Geometry · Triangle inequality · Antipodal points · Sums of distances · Combinatorial geometry · Sharp constants
Let \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which
\[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \]
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Serbian Republic Competition (Republicko takmicenje) 2005, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source