RO-NMO 2004 Problem 72

The sidelengths of a triangle are $a,b,c$ .

(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$ .

(b) Prove that $\sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$ .