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IMOSL 2015 C1 ( available here )

In Lineland there are $n$ 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 right and a left bulldozer confront each other, the larger bulldozer pushes the smaller one off the road.
On the other hand, the 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 $B$ being 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, $B$ can sweep $A$ away if the left bulldozer of $B$ can move to $A$ pushing off all bulldozers of all towns on its way.
Prove that there is exactly one town which cannot be swept away by any other one.

My question is about the interpretation of the problem statement of 'sweeping away' another town.

Suppose there are three towns from left to right: A, B and C.

A has the tanks, denoted by size:

1 (left), 15(right)

B has:

30(left), 10(right)

C has:

11(left), 1(right)

My question is, would C sweep A?

If we consider only C and A's trucks, C would not sweep A, but A would sweep C. However, if B's trucks are also moving, then B's left truck takes out A's right truck, and then C sweeps both B and A. What is the correct interpretation?

Also, please dont give the solution of the problem in your answers!

Prem
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Illusioner_
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  • The size of C-right bulldozer has no impact. This bulldozer will survive, or die if any right bulldozer can reach the right end of the road. Same for A-left bulldozer. At the end of the process only 1 Left-bulldozer and only 1 right-bulldozer will survive. The speed of each bulldozer has an impact : If C-left bulldozer can push out B-left bulldozer before B-left bulldozer cross A-right bulldozer, the size of b-left bulldozer will have no impact. – Lourrran Oct 20 '24 at 08:12
  • I think speed does not matter here, what came to my mind was: all the bulldozers are not moving except C's. But yes, if the problem allowed B to move while C was moving, speed would play a role – Illusioner_ Oct 20 '24 at 10:25
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    ((1)) Definition of "swept away" makes no mention of speeds , @Lourrran , while Definition mentions only a pair of towns. ((2)) I think OP is not looking for the Solutions here. while OP can ask for the Solutions via newer Posts , when that is necessary ! – Prem Oct 20 '24 at 11:02

2 Answers2

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OP : "What is the correct interpretation?"

(1) Definition indicates that we must consider only a pair of towns to make the eventual outcome. We are not concerned with whether some other towns can participate to alter the outcome.

(2) OP gave that $A$ is $(1,15)$ , while $C$ is $(11,1)$ where $1$ is occurring multiple times , which is not allowed.

With that , let me change it to :
$A$ is $(1,15)$ , $C$ is $(11,3)$
When $A$ is the left town , then $A$ will demolish $C$ using the right $15$.

Prem
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Let us look how the process can finish.

We are sure that exactly one left bulldozer and one right bulldozer will survive at the end of the process.

Introduce some notations :

  • $n$ = number of towns
  • $R_i$ and $L_i$ : represent the Right-bulldozer and Left-bulldozer from town n°$i$
  • $r_R$ and $r_L$ : the rank of the bulldozer that finishes alive at the Right end (and Left end) of the road.

$L_1$ bulldozer goes to the left, and stops at the end of the road. He stays here ; he can survive only if no other left-bulldozer survives. Same for $R_n$ bulldozer.

Is it possible to have $r_L < r_R$, Is it possible to have $r_L>r_R$, Is it possible to have $r_L=r_R$ ?

During the process, we have many fights : a fight happens when a left-bulldozer confront a right bulldozer.

Consider the last fight. If the winner was the right bulldozer $R_i$ ; this bulldozer will go ahead and it will eject the bulldozer which was laying at the right end of the road. So $r_R = i$

All left-bulldozer with rank greater than $i$ have been killed, either by this bulldozer $R_i$, either by another bulldozer.

So it is impossible to have $r_L > r_R$

Is it possible to have $r_L < r_R$ ?
If $r_L < r_R$ , it means that all Left bulldozer with a rank greater than $r_L$ have been killed, and all Right-Bulldozer with a rank lower than $r_R$ have been killed. and it is impossible. (maybe we need a better proove here ?).

So we are sure that at the end of the process, $r_L=r_R$. And the only remaining town is this one.

Lourrran
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  • Again, don't think this is correct. As @Prem stated, we only consider pairwise. This answer would be true only if all bulldozers moved at once! What we consider is a pair of bulldozers. Their left and rights are moving towards each other. Of course, other bulldozers in the way would impact the results but they can't move. – Illusioner_ Oct 20 '24 at 10:31