2023 CMO 第 2 题

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: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$

给定正整数 $m,n$ ，将规则 $(2m+2n)$ 边形的点着色为黑色和白色，将 $2m$ 着色为黑色，将 $2n$ 着色为白色。

两个黑点$B,C$的*着色距离*$d(B,C)$定义为连接这两个黑点的两条路径中白点的较少数量。

两个白点 $W,X$ 的*着色距离* $d(W,X)$ 定义为连接两个白点的两条路径中较少数量的黑点。

我们定义黑点的匹配 $\mathcal{B}$ ：用 $A_1,\cdots,A_m,B_1,\cdots,B_m$ 标记 $2m$ 黑点，满足多边形内不存在 $A_iB_i$ 相交的情况。

我们定义白点的匹配 $\mathcal{W}$ ：用 $C_1,\cdots,C_n,D_1,\cdots,D_n$ 标记 $2n$ 个白点，满足 gon 内不存在 $C_iD_i$ 相交的情况。

我们定义 $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j)$ 。

证明: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$