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(x + y + z)xyz= -a^2 if x^2-1/y = y^2 -1/z = z^2 -1/x=a

Source: 2014 Belarus TST 3.2

December 30, 2020
algebrasystem of equations

Problem Statement

Let x,y,zx,y,z be pairwise distinct real numbers such that x2−1/y=y2−1/z=z2−1/xx^2-1/y = y^2 -1/z = z^2 -1/x. Given z2−1/x=az^2 -1/x = a, prove that (x+y+z)xyz=−a2(x + y + z)xyz= -a^2.
(I. Voronovich)
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