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Ineq: √( (ab+bc+ca)/(a^2+b^2+c^2) ) + Σ ( a/(b+c) ) ≥ 5/2

Source: Spanish Mathematical Olympiad, National Stage, 2011

August 9, 2011
inequalitiesfunctioncalculusderivativeinequalities proposedalgebra

Problem Statement

Let aa, bb, cc be positive real numbers. Prove that ab+c+bc+a+ca+b+ab+bc+caa2+b2+c2≥52 \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52 and determine when equality holds.
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