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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.

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