JBMO-SL 2002 Problem 3

Let $a,b,c$ be positive real numbers such that $abc=\frac{9}{4}$ . Prove the inequality: $a^3 + b^3 + c^3 > a\sqrt {b + c} + b\sqrt {c + a} + c\sqrt {a + b}$

Jury's variant: Prove the same, but with $abc=2$