RU-ALLRUSSIAN 2009 Problem 112

Consider the sequence of numbers $(a_n)$ ( $n = 1, 2, \ldots$ ) defined as follows: $a_1\in (1, 2)$ , $a_{k + 1} = a_k + \frac{k}{a_k}$ ( $k = 1, 2, \ldots$ ). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.