2015 IMO Shortlist C1

In Lineland there are $n \geq 1$ 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 $2 n$ 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. (Estonia)

Lineland 有 $n \geq 1$ 个城镇，沿着从左到右的道路排列。每个城镇都有一台左推土机(放置在城镇左侧，面朝左)和一台右推土机(放置在城镇右侧，面朝右)。 $2 n$ 推土机的尺寸各不相同。每当左右推土机对峙时，较大的推土机就会将较小的推土机推离路面。另一方面，推土机的后部完全没有保护；因此，如果一台推土机到达另一台推土机的后端，第一台推土机会将第二台推土机推离路面，无论它们的尺寸如何。令 $A$ 和 $B$ 为两个城镇，$B$ 位于 $A$ 的右侧。我们说，如果 $A$ 的正确推土机可以移动到 $B$ 并推开它遇到的所有推土机，则 $A$ 镇可以扫除 $B$ 镇。同样，如果$B$的左侧推土机可以移动到$A$并推开沿途所有城镇的所有推土机，$B$就可以扫走$A$。证明确实有一个城镇不能被任何其他城镇冲走。 (爱沙尼亚)