Problem

Source: Ukraine 2005 grade 9

Tags: inequalities, triangle inequality, geometry unsolved, geometry



On the plane are given n3 points, not all on the same line. For any point M on the same plane, f(M) is defined to be the sum of the distances from M to these n points. Suppose that there is a point M1 such that f(M1)f(M) for any point M on the plane. Prove that if a point M2 satisfies f(M1)=f(M2), then M1M2.