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Easy inequality on a + b + c = 3

Source: Mexican Mathematical Olympiad 2014 Problem 5

November 15, 2014
inequalitiesinequalities proposedHolder

Problem Statement

Let a,b,ca, b, c be positive reals such that a+b+c=3a + b + c = 3. Prove:
a2a+bc3+b2b+ca3+c2c+ab3≥32 \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2}
And determine when equality holds.
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