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a+b+c=0 if unlikely inequalities hold

Source: Baltic Way 1995

October 8, 2011
inequalitiesalgebra proposedalgebra

Problem Statement

The real numbers a,ba,b and cc satisfy the inequalities ∣a∣≥∣b+c∣,∣b∣≥∣c+a∣|a|\ge |b+c|,|b|\ge |c+a| and ∣c∣≥∣a+b∣|c|\ge |a+b|. Prove that a+b+c=0a+b+c=0.
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