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IMO Shortlist 2014 C4

Source:

July 11, 2015
IMO Shortlistcombinatoricsalgebraprime numbers

Problem Statement

Construct a tetromino by attaching two 2×12 \times 1 dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them SS- and ZZ-tetrominoes, respectively. Assume that a lattice polygon PP can be tiled with SS-tetrominoes. Prove that no matter how we tile PP using only SS- and ZZ-tetrominoes, we always use an even number of ZZ-tetrominoes.
Proposed by Tamas Fleiner and Peter Pal Pach, Hungary
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