Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.
Source: 2011 Saudi Arabia IMO TST 1.3
Tags: number theory, floor function, algebra, Even
Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.