Belarus — Selection and Training Session

We call a coloring of an $m \times n$ table ($m, n \ge 5$) in three colors a good coloring if the following two conditions are satisfied:

1) Each cell has the same number of neighboring cells of two other colors;
2) Each corner cell has no neighboring cells of its color.

Find all pairs $(m, n)$ ($m, n \ge 5$) for which there exists a good coloring of $m \times n$ table.