IMO Shortlist 2003 problem N4

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Dodao/la: arhiva
2. travnja 2012.
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.
Izvor: Međunarodna matematička olimpijada, shortlist 2003