14 - Inequalities

Let $a,b,c$ be nonnegative real numbers such that

$a+b+c=1$.

Prove that

$\sqrt[3]{13a^3 + 14b^3} + \sqrt[3]{13b^3 + 14c^3} + \sqrt[3]{13c^3 + 14a^3} ≥ 3$.

Proposed by Titu Andreescu, University of Texas at Dallas, USA

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