Let $a_1,a_2,\cdots,a_n$ be $n$ numbers such that $a_i$ is either $1$ or $-1$. If $$a_1a_2a_3a_4+a_2a_3a_4a_5+\cdots+a_na_1a_2a_3=0$$ then prove that $4 \mid n$.
By multiplying all the terms, we get,
I think that I will be able to represent $4n$ a power of $1$, but getting no clue. Please help! I also think that this problem can be done with invariance and extremal principal too. Please help with these approaches too!