IMO Shortlist 2014 problem C4
Construct a tetromino by attaching 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 S- and Z-tetrominoes, respectively.
Assume that a lattice polygon can be tiled with S-tetrominoes. Prove that no matter how we tile using only S- and Z-tetrominoes, we always use an even number of Z-tetrominoes.