Question 126
Suppose $a_1, a_2, \dots, a_n$ are positive numbers. Prove that the square root of the arithmetic mean of the numbers $a_i a_j$, $1 \le i \lt j \le n$ doesn't exceed the arithmetic mean of $a_k$, $k = 1, 2, \dots, n$, with equality iff $a_1 = a_2 = \dots = a_n$.