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Convex subsets of R^3 every three of which intersect

Source: Iranian National Olympiad (3rd Round) 2004

January 9, 2009
geometrygeometry proposed

Problem Statement

Finitely many convex subsets of R3\mathbb R^3 are given, such that every three have non-empty intersection. Prove that there exists a line in R3\mathbb R^3 that intersects all of these subsets.
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