USAMO 2002 — Days 1 and 2 Question 5

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

Problem proposer credited in the source: Gabriel Carroll.