KURSCHAK 2007

We have placed $n>3$ cards around a circle, facing downwards. In one step we may perform the following operation with three consecutive cards. Calling the one on the center $B$ , the two on the ends $A$ and $C$ , we put card $C$ in the place of $A$ , then move $A$ and $B$ to the places originally occupied by $B$ and $C$ , respectively. Meanwhile, we flip the cards $A$ and $B$ .

Using a number of these steps, is it possible to move each card to its original place, but facing upwards?

Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$ , one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.

Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties:

• any vertical or horizontal line in the plane cuts $K$ in at most $2$ points,
• any point of $H\setminus K$ is contained by a segment with endpoints from $K$ .