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Prove that xz< 1/2

Source: Mediterranean MO 2007

October 31, 2010
inequalitiesinequalities unsolved

Problem Statement

Let x≥y≥zx \geq y \geq z be real numbers such that xy+yz+zx=1xy + yz + zx = 1. Prove that xz<12.xz < \frac 12. Is it possible to improve the value of constant 12 ?\frac 12 \ ?
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