18. (LUX 1) ${ }^{\mathrm{IMO}}$ Consider two concentric circles of radii $R$ and $r(R>r)$ with center $O$. Fix $P$ on the small circle and consider the variable chord $P A$ of the small circle. Points $B$ and $C$ lie on the large circle; $B, P, C$ are collinear and $B C$ is perpendicular to $A P$. (a) For what value(s) of $\angle O P A$ is the sum $B C^{2}+C A^{2}+A B^{2}$ extremal? (b) What are the possible positions of the midpoints $U$ of $B A$ and $V$ of $A C$ as $\angle O P A$ varies?
18. (i) Define $\angle A P O=\phi$ and $S=A B^{2}+A C^{2}+B C^{2}$. We calculate $P A=$ $2 r \cos \phi$ and $P B, P C=\sqrt{R^{2}-r^{2} \cos ^{2} \phi} \pm r \sin \phi$. We also have $A B^{2}=$ $P A^{2}+P B^{2}, A C^{2}=P A^{2}+P C^{2}$ and $B C=B P+P C$. Combining all these we obtain  
$$ 
\begin{aligned} 
S & =A B^{2}+A C^{2}+B C^{2}=2\left(P A^{2}+P B^{2}+P C^{2}+P B \cdot P C\right) \\ 
& =2\left(4 r^{2} \cos ^{2} \phi+2\left(R^{2}-r^{2} \cos ^{2} \phi+r^{2} \sin ^{2} \phi\right)+R^{2}-r^{2}\right) \\ 
& =6 R^{2}+2 r^{2} . 
\end{aligned} 
$$ 
Hence it follows that $S$ is constant; i.e., it does not depend on $\phi$. (ii) Let $B_{1}$ and $C_{1}$ respectively be points such that $A P B B_{1}$ and $A P C C_{1}$ are rectangles. It is evident that $B_{1}$ and $C_{1}$ lie on the larger circle and that $\overrightarrow{P U}=\frac{1}{2} \overrightarrow{P B_{1}}$ and $\overrightarrow{P V}=\frac{1}{2} \overrightarrow{P C_{1}}$. It is evident that we can arrange for an arbitrary point on the larger circle to be $B_{1}$ or $C_{1}$. Hence, the locus of $U$ and $V$ is equal to the circle obtained when the larger circle is shrunk by a factor of $1 / 2$ with respect to point $P$.