Question 4
Let $x_1,\ldots,x_8\ge0$ be real numbers such that $x_i+x_{i+1}+x_{i+2}\le1$ for $i=1,\ldots,8$, where $x_9=x_1$ and $x_{10}=x_2$. Prove
$$\sum_{i=1}^8x_ix_{i+2}\le1$$
and find all equality cases.
$$\sum_{i=1}^8x_ix_{i+2}\le1$$
and find all equality cases.