69th Belarusian Mathematical Olympiad

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 $$(a_1-b_1{)}^2+(a_2-b_2{)}^2+\cdots +(a_n-b_n{)}^2\ge C_n.$$