2018 IMO 第 5 题

Let $a_{1}$ , $a_{2}$ , ... be an infinite sequence of positive integers, and $N$ a positive integer. Suppose that for all integers $n\geq N$ , the expression

$$\frac{a_{1}}{a_{2}} +\frac{a_{2}}{a_{3}} +\dots +\frac{a_{n - 1}}{a_{n}} +\frac{a_{n}}{a_{1}}$$

is an integer. Prove that $(a_{n})$ is eventually constant.

令 $a_{1}$ 、 $a_{2}$ 、... 为正整数的无限序列，$N$ 为正整数。假设对于所有整数 $n\geq N$ ，表达式

$$\frac{a_{1}}{a_{2}} +\frac{a_{2}}{a_{3}} +\dots +\frac{a_{n - 1}}{a_{n}} +\frac{a_{n}}{a_{1}}$$

是一个整数。证明 $(a_{n})$ 最终是常数。