Olympiad problems

2009 India IMO Training Camp, 3

Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following: $ a_1 \equal{} a \\ a_2 \equal{} b \\ a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$. Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.

1984 USAMO, 2

Tags: AMC , USA(J)MO , USAMO , number theory unsolved , number theory

The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product. $\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer? $\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?

2012 USAMO, 1

Tags: USA(J)MO , USAMO , 2012 USAJMO , 2012 USAMO , geometry , Fibonacci

Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.