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How to prove this inequality?

Source: Romanian Mathematical Contest

January 19, 2020
inequalitiesinequalities unsolved

Problem Statement

How do I prove that, for every a,b,ca, b, c positive real numbers such that a+b+c=1a+b+c = 1 the following inequality holds: a3a2+b2+b3b2+c2+c3c2+a2≥12\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}?
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