Question 53
Let $n$ be a positive integer. Prove that at least one of the integers
$$
\left[2^{n} \cdot \sqrt{2}\right],\left[2^{n+1} \cdot \sqrt{2}\right], \ldots,\left[2^{2 n} \cdot \sqrt{2}\right]
$$
is even, where $[a]$ denotes the integer part of $a$.
$$
\left[2^{n} \cdot \sqrt{2}\right],\left[2^{n+1} \cdot \sqrt{2}\right], \ldots,\left[2^{2 n} \cdot \sqrt{2}\right]
$$
is even, where $[a]$ denotes the integer part of $a$.