CONO-SUR 2013 Problem 3

Nocycleland is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ neighbors $B$ if there is one road that connects them, and a city $A$ quasi-neighbors $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$ . It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$ , $B$ , $C$ and $D$ such that $A$ neighbors $B$ , $B$ neighbors $C$ , $C$ neighbors $D$ , and $D$ neighbors $A$ . Show that there is at least one city that quasi-neighbors at least $57$ other cities.