[ Extreme properties (continued).]

Inside a circle with center $O$, a point $A$ is given. Find the point $M$ on the circle for which the angle $OMA$ is maximized.
The geometric locus of points $X$, for which the angle $OXA$ is constant, consists of two arcs of circles $S_{1}$ and $S_{2}$ symmetric with respect to the line $OA$. Consider the case when the diameter of the circles $S_{1}$ and $S_{2}$ is equal to the radius of the original circle, i.e., these circles touch the original circle at points $M_{1}$ and $M_{2}$, for which $\angle O A M_{1}=\angle O A M_{2}=90^{\circ}$. Points $M_{1}$ and $M_{2}$ are the desired points, since if $\angle O X A>\angle O M_{1} A=\angle O M_{2} A$, then point $X$ lies strictly inside the figure formed by the circles $S_{1}$ and $S_{2}$, i.e., it cannot lie on the original circle.