CGMO 2022 Problem 8

Let $x_1, x_2, \ldots, x_{11}$ be nonnegative reals such that their sum is $1$ . For $i = 1,2, \ldots, 11$ , let $$y_i = \begin{cases} x_{i} + x_{i + 1}, & i \textup{odd} , x_{i} + x_{i + 1} + x_{i + 2}, & i \textup{even} ,\end{cases}$$ where $x_{12} = x_{1}$ . And let $F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}$ . Prove that $x_6 < x_8$ when $F$ achieves its maximum.