# The paper behind an olympiad problem

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?

• I assume the notation $\overline{a_ka_{k-1}...a_1}$ means the number $\sum_{i=1}^k a_i10^{i-1}$.
– YCor
Sep 28, 2022 at 6:43
• @YCor I would assume that it means the repeating decimal, i.e. $(1-10^{-k})^{-1}\sum_{i=1}^ka_i10^{1-i}$, and that the OP wants to be a square in $\mathbb{Q}$. But it would be best for the OP to clarify. Sep 28, 2022 at 9:15
• @NeilStrickland I think YCor's interpretation is correct. The overline notation is pretty standard in olympiad circles to denote the number represented by a decimal string. Sep 28, 2022 at 10:40

There it was proven by L. Csirmaz that the longest possible sequence is $$25, 625, 5625, 75625,275625$$. 