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A hard problem with triangle inequality

Source: 2018 JBMO TST - Turkey, P8

March 27, 2020
algebratriangle inequalitypositive real numbersinequalitiesTurkey

Problem Statement

Let x,y,zx, y, z be positive real numbers such that x,y,z\sqrt {x}, \sqrt {y}, \sqrt {z} are sides of a triangle and xy+yz+zx=5\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5. Prove that x(y2−2z2)z+y(z2−2x2)x+z(x2−2y2)y⩾0\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0
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