Find all positive integers $a$ such that $a^{2}+2 a$ is a perfect square.
There are two cases. If $a>0$, we note that $a^{2}<a^{2}+2a<$ $a^{2}+2a+1=(a+1)^{2}$. Therefore, $a^{2}+2a$ cannot be a perfect square. If $a=0$, we find $a^{2}+2a=0$ which is a perfect square. This is thus the only one.