UA-UMO 2005 Problem 93

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 $M_1 \equiv M_2.$