Question 87
Let $\alpha \ge \frac{1}{2}$ be a real number. Prove that the following inequality holds for all positive real numbers $x$, $y$, $z$:
$$
x(x - y)(\alpha x - y) + y(y - z)(\alpha y - z) + z(z - x)(\alpha z - x) \ge 0.
$$
$$
x(x - y)(\alpha x - y) + y(y - z)(\alpha y - z) + z(z - x)(\alpha z - x) \ge 0.
$$