IrMO 2007 Problem 2

Suppose that $a,b,$ and $c$ are positive real numbers. Prove that:

$\frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \le \frac {\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}}{3}$ .

For each of the inequalities, find the conditions on $a,b,$ and $c$ such that equality holds.