Question 32
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. Denote $P$ as the midpoint of the segment $AO$ and $S$ as the intersection of the perpendicular bisector of the segment $AO$ with $BC$. The circumcircle of the triangle $APS$ intersects the segment $OH$ at $Z$. Prove that
$$
\vec{OZ} = \frac{1}{2} (\vec{OA} + \vec{OB} + \vec{OC}) .
$$
$$
\vec{OZ} = \frac{1}{2} (\vec{OA} + \vec{OB} + \vec{OC}) .
$$