PE-MO 2014 Problem 2

The $U$ -tile is made up of $1 \times 1$ squares and has the following shape: where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$ .

Notice that there are different types of tile $U$ . For example, some types of $U$ tiles are as follows:

Prove that for each integer $n \ge 6$ , the board of $n\times n$ can be completely covered with $U$ -tiles , with no gaps and no overlapping clicks.

Clarifications: The $U$ -tiles can be rotated. Any amount can be used in the covering of tiles of each type.