MathDB
Distance inequality holds for all interior points

Source: Bulgaria MO 2011

May 30, 2011
functiongeometrylimitgeometry proposed

Problem Statement

Triangle ABCABC and a function f:R+→Rf:\mathbb{R}^+\to\mathbb{R} have the following property: for every line segment DEDE from the interior of the triangle with midpoint MM, the inequality f(d(D))+f(d(E))≤2f(d(M))f(d(D))+f(d(E))\le 2f(d(M)), where d(X)d(X) is the distance from point XX to the nearest side of the triangle (XX is in the interior of △ABC\triangle ABC). Prove that for each line segment PQPQ and each point interior point NN the inequality ∣QN∣f(d(P))+∣PN∣f(d(Q))≤∣PQ∣f(d(N))|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N)) holds.
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