2019 IMO Shortlist C4

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L}$, Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n$, what are all possible values for $k(\mathfrak{L})$, where $\mathfrak{L}$ is a labyrinth with $n$ walls? (Canada)

在卡米洛特的一个平面上，亚瑟王建造了一个由 $n$ 个墙组成的迷宫 $\mathfrak{L}$，每面墙都是一条无限直线。没有两堵墙是平行的，也没有三堵墙有公共点。然后梅林将每面墙的一侧全部涂成红色，另一侧全部涂成蓝色。在两堵墙的交叉处有四个角：两个对角相对的角是红边和蓝边相交的角，一个角是两个红边相交的角，一个角是两个蓝边相交的角。在每个这样的十字路口，都有一个双向门连接两个对角线相对的角，不同颜色的边在这两个角相交。梅林粉刷墙壁后，莫甘娜将一些骑士放入迷宫中。骑士可以穿过门，但不能穿过墙。令$k(\mathfrak{L})$为最大的数字$k$，这样，无论梅林如何绘制迷宫$\mathfrak{L}$，莫甘娜总能放置至少$k$个骑士，使得他们中的任何两个都无法相遇。对于每个 $n$，$k(\mathfrak{L})$ 的所有可能值是多少，其中 $\mathfrak{L}$ 是一个有 $n$ 个墙壁的迷宫？ (加拿大)