Generalization of IMO Shortlist 2010 A3

[Thanic Nur Samin]

Let $x_1,x_2 \cdots x_n$ be non-negative real numbers such that $x_i+x_{i+1}+x_{i+2} \le 1$ for all integers $i$ such that $1 \le i \le n$(indices are taken modulo $n$) and $n$ is even. Prove that, $\sum\limits_{i=1}^{n-2}x_ix_{i+2} \le \dfrac{n-2}{8}$.

(The original shortlist problem will be solved after taking the sequence $x_1,x_2\cdots x_{100},x_1,x_2$ in this generalization.)