International Mathematical Olympiad (IMO) $2018$ , Day $2$ , Problem $5$ of $6$

Let $a_1, a_2 ...$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$ , the number

$\frac{a_1}{a_2} + \frac{a_2}{a_3} + ... + \frac{a_{n - 1}}{a_n} +\frac{a_n}{a_1}$

is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m + 1}$ for all $m \geq M$ .

The person that answers correctly and gives the official solution first - go to International Mathematical Olympiad (IMO) Hall of Fame