IMO 2018 P5

Discussion on International Mathematical Olympiad (IMO)
M Ahsan Al Mahir
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Joined: Wed Aug 10, 2016 1:29 am

IMO 2018 P5

Unread post by M Ahsan Al Mahir » Wed Jan 09, 2019 11:45 pm

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

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