FKMO 2013 Problem 3

For a positive integer $n \ge 2$ , define set $T = \{ (i,j) | 1 \le i < j \le n , i | j \}$ . For nonnegative real numbers $x_1 , x_2 , \cdots , x_n$ with $x_1 + x_2 + \cdots + x_n = 1$ , find the maximum value of $$\sum_{(i,j) \in T} x_i x_j$$ in terms of $n$ .