IZhO 2024 Problem 5

A table $m \times n$ is given, where $mn$ is divisible by 6. In this table a stripe is any $1 \times 3$ or $3 \times 1$ rectangle, and a domino is any $1 \times 2$ or $2 \times 1$ rectangle. The table is tiled with stripes. Prove that on top of this tiling the table can be tiled with dominoes so that in each stripe some two cells are covered by one domino and the remaining cell is covered by another domino. (The table is tiled by rectangles if the rectangles cover the entire table and do not overlap with each other.)