mavropnevma wrote:

Of course, the question that immediately comes to mind is if one can find infinitely many consecutive triplets equal to $+1$ (respectively $-1$ ); then groups of four, etc., $k\geq 3$ . We have no answer as of yet for this ...

This is a fairly state-of-the-art question. For $k=3$ Hildebrand [1] showed that all eight patters on $\pm 1$ occur infinitely often and much later Matomaki et al. [2] have shown that these patters occur with a positive frequency (I am talking about the generalised problem, but it turns our that the hardest part of the proof is dealing with the runs in question, namely $k$ consecutive 1's or -1's).

Unless some major advancements have been made in the last 5-ish years, it is still unkown whether all the patters for $k=4$ are present infinitely often. Of course, it is expected that for any $k{}$ , any pattern of signs occurs infinitely often - even more so, these patterns occur with equal frequencies - as stated by Chowla's conjecture.

I'm not entirely sure, but there are some connections between Chowla's conjecture and the Riemann Hypothesis, so perhaps it is completely out of reach as of now, let alone for an olympiad problem.

[1] A. Hildebrand – On consecutive values of the Liouville function. Enseign. Math. (2) 32 (1986), 219–226.
[2] K. Matomaki, M. Radziwill and T. Tao – Sign patterns of the Mobus and Liouville functions. Preprint, arXiv:1509:01545.

Edit: I just realised that the second paper I linked wasn't even published at the time of the competition. How time flies...