1999 APMO 第 5 题

Let $S$ be a set of $2 n+1$ points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ in its exterior. Prove that the number of good circles has the same parity as $n$.

令 $S$ 为平面上 $2 n+1$ 个点的集合，其中没有三个点共线，也没有四个点同环。如果一个圆的圆周上有 3 个 $S$ 点，内部有 $n-1$ 个点，外部有 $n-1$ 个点，则该圆被称为好圆。证明好圈数与$n$具有相同的奇偶性。