Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
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A version of this question was asked on MO a while back: "Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?." It was determined that that particular problem (which I now think is identical to yours as currently posted) is open, and appears in Unsolved Problems in Geometry.
There Anton Geraschenko posted this interesting dissection: