IMO 2022 Shortlist C3
In each square of a garden shaped like a 2022 ˆ 2022 board, there is initially a tree of height 0. A gardener and a lumb...
Category: Combinatorics
Problem
In each square of a garden shaped like a 2022 ˆ 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 106 . 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)