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Inequality of sum of fractional parts

Source: All-Russian MO 1999

December 31, 2012
inequalitiesinductionfloor functionalgebra unsolvedalgebranumber theory

Problem Statement

Prove that for all natural numbers nn, ∑k=1n2{k}≤n2−12. \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. Here, {x}\{x\} denotes the fractional part of xx.
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