US-TST 2003 Problem 5

Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$ . Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$ .