Question 164
Let $x_1, x_2, \dots, x_{11}$ be nonnegative real numbers which add up to one. For $i = 1, 2, \dots, 11$, let
$$
y_i = \begin{cases} x_i + x_{i+1}, & \text{if } i \text{ is odd,} \\ x_i + x_{i+1} + x_{i+2}, & \text{if } i \text{ is even,} \end{cases}
$$
where $x_{12} = x_1$. Set $F(x_1, x_2, \dots, x_{11}) = y_1y_2\cdots y_{11}$.
*Prove that if $F$ achieves maximum, then $x_6 \lt x_8$.*
$$
y_i = \begin{cases} x_i + x_{i+1}, & \text{if } i \text{ is odd,} \\ x_i + x_{i+1} + x_{i+2}, & \text{if } i \text{ is even,} \end{cases}
$$
where $x_{12} = x_1$. Set $F(x_1, x_2, \dots, x_{11}) = y_1y_2\cdots y_{11}$.
*Prove that if $F$ achieves maximum, then $x_6 \lt x_8$.*