XXI OBM

Planet Zork is spherical and has many towns. For each town there is a corresponding antipodal town (i.e. symmetric in relation to the centre of the planet). There are roads connecting pairs of towns in Zork. If there is a road connecting towns $P$ and $Q$ then there is also a road connecting towns $P^{\prime }$ and $Q^{\prime }$, where $P^{\prime }$ is the antipode of $P$ and $Q^{\prime }$ is the antipode of $Q$. Besides the roads do not cross each other and for any given two towns $P$ and $Q$ it is possible to travel from $P$ to $Q$ through some sequence of roads. The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by no more than 100 Urghs. Prove that there exist two antipodal towns such that the prices of Kriptonita in these towns differ by no more than 100 Urghs.