In IMO Shortlist 2013, there is a number theory problem:

Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $k>N$, the number $\overline{a_ka_{k-1}...a_1}=\sum_{i=1}^ka_i10^{i-1}$ is a perfect square.

This is a very interesting problem and the generalizations of this problem can have research value. So I guess that this problem is based on some research paper.

Is there any paper about or relates to this problem?