ELMO 2011 Problem 3

Determine whether there exists a sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers such that the following holds:

• For all $n \ge 0$, $a_n \neq 0$.
• There exist real numbers $x$ and $y$ such that $a_{n+2} = xa_{n+1} + ya_n$ for all $n \ge 0$.
• For all positive real numbers $r$, there exists positive integers $i$ and $j$ such that $|a_i| < r < |a_j|$.