Question 46
Let $n \ge 2$ be a positive integer and $a_1 \lt a_2 \lt \dots \lt a_{2n}$ be real numbers. If $S = \sum_{i=1}^{2n} a_i$, $A_1 = \sum_{i,j,i\lt j} a_{2i}a_{2j}$ and $A_2 = \sum_{i,j,i\lt j} a_{2i-1}a_{2j-1}$, prove the inequality
$$
(n-1)S^2 \gt 4n(A_1 + A_2).
$$
$$
(n-1)S^2 \gt 4n(A_1 + A_2).
$$