2018 IMO Shortlist N7

Let $n \geq 2018$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be pairwise distinct positive integers not exceeding $5 n$. Suppose that the sequence $$\frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \ldots, \frac{a_{n}}{b_{n}}$$ forms an arithmetic progression. Prove that the terms of the sequence are equal.

令 $n \geq 2018$ 为整数，并令 $a_{1}、a_{2}、\ldots、a_{n}、b_{1}、b_{2}、\ldots、b_{n}$ 为不超过 $5 n$ 的成对不同正整数。假设序列 $$\frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \ldots, \frac{a_{n}}{b_{n}}$$ 形成一个算术级数。证明数列的项相等。