IMO 1984 LL POL45

Let X be an arbitrary nonempty set contained in the plane and

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IMO 1984 LL POL45

Origin: POL

Problem

Let X be an arbitrary nonempty set contained in the plane and let sets A1, A2, . . . , Am and B1, B2, . . . , Bn be its images under parallel translations. Let us suppose that A1 \cupA2 \cup\cdot \cdot \cdot \cupAm \subsetB1 \cupB2 \cup\cdot \cdot \cdot \cupBn and that the sets A1, A2, . . . , Am are disjoint. Prove that m \leqn.