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Quadrilateral condition in inequality terms

Source: Romanian IMO Team Selection Test TST 1996, problem 14

September 27, 2005
inequalitiesinequalities proposed

Problem Statement

Let x,y,z x,y,z be real numbers. Prove that the following conditions are equivalent: (i) x,y,z x,y,z are positive numbers and 1x+1y+1z≤1 \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 ; (ii) a2x+b2y+c2z>d2 a^2x+b^2y+c^2z>d^2 holds for every quadrilateral with sides a,b,c,d a,b,c,d .
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