The Attempt at a Solution
So my first thought is that the only way to solve this problem is to apply a characterization of a cyclic quadrilateral. We know that the perpendicular bisectors of a cyclic quadrilateral are concurrent. So here's my thoughts: Construct triangle OPQ. The perpendicular bisector of OP is located on the midpoint of the radius of C(O, OP) and the perpendicular bisector of OQ is located on the midpoint of C(O, OQ). Since OPQ is a triangle, we know these perpendicular bisectors have to be concurrent, and the intersection is the center of the circumcircle. Now we just have to show that R is also located on the circumcircle of OPQ.
The part in the hypothesis about the circles being concentric seems like it is relevant, but I'm not sure how to incorporate it other than what I mentioned earlier.