TUYMAADA 2000

Are there prime $p$ and $q$ larger than $3$ , such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$ ?

Let $O$ be the center of the circle circumscribed around the the triangle $ABC$ . The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

Every two of five regular pentagons on the plane have a common point. Is it true that some of these pentagons have a common point?

There are $2000$ cities in the country, each of which has exactly three roads to other cities. Prove that you can close $1000$ roads, so that there is not a single closed route in the country, consisting of an odd number of roads.