Question 5
A rational number $x$ is called beautiful if it has a finite expansion in some base $b$, where $b$ is an integer in $[2,2022]$. Prove that there are only finitely many positive integers $n\ge4$ such that, for every integer $m$ in the open interval $(2n/3,n)$, at least one of $$\frac m{n-m},\qquad\frac{n-m}{m}$$ is beautiful.