ELMO 2012 Problem 4

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$ . Show that there exists a positive integer $n$ such that $a_n=b_n$ .

David Yang.