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3(x^2 + y^2 + z^2) = 1 , x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3, system 3x3

Source: 2010 Dutch IMO TST1 P5

January 10, 2020
system of equationsalgebra

Problem Statement

Find all triples (x,y,z)(x,y, z) of real (but not necessarily positive) numbers satisfying 3(x2+y2+z2)=13(x^2 + y^2 + z^2) = 1 , x2y2+y2z2+z2x2=xyz(x+y+z)3x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3.
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