Question 46
Let $n \in \mathbb{N}$, $n \gt 2$, and suppose $a_1, a_2, \dots, a_{2n}$ is a permutation of the numbers $1, 2, \dots, 2n$ such that $a_1 \lt a_3 \lt \dots \lt a_{2n-1}$ and $a_2 \gt a_4 \gt \dots \gt a_{2n}$. Prove that
$$
(a_1 - a_2)^2 + (a_3 - a_4)^2 + \dots + (a_{2n-1} - a_{2n})^2 \gt n^3.
$$
$$
(a_1 - a_2)^2 + (a_3 - a_4)^2 + \dots + (a_{2n-1} - a_{2n})^2 \gt n^3.
$$