11. Given a circle $\gamma$ with center $O$ and diameter $A B$ of length $16 \mathrm{~cm}$, let $P$ be a point on the extension of $A B$ beyond $B$, and let $r$ be a line passing through $P$ that intersects $\gamma$ at points $C$ and $D$ (with $D$ between $C$ and $P$). Suppose, furthermore, that $O D=D P$ and $A \widehat{P} C=18^{\circ}$. What is the measure of the angle $A \widehat{O} C$?
(A) $48^{\circ}$
(B) $54^{\circ}$
(C) $60^{\circ}$
(D) $45^{\circ}$
(E) $72^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_04_17_ccb981cb4d80b90fdff2g-2.jpg?height=239&width=335&top_left_y=462&top_left_x=1067)
Problem 11. The answer is $\mathbf{( B )}$.

Let $\alpha=\widehat{D C O}=\widehat{O D C}$ and $\beta=\widehat{O P D}=\widehat{D O P}$. By the exterior angle theorem applied to the angle at $O$ of triangle $C O P$, we have $\widehat{A O C}=\alpha+\beta$; and by the same theorem applied to the angle at $D$ of triangle $O D P$, we have $\alpha=2 \beta$. Therefore, $\widehat{A O C}=3 \beta=54^{\circ}$.

(Problem proposed by F. Caceffo)