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Cyclic inequality in x,y,z

Source: Swiss Math Olympiad 2010 - final round, problem 4

March 16, 2010

inequalitiesalgebrafunctiontriangle inequalitythree variable inequality

Problem Statement

Let

x

y

z \in\mathbb{R}^+

satisfying

xyz = 1

. Prove that \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}