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Prove that expression is always even.

Source: INMO 2014- Problem 2

February 2, 2014
floor functionnumber theory unsolvednumber theory

Problem Statement

Let nn be a natural number. Prove that, ⌊n1⌋+⌊n2⌋+⋯+⌊nn⌋+⌊n⌋ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor is even.
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