Question 6
(GRE 4) Show that if $a,b,c$ are the lengths of the sides of a triangle and if $2S=a+b+c$, then
$$\frac{a^n}{b+c}+\frac{b^n}{c+a}+\frac{c^n}{a+b}\ge\left(\frac23\right)^{n-2}S^{n-1},\qquad n\ge1.$$
$$\frac{a^n}{b+c}+\frac{b^n}{c+a}+\frac{c^n}{a+b}\ge\left(\frac23\right)^{n-2}S^{n-1},\qquad n\ge1.$$