Question 188
A fraction $x$ is called ‘pretty’ if it has a finite expansion in the base-$b$ numeral system for some positive integer $b$ in $[2; 2022]$. Prove that there are only finitely many positive integers $n \ge 4$ such that for every $m$ in $\left(\frac{2n}{3}, n\right)$, at least one of the two numbers $\frac{m}{n-m}$ and $\frac{n-m}{m}$ is pretty.