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Part of 2004 Iran MO (3rd Round)

Problems(1)

Convex subsets of R^3 every three of which intersect

Source: Iranian National Olympiad (3rd Round) 2004

Finitely many convex subsets of

\mathbb R^3

are given, such that every three have non-empty intersection. Prove that there exists a line in

\mathbb R^3

that intersects all of these subsets.

geometrygeometry proposed