $\{0,1,2, \ldots, n\}$. Consider the polynomial $P(x)=\sum_{i \in S} x^{i}$. Let $X_{n}$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most 3 with integer coefficients satisfying $Q(0) \neq 0$. Find the limit of $X... | math | {
"ground_truth": "\\frac{10015}{20736}",
"style": "rule"
} | {
"index": "DeepScaleR-4296"
} |
DeepScaleR-Qwen-base | Let $A B C$ be a triangle with $A B=A C=5$ and $B C=6$. Denote by $\omega$ the circumcircle of $A B C$. We draw a circle $\Omega$ which is externally tangent to $\omega$ as well as to the lines $A B$ and $A C$ (such a circle is called an $A$-mixtilinear excircle). Find the radius of $\Omega$. Let's think step by step a... | math | {
"ground_truth": "\\frac{75}{8}",
"style": "rule"
} | {
"index": "DeepScaleR-4297"
} |
DeepScaleR-Qwen-base | The equation $x^{2}+2 x=i$ has two complex solutions. Determine the product of their real parts. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{1-\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "DeepScaleR-4298"
} |
DeepScaleR-Qwen-base | A circle passes through the points $(2,0)$ and $(4,0)$ and is tangent to the line $y=x$. Find the sum of all possible values for the $y$-coordinate of the center of the circle. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "DeepScaleR-4299"
} |
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