2002 USAMO 第 5 题

Let $a, b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \ldots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_in_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \le i \le k$).

设$a，b$为大于2的整数。证明存在一个正整数$k$和一个正整数的有限序列$n_1, n_2, \ldots, n_k$，使得$n_1 = a$，$n_k = b$，并且对于每个$i$ ($1 \le i \le k$)，$n_in_{i+1}$可被$n_i + n_{i+1}$整除。