Question 18
(LUX 1) IMO1 Consider two concentric circles of radii $R$ and $r$ $(R\gt r)$ with center $O$. Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large circle; $B,P,C$ are collinear and $BC$ is perpendicular to $AP$.
(a) For what value(s) of $\angle OPA$ is the sum $BC^2+CA^2+AB^2$ extremal?
(b) What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies?
(a) For what value(s) of $\angle OPA$ is the sum $BC^2+CA^2+AB^2$ extremal?
(b) What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies?