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Given two conditions related to x,y,z and min of a ratio wanted

Source: Turkey EGMO TST 2022 P6

March 16, 2022
inequalitiesminimum valueratio

Problem Statement

Let x,y,zx,y,z be positive real numbers satisfying the equations xyz=1 and yz(y−x2)+zx(z−y2)+xy(x−z2)=0xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0 What is the minimum value of the ratio of the sum of the largest and smallest numbers among x,y,zx,y,z to the median of them.
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