\noindent\textbf{4} \hfill \textit{(See the solution/explanation page)}

\medskip
\noindent Consider the four points $\mathrm{O}(0,\ 0,\ 0)$, $\mathrm{A}(2,\ 0,\ 0)$, $\mathrm{B}(1,\ 1,\ 1)$, $\mathrm{C}(1,\ 2,\ 3)$ in coordinate space.

\medskip
\noindent (1) Find the coordinates of the point P satisfying $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OA}}$, $\overrightarrow{\mathrm{OP}} \perp \overrightarrow{\mathrm{OB}}$, $\overrightarrow{\mathrm{OP}} \cdot \overrightarrow{\mathrm{OC}} = 1$.

\medskip
\noindent (2) Drop a perpendicular from point P to line AB, and let H be the intersection of that perpendicular with line AB. Express $\overrightarrow{\mathrm{OH}}$ in terms of $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$.

\medskip
\noindent (3) Define point Q by $\overrightarrow{\mathrm{OQ}} = \dfrac{3}{4}\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OP}}$, and consider the sphere $S$ centered at Q with radius $r$.

\medskip
\noindent Find the range of $r$ such that $S$ has a common point with triangle OHB. Here, triangle OHB lies in the plane containing the three points O, H, B, and consists of the boundary and its interior.



%% Page 5