Question 6
Let $x_1, \ldots, x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \le 1$ for all $i = 1, \ldots, 100$ (we put $x_{101} = x_1, x_{102} = x_2$).
Find the maximal possible value of the sum $S = \sum_{i=1}^{100} x_i x_{i+2}$.
Find the maximal possible value of the sum $S = \sum_{i=1}^{100} x_i x_{i+2}$.