INMO 2014 Problem 2

Let $n$ be a natural number. Prove that

$$\left[\frac{n}{1}\right]+\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\cdots\left[\frac{n}{n}\right]+[\sqrt{n}]$$

is even. (Here $[x]$ denotes the largest integer smaller than or equal to $x$.)