A gardener and a woodpecker play the following game, in their garden which is in the shape of a $2022 \times 2022$ grid consisting of $2022^{2}$ cells. Two cells are considered neighbors if they share a vertex or an edge. Initially, each cell houses a tree of size 0. Then, on each turn of the game,
$\triangleright$ the gardener chooses a cell; the trees in this cell and in the adjacent cells (a total of four to nine cells) all increase in size by 1;
$\triangleright$ the woodpecker then chooses four cells; the trees in these cells all decrease in size by 1 (or remain at 0 if the woodpecker has chosen a cell with a tree of size 0).
A tree is said to be resplendent if its size is at least $10^{6}$. Find the largest integer $A$ for which the gardener can ensure, in a finite number of turns, regardless of the woodpecker's choices, to have grown at least $A$ resplendent trees.
We will demonstrate that the sought integer is $A=5 n=2271380$, where we have set $n=674^{2}=(2022 / 3)^{2}$.
First, here is a strategy for the woodpecker. He numbers the rows and columns from 1 to 2022, then colors black each cell located in a column $c$ and a row $\ell$ for which 3 does not divide either $c$ or $\ell$. Among the four to nine cells on which the gardener acts each turn, at most four are black. The woodpecker then selects the black cells in question; if there are fewer than four cells, he chooses other cells at random.
By doing so, he ensures that, at the end of the turn, no tree located in a black cell will have grown taller. An immediate induction indicates that no tree located in a black cell will ever exceed a height of 1, and since there are $4 n$ black cells, it will never be possible to exceed a total of $2022^{2}-4 n=5 n$ splendid trees.
Conversely, here is a strategy for the gardener; we set $m=10^{6}$. Ignoring the coloring imposed by the woodpecker, he colors red each cell located in a column $c$ and a row $\ell$ for which $c \equiv \ell \equiv 2(\bmod 3)$. There are $n$ red cells, and each cell is either red or adjacent to a red cell. The gardener then sets $k=(4 n+1) m+1$ and, during the first $k n$ turns of the game, he chooses each red cell exactly $k$ times. He has thus increased the height of each tree in the garden by $k$.
Let $a$ be the number of trees that are not splendid at the end of this first part of the game: the woodpecker must have chosen at least $k-m$ times each of these $a$ trees. Since he chose $4 k n$ trees during the $k n$ turns of the game, we deduce that $a(k-m) \leqslant 4 k n$, hence

$$
(k-m)(a-4 n-1) \leqslant 4 k n-(k-m)(4 n+1)=(4 n+1) m-k<0 .
$$

Since $k \geqslant m$, this means that $a<4 n+1$, therefore that $a \leqslant 4 n=9 n-A$.
In conclusion, the sought integer is indeed $A=5 n$.
Comment from the graders: The problem was difficult and many students did not fully understand what was being asked of them, including some who received points for having good ideas. Too many copies resemble the following:
"The gardener divides the grid into $3 \times 3$ squares and plays in the centers of the squares because it is the most interesting for him. In each of these squares, the woodpecker has the choice of either always attacking the same trees or spreading out; he should always attack the same trees, otherwise all the trees will become splendid. He can attack 4 trees per square; consequently, the gardener, by playing enough times in each center, manages to make 5 trees per square splendid, which is $5 \times 2022^{2} / 9$ trees."
Such a copy essentially combines everything that should not be done:
$\triangleright$ What does the phrase "it is more interesting to play in the center" mean? It needs to be justified even if it seems intuitive. Similarly, many students claimed that the gardener had no interest in playing on the edges because "it grows fewer trees," but only one student managed to justify it correctly. Indeed, growing these trees on the edges might be more interesting than other already immense trees, so the correct way to justify this statement was to say that the gardener can replace an edge move with an adjacent move, which will grow the same trees and others in addition, and which is therefore always better. In summary: to say a move is bad, it is necessary to propose a better alternative move that fulfills all the objectives achieved by the first move.
$\triangleright$ Many students made false dichotomies for the woodpecker's strategy: either he always attacks the same trees, or he flutters uniformly. Intermediate options are of course possible, where the woodpecker mainly targets certain trees but sometimes goes to others.
$\triangleright$ The main error, however, is this: once a good strategy for one of the two players has been found, it is absolutely forbidden to use it to find a "reactive" strategy for the other player. Even if you are convinced that the gardener has an interest in playing in the centers of the $3 \times 3$ squares, it is impossible to assume this to describe the woodpecker's strategy. Even if you are convinced that the woodpecker has an interest in choosing a certain number of trees against which he harbors a millennium-old grudge and to attack them whenever the gardener tries to make them grow, it is impossible to assume this to describe the gardener's strategy. Indeed, even if there are optimal strategies for each of the two players that meet these reasonable constraints, there are also equally effective strategies that do not work this way.
Here is how to proceed for a confrontation between two players: the proof must be divided into two completely separate parts that share no common argument. On the one hand, a strategy for the woodpecker must be given that ensures a certain score even if the gardener plays chaotically. It is then crucial to make no assumptions about the gardener's strategy and to avoid false dichotomies; the gardener should be allowed to "play badly" in this part if he wishes, as it is always simpler than justifying that his "bad" move could not become good in a distant future. In the second half of the proof, everything is reversed, a strategy for the gardener is given that remains valid even if the woodpecker does not follow the strategy just described, even if he also plays chaotically, without making any assumptions about his strategy.