Školjka

%V0 Let $b$ be an integer greater than $5$. For each positive integer $n$, consider the number $$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5,$$ written in base $b$. Prove that the following condition holds if and only if $b = 10$: there exists a positive integer $M$ such that for any integer $n$ greater than $M$, the number $x_n$ is a perfect square.