Combinatorics · Helly theorem · Convex sets · Arcs and chords · Covering a circle · Antipodal points
Finitely many arcs are marked on a circle. The length of each of them is smaller than half of the circumference, and any three of the marked arcs have a common point.
Prove that there is a point of the circle which lies on none of the marked arcs.
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Serbian National Competition (Drzavno takmicenje) 2009, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source