OBM 1999 Problem 4

On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$ , then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$ . In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$ .

The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.