1972 IMO 第 4 题

(GDR 1) Let $n_{1}, n_{2}$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_{1}$ and $M_{2}$ consisting of $2 n_{1}$ and $2 n_{2}$ points, respectively, and such that no three points of the union $M_{1} \cup M_{2}$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ( $g$ not being included in either) contains exactly half of the points of $M_{1}$ and exactly half of the points of $M_{2}$.

(GDR 1) 令 $n_{1}, n_{2}$ 为正整数。考虑在平面 $E$ 中，两个不相交的点集 $M_{1}$ 和 $M_{2}$ 分别由 $2 n_{1}$ 和 $2 n_{2}$ 点组成，并且并集 $M_{1} \cup M_{2}$ 中没有三个点共线。证明存在一条直线$g$，其性质如下：$E$ 上由$g$ 确定的两个半平面($g$ 均不包含在内)中的每一个都恰好包含$M_{1}$ 的一半点和$M_{2}$ 的一半点。