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Asymetric inequality with determinant-like condition

Source: 5th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Senior - Problem 2

January 29, 2024
inequalitiesalgebra

Problem Statement

Let x,yx,y and zz be positive real numbers such that xy+z2=8xy+z^2=8. Determine the smallest possible value of the expression x+yz+y+zx2+z+xy2.\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.
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