Školjka

%V0 Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then $$\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N$$ Proposed by Greece.