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n points

Source: Ukraine 2005 grade 9

July 26, 2009
inequalitiestriangle inequalitygeometry unsolvedgeometry

Problem Statement

On the plane are given n≥3 n \ge 3 points, not all on the same line. For any point M M on the same plane, f(M) f(M) is defined to be the sum of the distances from M M to these n n points. Suppose that there is a point M1 M_1 such that f(M1)≤f(M) f(M_1)\le f(M) for any point M M on the plane. Prove that if a point M2 M_2 satisfies f(M_1)\equal{}f(M_2), then M1≡M2. M_1 \equiv M_2.
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