Problem

Source: Ukraine 2005 grade 9

Tags: inequalities, triangle inequality, geometry unsolved, geometry



On the plane are given $ n \ge 3$ 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 $ M_1$ such that $ f(M_1)\le f(M)$ for any point $ M$ on the plane. Prove that if a point $ M_2$ satisfies $ f(M_1)=f(M_2),$ then $ M_1 \equiv M_2.$