Problem

Source: Mongolia MO 2000 Teachers P5

Tags: inequalities, number theory



Let $m,n,k$ be positive integers with $m\ge2$ and $k\ge\log_2(m-1)$. Prove that $$\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.$$