Question 18
On a rectangular board consisting of $m\times n$ squares, where $m,n\ge3$, dominos ($2\times1$ or $1\times2$ tiles) are placed without overlapping or extending beyond the board. Each domino covers exactly two squares. Suppose no additional domino can be placed on the board and the four corner squares are not all empty. Prove that at least $\frac23$ of the squares are covered by dominos.