2017 IMO Shortlist C8

Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2 n+1) \times(2 n+1)$ square centered at $c$, apart from $c$ itself. We call a butterfly lonely, crowded, or comfortable, depending on whether the number of butterflies in its neighborhood $N$ is respectively less than, greater than, or equal to half of the number of lattice points in $N$. Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state.

令 $n$ 为给定的正整数。在笛卡尔平面中，每个具有非负坐标的格点最初包含一只蝴蝶，并且不存在其他蝴蝶。格点 $c$ 的邻域由以 $c$ 为中心的轴对齐 $(2 n+1) \times(2 n+1)$ 正方形内的所有格点组成，除了 $c$ 本身。我们称一只蝴蝶是孤独的、拥挤的还是舒适的，取决于它的邻域 $N$ 中的蝴蝶数量是否分别小于、大于或等于 $N$ 中格点数量的一半。每分钟，所有孤独的蝴蝶同时飞走。只要还有孤独的蝴蝶存在，这个过程就会持续下去。假设该过程最终停止，确定最终状态下舒适蝴蝶的数量。