Question 15
On a rectangular board consisting of $m \times n$ squares ($m, n \ge 3$), dominos have been placed (2 × 1- or 1 × 2-tiles), not overlapping each other. Each domino covers exactly two squares of the board. Suppose that the placement of the dominos has the property that no extra domino can be placed on the board, and the four corners of the board are not all empty. Prove that at least $\frac{2}{3}$ of the squares of the board is covered by dominos.