Increments and Decrements in Square Grid
Source: ISL 2022/C3
July 9, 2023
combinatoricsIMO ShortlistAZE IMO TST
Problem Statement
In each square of a garden shaped like a board, there is initially a tree of height . 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 . Determine the largest such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.