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IMO Shortlist 2011, Combinatorics 7

Source: IMO Shortlist 2011, Combinatorics 7

July 12, 2012
inequalitiescombinatoricsIMO ShortlistExtremal combinatorics

Problem Statement

On a square table of 20112011 by 20112011 cells we place a finite number of napkins that each cover a square of 5252 by 5252 cells. In each cell we write the number of napkins covering it, and we record the maximal number kk of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of kk?
Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia