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Omni-MATH / In each square of a garden shaped like a 2022×2022 board, there is initially a tree of height 0. A gardener and a lumberjack alternate turns playing the following game, with the ga…

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problem

In each square of a garden shaped like a 2022×20222022 \times 2022 board, there is initially a tree of height 00. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] 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. [/list] We say that a tree is [i]majestic[/i] if its height is at least 10610^6. Determine the largest KK such that the gardener can ensure there are eventually KK majestic trees on the board, no matter how the lumberjack plays.
Plain-text mathematical notation (without MathML)
In each square of a garden shaped like a 2022×2022 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:
[list]
[*] 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.
[/list]
We say that a tree is [i]majestic[/i] if its height is at least 10⁶. Determine the largest K such that the gardener can ensure there are eventually K majestic trees on the board, no matter how the lumberjack plays.
Original LaTeX notation
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:
[list]
[*] 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.
[/list]
We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

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Source and history

task_id
37401acb-2c1a-56c5-bdb7-acabde535384
task_key
test--37401acb-2c1a-56c5-bdb7-acabde535384
task_revision_id
1
config / split
/ test

Official source

initial import