sharker

$\sum {\frac{a}{{bc + 1}} = \sum {a - \frac{{abc}}{{1 + bc}} \ge 3 - \sum {\frac{{a\sqrt {bc} }}{2} = 3 - \sum {\frac{{\sqrt a .\sqrt {abc} }}{2} \ge 3 - \sum {\frac{{\sqrt a \sqrt {\frac{{{{(a + b + c)}^3}}}{{27}}} }}{2} = 3 - \sum {\frac{{\sqrt a }}{2}} } } } } }$

 $\ge 3 - \frac{1}{2}\sqrt {(a + b + c).3}  = \frac{3}{2}$