[จูกัดเหลียง]

Let $x_1=\max\left\{\,x_1,x_2,...x_8\right\} $
$$x_1x_2+x_2x_3+...x_7x_8\le x_1(x_2+x_3+...+x_8)=x_1(8-x_1)\le \Big(\frac{x_1+(8-x_1)}{2}\Big)^2=16$$

The equation occurs at $(4,4,0,0...,0)$or its permutations.