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Found problems: 1

2018 China National Olympiad, 3
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$.
approximation inequality diophantine approximation number theory fractional part AM-GM inequality fractional parts inequalities proof by contradiction