Problem

Source: ELMO Shortlist 2011, A7; also ELMO #3

Tags: quadratics, search, trigonometry, algebra proposed, algebra



Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. Alex Zhu.