MathDB

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})

Problem Statement