4B. Find all five-digit numbers $\overline{a b c d e}$ such that $\overline{a b}$, $\overline{b c}$, $\overline{c d}$, and $\overline{d e}$ are perfect squares.
Solution. The only two-digit numbers that are perfect squares are 16, 25, 36, 49, 64, and 81. They start with the digits $1,2,3,4,6,8$, and end with the digits 1, 4, 5, 6. Therefore, the digits $b, c, d$ can only be the digits $1,4,6$. The numbers $\overline{b c}$ and $\overline{c d}$ are perfect squares, which is only possible if $\overline{b c}=16$ and $\overline{c d}=64$. According to this, $\overline{a b}=\overline{a 1}$, from which we get $a=8$, and $\overline{d e}=\overline{4 e}$, from which we get $e=9$. Finally, the desired number is 81649, and it is the only one with the required property.

## 2nd year