Question 101
Let $n$ be a positive integer. There are $n$ towns arranged on an East-West road. Each town has two rhinoceroses, one heading East from the town, and the other heading West from the town (rhinoceroses cannot turn into another direction.) These $2n$ rhinoceroses all have different sizes. When two rhinoceroses confront face to face, the larger one would knock the smaller one out of the road. However, if a rhinoceros is bumped by another from its rear end, the bumped rhinoceros is knocked out of the road, regardless of their sizes.
Let $A$ and $B$ be two towns, with $B$ being East to $A$. We say that $A$ tramples $B$, if the East-heading rhinoceros of $A$ can reach $B$ by knocking off all rhinoceroses in between. Similarly, we say that $B$ tramples $A$ if the West-heading rhinoceros of $B$ can reach $A$ by knocking off all rhinoceroses in between.
Prove that there is exactly one town that would not be trampled by any other town.
Let $A$ and $B$ be two towns, with $B$ being East to $A$. We say that $A$ tramples $B$, if the East-heading rhinoceros of $A$ can reach $B$ by knocking off all rhinoceroses in between. Similarly, we say that $B$ tramples $A$ if the West-heading rhinoceros of $B$ can reach $A$ by knocking off all rhinoceroses in between.
Prove that there is exactly one town that would not be trampled by any other town.