Question 47
In the acute angled triangle $ABC$ consider the altitudes $BB'$ and $CC'$. The half-line $C'B'$ intersects the circumcircle of triangle at $B''$ and denote by $\alpha_A$ the angle $\widehat{ABB''}$. In a similar way define the angles $\alpha_B$ and $\alpha_C$. Prove the inequality
$$
\sin \alpha_A \sin \alpha_B \sin \alpha_C \le \frac{3\sqrt{6}}{32}.
$$
$$
\sin \alpha_A \sin \alpha_B \sin \alpha_C \le \frac{3\sqrt{6}}{32}.
$$