$\overline{a_{1} a_{2} a_{3}}$ and $\overline{a_{3} a_{2} a_{1}}$ are two three-digit decimal numbers, with $a_{1}, a_{3}$ being different non-zero digits. The squares of these numbers are five-digit numbers $\overline{b_{1} b_{2} b_{3} b_{4} b_{5}}$ and $\overline{b_{5} b_{4} b_{3} b_{2} b_{1}}$ respectively. Find all such three-digit numbers.
Assume $a_{1}>a_{3}>0$. As the square of $\overline{a_{1} a_{2} a_{3}}$ must be a five-digit number we have $a_{1} \leq 3$. Now a straightforward case study shows that $\overline{a_{1} a_{2} a_{3}}$ can be 301, 311, 201, 211 or 221.