2014 IMO Shortlist C5

Consider $n \geq 3$ lines in the plane such that no two lines are parallel and no three have a common point. These lines divide the plane into polygonal regions; let $\mathcal{F}$ be the set of regions having finite area. Prove that it is possible to colour $\lceil\sqrt{n / 2}\rceil$ of the lines blue in such a way that no region in $\mathcal{F}$ has a completely blue boundary.

考虑平面中的 $n \geq 3$ 条线，使得没有两条线平行，并且没有三条线有公共点。这些线将平面划分为多边形区域；令 $\mathcal{F}$ 为具有有限面积的区域的集合。证明可以将线的 $\lceil\sqrt{n / 2}\rceil$ 着色为蓝色，使得 $\mathcal{F}$ 中没有区域具有完全蓝色的边界。