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Inequality of decimal part

Source: 2023 China Southeast MO Grade11 Day1 Problem 4

July 30, 2023
inequalitiesnumber theory

Problem Statement

Find the largest real number cc, such that for any integer s>1s>1, and positive integers m,nm, n coprime to ss, we have∑j=1s−1{jms}(1−{jms}){jns}(1−{jns})≥cs \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs where {x}=x−⌊x⌋\{ x \} = x - \lfloor x \rfloor .
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