Question 128
A sequence $a_1, a_2, \dots$ satisfies the following conditions: $a_1 \in (1, 2)$, and $a_{k+1} = a_k + \frac{k}{a_k}$ for every positive integer $k$. Prove that there is at most one (unordered) pair of distinct indices $(i, j)$ such that $a_i + a_j$ is an integer. (A. Golovanov)