IrMO 2019 Problem 9

Suppose $x, y, z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$ . Prove that $8xyz \le 1$ , with equality if and only if $(x, y,z)$ is one of the following: $$\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)$$