KURSCHAK 2013 Problem 1

Let $a,b$ be positive real numbers satisfying $2ab=a-b$ . Denote for any positive integer $k$ $x_k$ and $y_k$ to be the closest integer to $ak$ and $bk$ , respectively (if there are two closest integers, choose the larger one). Prove that any positive integer $n$ appears in the sequence $(x_k)_{k\ge 1}$ if and only if it appears at least three times in the sequence $(y_k)_{k\ge 1}$ .