Question 158
Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then
$$
k(x^2 - x + 1)(y^2 - y + 1)(z^2 - z + 1) \geq (xyz)^2 - xyz + 1.
$$
$$
k(x^2 - x + 1)(y^2 - y + 1)(z^2 - z + 1) \geq (xyz)^2 - xyz + 1.
$$