Problem

Source: 2023 China Southeast MO Grade11 Day1 Problem 4

Tags: inequalities, number theory



Find the largest real number $c$, such that for any integer $s>1$, and positive integers $m, n$ coprime to $s$, we have$$ \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs$$where $\{ x \} = x - \lfloor x \rfloor $.