HK-PC 2018

In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.

A triangle $ABC$ has its orthocentre $H$ distinct from its vertices and from the circumcenter $O$ of $\triangle ABC$ . Denote by $M, N$ and $P$ respectively the circumcenters of triangles $HBC, HCA$ and $HAB$ . Show that the lines $AM, BN, CP$ and $OH$ are concurrent.