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Bosnia and Herzegovina JBMO TST 2019 Q1

Source: https://artofproblemsolving.com/community/c6h1870126p12682434

July 6, 2019
algebra

Problem Statement

Let x,y,zx,y,z be real numbers ( x≠yx \ne y, y≠zy\ne z, x≠zx\ne z) different from 00. If x2−yzx(1−yz)=y2−xzy(1−xz)\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}, prove that the following relation holds: x+y+z=1x+1y+1z.x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.
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