2022 IMO Shortlist C3

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: - The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. - The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. We say that a tree is majestic if its height is at least $10^{6}$. Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays. (Colombia)

在形状像 $2022 \times 2022$ 板的花园的每个正方形中，最初有一棵高度为 0 的树。园丁和伐木工人轮流玩以下游戏，园丁先玩： - 园丁在花园中选择一个方格。该方格和周围所有方格(最多有八棵)上的每棵树都会变高一个单位。 - 然后伐木工人在板上选择四个不同的方块。这些方块上每棵正高度的树都会变短一个单位。如果一棵树的高度至少为 $10^{6}$，我们就说它是雄伟的。确定最大的数字$K$，这样无论伐木工人如何玩，园丁都可以确保棋盘上最终有$K$棵雄伟的树。 (哥伦比亚)