2014 IMO 第 6 题

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its finite regions. Prove that for all sufficiently large $n$ , in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ lines blue in such a way that none of its finite regions has a completely blue boundary.

平面上的一组直线如果没有两条直线平行并且没有三条直线经过同一点，则该组直线处于一般位置。一组处于一般位置的线将平面切割成多个区域，其中一些区域的面积是有限的；我们称这些为有限区域。证明对于所有足够大的 $n$ ，在一般位置的任何一组 $n$ 线中，可以将至少 $\sqrt{n}$ 线着色为蓝色，使其有限区域没有一个完全蓝色的边界。