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2022 Tuymaada Olympiad, 7
A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]
rectangle domino combinatorics tiels