2018 IMO 第 4 题

A site is any point $(x,y)$ in the plane for which $x,y\in \{1,\ldots ,20\}$ . Initially, each of the 400 sites is unoccupied. Any and Ben take turns placing stones on unoccupied sites, with Amy going first; Amy has the additional restriction that no two of her stones may be at a distance equal to $\sqrt{5}$ . They stop once either player cannot move. Find the greatest $K$ such that Amy can ensure that she places at least $K$ stones.

站点是平面上的任意点 $(x,y)$，其中 $x,y\in \{1,\ldots ,20\}$ 。最初，400 个地点均无人居住。 Any 和 Ben 轮流在无人居住的地方放置石头，Amy 先走；艾米还有一个额外的限制，即她的两颗宝石之间的距离不得等于 $\sqrt{5}$ 。一旦任何一个玩家无法移动，他们就会停止。找到最大的 $K$，这样艾米就可以确保她放置至少 $K$ 的石头。