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IMO ShortList 2003, combinatorics problem 2

Source:

May 17, 2004
geometrycirclesIntersectionIMO Shortlist

Problem Statement

Let D1D_1, D2D_2, ..., DnD_n be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 20032003 discs DiD_i. Prove that there exists a disc DkD_k which intersects at most 7⋅2003−1=140207\cdot 2003 - 1 = 14020 other discs DiD_i.
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