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A table and a restless mouse

Source: 2021 Centroamerican and Caribbean Mathematical Olympiad, P3

August 11, 2021
Coloringcombinatoricsgame

Problem Statement

In a table consisting of 2021×20212021\times 2021 unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?