10 - Inequalities

Let $a, b, c$ be real numbers greater than or equal to $1$.

Prove that

$\dfrac{a^ 3 + 2}{b^ 2 − b + 1} + \dfrac{b^ 3 + 2}{ c^ 2 − c + 1} + \dfrac{c^ 3 + 2}{ a^ 2 − a + 1} ≥ 9$.

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

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