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China Mathematics Olympiads (National Round) 2007 Problem 1

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November 28, 2010
complex numbersinequalities proposedinequalities

Problem Statement

Given complex numbers a,b,ca, b, c, let ∣a+b∣=m,∣a−b∣=n|a+b|=m, |a-b|=n. If mn≠0mn \neq 0, Show that max⁡{∣ac+b∣,∣a+bc∣}≥mnm2+n2\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}
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