11th European Girls' Mathematical Olympiad | EGMO 2022

Problem 5. For all positive integers $n$ , $k$ , let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$ . (For example, $f(2, 2)=2$ and $f(3, 2)=3$ .) Find all positive integers $n$ such that for every positive integer $k$ , the number $f(n, 2k)$ is odd.