US-TST 2001 Problem 6

Let $a,b,c$ be positive real numbers such that $$a+b+c\geq abc.$$ Prove that at least two of the inequalities $$\frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6$$ are true.