2005 USAMO 第 5 题

Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.

Prove that there exist at least two balancing lines.

令 $n$ 为大于 1 的整数。假设平面上有 $2n$ 个点，其中没有三个点共线。假设给定的 $2n$ 点中的 $n$ 为蓝色，其他 $n$ 为红色。平面中的一条线如果经过一个蓝点和一个红点，并且对于线的每一侧，该侧的蓝点数量等于同一侧的红点数量，则该线称为平衡线。

证明至少存在两条平衡线。