Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying $$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$ are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$. a) Prove that $A_n$ is finite if and only if $n \not = 2$. b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that $$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Consider the set of all fractions $\tfrac{x}{y},$ 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 $10\%$? $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinitely many} $

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?

The diagram below shows circles radius $1$ and $2$ externally tangent to each other and internally tangent to a circle radius $3$. There are relatively prime positive integers $m$ and $n$ so that a circle radius $\frac{m}{n}$ is internally tangent to the circle radius $3$ and externally tangent to the other two circles as shown. Find $m+n$. [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw(circle((8,2), 3)); draw(circle((8,1), 2)); draw(circle((8,4), 1)); draw((8,-1)--(8,5)); draw(circle((9.72,3.28), 0.86)); label("$ 2 $",(7.56,1.38),SE*labelscalefactor); label("$ 1 $",(7.6,4.39),SE*labelscalefactor); [/asy]

The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

Let $ n, m$ be positive integers of different parity, and $ n > m$. Find all integers $ x$ such that $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$ is a perfect square.

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\tfrac23,$ and each of the other five sides has probability $\tfrac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a>1$ and $n>0$ be integers such that $a^n+1$ is a prime. Prove that $d(a^n-1)\ge n$.

Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

Natural numbers $a$ and $b$ are relatively prime. Prove that the greatest common divisor of $(a+b)$ and $(a^2+b^2)$ is either $1$ or $2$.

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that \[nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.\]

Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

A drawer has $5$ pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$? [i]Author: Ray Li[/i]