2020/21 AHSMC Part 1 Problem 11

If $n$ is even, then $n^n$ is a perfect square. So all $50$ even choices work.

If $n$ is odd, then $n^n$ is a perfect square exactly when $n$ is a perfect square. Thus, only $n \in \{1^2, 3^2, 5^2, 7^2, 9^2\}$ work in the odd case. Therefore there are $55$ integers having the required property. The answer is (C).