On the squares of an $8 \times 8$ chessboard, we wrote numbers, with 0 in every corner, and the number on any square is not greater than the average of the numbers written on the squares adjacent to it by an edge. What numbers did we write on the chessboard?
Since we wrote only a finite number of numbers on the board, there is a largest one among them; let this be $M$. Consider one of the cells where we wrote $M$. The numbers written in the adjacent cells are less than or equal to $M$, yet according to the condition, their average is at least $M$. This is only possible if each of them is equal to $M$. Therefore, if we wrote $M$ in a cell, we must have also written $M$ in the adjacent cells.

Saying the same for the adjacent cells, we get that in the neighbors of the adjacent cells, their neighbors, and so on, we must have also written $M$, which means we wrote $M$ in every cell. Since zeros are in the corners, this means we filled the table with all zeros.