Problem

Source: Croatia TST 2009

Tags: inequalities, algebra unsolved, algebra



Solve in the set of real numbers: \[ 3\left(x^2 + y^2 + z^2\right) = 1, \] \[ x^2y^2 + y^2z^2 + z^2x^2 = xyz\left(x + y + z\right)^3. \]