MathDB

4

Part of 2001 Mediterranean Mathematics Olympiad

Problems(1)

Problem with function on points

Source: Mediterranean MO 2001

10/31/2010
Let SS be the set of points inside a given equilateral triangle ABCABC with side 11 or on its boundary. For any M∈S,aM,bM,cMM \in S, a_M, b_M, c_M denote the distances from MM to BC,CA,ABBC,CA,AB, respectively. Define f(M)=aM3(bM−cM)+bM3(cM−aM)+cM3(aM−bM).f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M). (a) Describe the set {M∈S∣f(M)≥0}\{M \in S | f(M) \geq 0\} geometrically.
(b) Find the minimum and maximum values of f(M)f(M) as well as the points in which these are attained.
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