CMO 2023 Problem 3

Given positive integer $m,n$ , color the points of the regular $(2m+2n)$ -gon in black and white, $2m$ in black and $2n$ in white. The coloring distance $d(B,C)$ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points. The coloring distance $d(W,X)$ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points. We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon. We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon. We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j)$ . Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$