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System of equations implies greatest value of an expression

Source: 2022 Bulgarian Spring Math Competition, Problem 10.1

March 27, 2022
system of equationsInequalityalgebra

Problem Statement

If x,y,z∈Rx, y, z \in \mathbb{R} are solutions to the system of equations {x−y+z−1=0xy+2z2−6z+1=0\begin{cases} x - y + z - 1 = 0\\ xy + 2z^2 - 6z + 1 = 0\\ \end{cases} what is the greatest value of (x−1)2+(y+1)2(x - 1)^2 + (y + 1)^2?
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