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Union of A_i s is a subset of union of B_i s

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October 13, 2010
algebra unsolvedalgebra

Problem Statement

Let XX be an arbitrary nonempty set contained in the plane and let sets A1,A2,⋯ ,AmA_1, A_2,\cdots,A_m and B1,B2,⋯ ,BnB_1, B_2,\cdots, B_n be its images under parallel translations. Let us suppose that A1∪A2∪⋯∪Am⊂B1∪B2∪⋯∪BnA_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n and that the sets A1,A2,⋯ ,AmA_1, A_2,\cdots,A_m are disjoint. Prove that m≤nm \le n.
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