IMO Shortlist 1987 problem 6
Dodao/la:
arhiva2. travnja 2012. Show that if
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
are the lengths of the sides of a triangle and if
![2S = a + b + c](/media/m/8/1/a/81a3227c511fdb41b6d505d09a0ec89f.png)
, 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](/media/m/7/e/2/7e2119d0c4672eece5fb96b61fe31647.png)
Proposed by Greece.
%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.
Izvor: Međunarodna matematička olimpijada, shortlist 1987