Show that if a,b,c are the lengths of the sides of a triangle and if 2S=a+b+c, then
\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \forall n \in \mathbb N Proposed by Greece. rearrangement inequalitythree variable inequalitytriangle inequalityInequalityIMO Shortlist