2015 APMO 第 4 题

Let $n$ be a positive integer. Consider $2 n$ distinct lines on the plane, no two of which are parallel. Of the $2 n$ lines, $n$ are colored blue, the other $n$ are colored red. Let $\mathcal{B}$ be the set of all points on the plane that lie on at least one blue line, and $\mathcal{R}$ the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects $\mathcal{B}$ in exactly $2 n-1$ points, and also intersects $\mathcal{R}$ in exactly $2 n-1$ points.

令 $n$ 为正整数。考虑平面上 $2 n$ 条不同的线，其中没有两条是平行的。在 $2 n$ 行中，$n$ 为蓝色，其他 $n$ 为红色。设 $\mathcal{B}$ 为平面上至少位于一条蓝线上的所有点的集合，$\mathcal{R}$ 为平面上至少位于一条红线上的所有点的集合。证明存在一个圆，它与 $\mathcal{B}$ 恰好在 $2 n-1$ 点处相交，并且也与 $\mathcal{R}$ 恰好在 $2 n-1$ 点处相交。