2017 USAMO 第 4 题

Let $P_1, \ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \ldots, R_n$ of the red points.

令 $P_1, \ldots, P_{2n}$ 为单位圆 $x^2 + y^2 = 1$ 上除 $(1,0)$ 之外的 $2n$ 个不同点。每个点的颜色要么是红色，要么是蓝色，其中恰好 $n$ 为红色，恰好 $n$ 为蓝色。令 $R_1, \ldots, R_n$ 为红点的任意顺序。令 $B_1$ 为距 $R_1$ 最近的蓝点，从 $R_1$ 开始逆时针绕圆旋转。然后让 $B_2$ 为距离 $R_2$ 最近的剩余蓝点，从 $R_2$ 逆时针绕圆移动，依此类推，直到我们标记了所有蓝点 $B_1、\ldots、B_n$。证明包含点 $(1,0)$ 的、形式为 $R_i \rightarrow B_i$ 的逆时针弧的数量与我们选择红点的顺序 $R_1, \ldots, R_n$ 的方式无关。