VN-TST 2022 Problem 5

A fractional number $x$ is called pretty if it has finite expression in base $-b$ numeral system, $b$ is a positive integer in $[2;2022]$ . Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$