PE-MO 2014

Find all triples ( $\alpha, \beta,\theta$ ) of acute angles such that the following inequalities are fulfilled at the same time $$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$ $$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$ $$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$

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.

a) Let $a, b, c$ be positive integers such that $ab + b + 1$ , $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$ , prove that $a = b = c$ .

b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$ .