Let $\{x_n\}_{n\geq 1}$ be a sequence of positive integers such that the there is a $k\in\mathbb{N}$ such that the sum $$\frac{x_1}{x_2}+\frac{x_2}{x_{3}}+\cdots + \frac{x_n}{x_1}\in\mathbb{N}$$ for all $n\geq k$. I have to show that there is an $k_0\in\mathbb{N}$ such that $x_{n}=x_{n+1}$ for all $n\geq k_{0}$. Basically i tried look at the quantity $$\left(\frac{x_1}{x_{2}}+\frac{x_{2}}{x_{3}}+\cdots+\frac{x_{k+1}}{x_1}\right)-\left(\frac{x_{1}}{x_{2}}+\frac{x_2}{x_3}+\cdots+\frac{x_k}{x_{1}}\right)\in \mathbb{N}\implies \frac{x_{k}}{x_{k+1}}+\frac{x_{k+1}-x_{k}}{x_{1}}\in\mathbb{N}$$ Not sure what to do afterwards. A full solution would be nice. Thanks.
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This is IMO 2018 problem 5. – timon92 Nov 10 '20 at 12:34
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1Solution can be found here, problem N4 on page 62. – WA Don Nov 10 '20 at 13:00
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Emm. Thank you. – Ramana Nov 10 '20 at 13:29