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Part of 2018 IOM

Problems(1)

Three variable, two equation real nonlinear system

Source: IOM 2018 #1, Vladimir Bragin

Solve the system of equations in real numbers: \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} Vladimir Bragin