Question 152
Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S = \sum_{1 \le i \lt j \le 5} x_i x_j$. We ask:
(1) What value of $x_1, x_2, x_3, x_4, x_5$ will make $S$ the maximum?
(2) Further, if $|x_i - x_j| \le 2$ for any $1 \le i, j \le 5$, then what value of $x_1, x_2, x_3, x_4, x_5$ will make $S$ the minimum? You should prove your answer.
(1) What value of $x_1, x_2, x_3, x_4, x_5$ will make $S$ the maximum?
(2) Further, if $|x_i - x_j| \le 2$ for any $1 \le i, j \le 5$, then what value of $x_1, x_2, x_3, x_4, x_5$ will make $S$ the minimum? You should prove your answer.