Question 156
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1, 2, \dots, 2n\}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1, \dots, a_n$ and $b_1, \dots, b_n$ of $A$ and $B$, respectively, such that
$$
(a_1 - b_1)^2 + (a_2 - b_2)^2 + \dots + (a_n - b_n)^2 \ge C_n.
$$
$$
(a_1 - b_1)^2 + (a_2 - b_2)^2 + \dots + (a_n - b_n)^2 \ge C_n.
$$