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Bosnia and Herzegovina JBMO TST 2013 Problem 2

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013

September 16, 2018

Inequalitypositive realalgebrainequalities

Problem Statement

Let

a

b

and

c

be positive real numbers such that

a^2+b^2+c^2=3

. Prove the following inequality:

\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}