IrMO 2011 Problem 2

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$ , respectively, the lengths of the medians which connect $A,B,C$ , respectively, with the centers of the corresponding opposite sides.

(a) Prove that $2m_a<b+c$ . Deduce that $m_a+m_b+m_c<a+b+c$ . (b) Give an example of (i) a triangle in which $m_a>\sqrt{bc}$ ; (ii) a triangle in which $m_a\le \sqrt{bc}$ .