OIM 2007

In a $19\times 19$ board, a piece called dragon moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board.

The draconian distance between two squares is defined as the least number of moves a dragon needs to move from one square to the other.

Let $C$ be a corner square, and $V$ the square neighbor of $C$ that has only a point in common with $C$ . Show that there exists a square $X$ of the board, such that the draconian distance between $C$ and $X$ is greater than the draconian distance between $C$ and $V$ .

Let's say a positive integer $n$ is atresvido if the set of its divisors (including 1 and $n$ ) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

Let $\mathcal{F}$ be a family of hexagons $H$ satisfying the following properties:

i) $H$ has parallel opposite sides.

ii) Any 3 vertices of $H$ can be covered with a strip of width 1.

Determine the least $\ell\in\mathbb{R}$ such that every hexagon belonging to $\mathcal{F}$ can be covered with a strip of width $\ell$ .

Note: A strip is the area bounded by two parallel lines separated by a distance $\ell$ . The lines belong to the strip, too.