SMO 2019 Problem 5

In a $m\times n$ chessboard ( $m,n\ge 2$ ), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.

Proposed by DVDthe1st, mzy and jjax