US-EGMO-TST 2019 Problem 4

For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that $$a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}$$ for all positive integers $m$ and $n$ . Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$ . Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$ .