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Difference between revisions of "2020 USOJMO Problems/Problem 1"

(Created page with "Let <math>n \geq 2</math> be an integer. Carl has <math>n</math> books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no t...")
 
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==Problem==
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Let <math>n \geq 2</math> be an integer. Carl has <math>n</math> books arranged on a bookshelf.
 
Let <math>n \geq 2</math> be an integer. Carl has <math>n</math> books arranged on a bookshelf.
 
Each book has a height and a width. No two books have the same height, and no two
 
Each book has a height and a width. No two books have the same height, and no two

Revision as of 17:05, 23 June 2020

Problem

Let $n \geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right.

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