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3x3 system, x^2 + y^2 = -x + 3y + z, x^2 + y^2 = -x + 3y + z, x^2 + z^2 = 2x + 2

Source: Dutch NMO 2018 p3

September 7, 2019
algebrasystem of equations

Problem Statement

Determine all triples (x,y,z)(x, y,z) consisting of three distinct real numbers, that satisfy the following system of equations: {x2+y2=−x+3y+zy2+z2=x+3y−zx2+z2=2x+2y−z\begin {cases}x^2 + y^2 = -x + 3y + z \\ y^2 + z^2 = x + 3y - z \\ x^2 + z^2 = 2x + 2y - z \end {cases}
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