Question 24
In Rectilandia there are $n$ cities located along a straight road. Each city has two robots assigned: an R-robot, located to its right and moving only in that direction, and an L-robot, located to its left and moving only in that direction. Each of the $2n$ robots has a number assigned, and these $2n$ numbers are all distinct. Two robots confront each other according to the following rules:
* If an L-robot meets an R-robot, the one with the larger number wins, and the one with the smaller number leaves the road.
* The robots are very vulnerable from behind, so if two R-robots meet, with no other robot between them, the one situated on the right leaves the road. Analogously, if two L-robots meet, the one on the left leaves the road, regardless, in both cases, of the numbers assigned to them.
Suppose that cities $A$ and $B$ are on the road in that order: $B$ to the right of $A$. We say that $A$ defeats $B$ if the R-robot of $A$ can advance toward $B$ expelling any robot it meets along the way. Analogously, we say that $B$ defeats $A$ if the L-robot of $B$ can advance toward $A$ expelling any robot it meets along the way. Prove that there exists a unique city that cannot be defeated by any other city.
* If an L-robot meets an R-robot, the one with the larger number wins, and the one with the smaller number leaves the road.
* The robots are very vulnerable from behind, so if two R-robots meet, with no other robot between them, the one situated on the right leaves the road. Analogously, if two L-robots meet, the one on the left leaves the road, regardless, in both cases, of the numbers assigned to them.
Suppose that cities $A$ and $B$ are on the road in that order: $B$ to the right of $A$. We say that $A$ defeats $B$ if the R-robot of $A$ can advance toward $B$ expelling any robot it meets along the way. Analogously, we say that $B$ defeats $A$ if the L-robot of $B$ can advance toward $A$ expelling any robot it meets along the way. Prove that there exists a unique city that cannot be defeated by any other city.