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Proof of ineqality

Source: Japan Mathematical Olympiad Finals 2005, Problem 3

March 30, 2005
inequalitiesfunctioninequalities proposed

Problem Statement

Let a,b,ca,b,c be positive real numbers such that a+b+c=1.a+b+c=1.Prove the following inequality. a1+b−c3+b1+c−a3+c1+a−b3≦1 a\sqrt[3]{1+b-c}+b\sqrt[3]{1+c-a}+c\sqrt[3]{1+a-b}\leqq 1
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