MathDB

In Lineland there are

n\geq1

towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the

2n

bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.
Let

A

and

B

be two towns, with

B

to the right of

A

. We say that town

A

can sweep town

B

away if the right bulldozer of

A

can move over to

B

pushing off all bulldozers it meets. Similarly town

B

can sweep town

A

away if the left bulldozer of

B

can move over to

A

pushing off all bulldozers of all towns on its way.
Prove that there is exactly one town that cannot be swept away by any other one.

Problem Statement