2023 WSMO Team Round Problems/Problem 14

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Problem

Consider quadrilateral $ABCD$ with circumcircle centered at $O$. Given $AB=2<CD$, the circumcircle of $\triangle{ABO}$ is tangent to the circumcircle of $\triangle{COD}$. The circumcircle of $\triangle{AOD}$ passes through the center of the circumcircle of $\triangle{ABO}$. Given the radius of the circumcircle of $ABCD$ is $6$. Denote $M,N$ as the area and perimeter of $ABCD$ respectively, compute $\frac{M}{N}$. The answer can be expressed in the simplest form of $\frac{p\sqrt{q}}{r}$. Find $p+q+r$

Solution