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

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2013 3.1

September 25, 2021
algebrasystem of equations

Problem Statement

Determine and justify all solutions (x,y,z)(x,y, z) of the system of equations:
x2=y+zx^2 = y + z y2=x+zy^2 = x + z z2=x+yz^2 = x + y
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