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a,b,c lengths of the sides of a triangle (IMO SL 1987-P6)

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August 19, 2010
rearrangement inequalitythree variable inequalitytriangle inequalityInequalityIMO Shortlist

Problem Statement

Show that if a,b,ca, b, c are the lengths of the sides of a triangle and if 2S=a+b+c2S = 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.
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