Question 23
Given a positive integer $n \ge 2$, define the set $T$ by
$$
T = \{ (i, j) : 1 \le i \lt j \le n \text{ and } i \ne j \}.
$$
For nonnegative real numbers $x_1, x_2, \dots, x_n$ satisfying $x_1 + x_2 + \dots + x_n = 1$, find the maximum (as a function of $n$) of
$$
\sum_{(i,j) \in T} x_i x_j.
$$
$$
T = \{ (i, j) : 1 \le i \lt j \le n \text{ and } i \ne j \}.
$$
For nonnegative real numbers $x_1, x_2, \dots, x_n$ satisfying $x_1 + x_2 + \dots + x_n = 1$, find the maximum (as a function of $n$) of
$$
\sum_{(i,j) \in T} x_i x_j.
$$