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Geometry Difficulty 5.0 AIME, harder Prove it United Kingdom

The obtuse-angled triangle $ABC$ has sides of length $a$ , $b$ and $c$ opposite the angles $\angle A$ , $\angle B$ and $\angle C$ respectively. Prove that
$a^3 \cos A + b^3 \cos B + c^3 \cos C < abc.$

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