67th Romanian Mathematical Olympiad

If $A$ and $B$ are positive integers, then $\overline{AB}$ will denote the number obtained by writing, in order, the digits of $B$ after the digits of $A$. For instance, if $A=193$ and $B=2016$, then $\overline{AB}=1932016$. Prove that there are infinitely many perfect squares of the form $\overline{AB}$ in each of the following situations: a) $A$ and $B$ are perfect squares; b) $A$ and $B$ are perfect cubes; c) $A$ is a perfect cube and $B$ is a perfect square; d) $A$ is a perfect square and $B$ is a perfect cube.