JEE preps are adversely impacting my thinking ability.
So I will try and do something that I didn't even do when I was actually preparing for the olympiads
Actually think !
It's trivial I know I know...
But it was fun !

Problem (ISL 2015 C1) :
In Lineland there are 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 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 and be two towns, with to the right of . We say that town can sweep town away if the right bulldozer of can move over to pushing off all bulldozers it meets. Similarly town can sweep town away if the left bulldozer of can move over to 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.

Solution :
Let the statement be true for .

Let the towns be labelled from left to right and their left and right bulldozer respectively.

Now we have to prove the statement for towns..
Consider the rightmost town and let some collide with

Then there are two cases :
derails all such . Then obviously is the new winner town !

If some derails . Then obviously since there is no other bulldozer between this point and ,
sweeps
Since there are no bulldozers between and , The first towns live unaffected by the remaining towns. And hence by inducton we are done.