We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $$a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$$

This problem was given on contest, but I don't know how to solve it.

Clearly we must have $5$ terms $a_ia_{i+1}$ equal $-1 $ and other $5$ equal $1$. I have created a graph in which $a_i$ is connected with $a_{i+1}$ (modulo 10) if their product is -1. So we have $5$ edges and we can write handshake lemma $$\sum_{i=0}^9 d_i=10$$ where $d_i \in \{0,1,2\}$, but all this is usless.

I tried to find a configuration but failed every time. Any idea? For sure there must be simple argumentation why this does not hold or simple configuration why it does. Just don't see.