11. Along the coast of a circular lake, there are three ports. On a map, the ports are located at three positions $A, B, C$, and measuring the angles, it is found that $C \widehat{A} B=57^{\circ}$, $A \widehat{B} C=48^{\circ}, B \widehat{C} A=75^{\circ}$. In case of an emergency, assistance within the lake is provided by the nearest port. Which portion of the lake is assisted by the port that must cover the largest area?
(A) $5 / 12$
(B) $11 / 30$
(C) $41 / 120$
(D) $7 / 18$
(E) $9 / 25$
Problem 11. The correct answer is $(\mathrm{B})$.

The different intervention zones of the three ports are delimited by the angle bisectors of the sides of triangle $A B C$, which concur at the center of the lake $O$. These are three circular sectors, the amplitudes of which are given by angles at the center that are supplementary to those of triangle $A B C$. Among these central angles, the largest is $\widehat{A^{\prime} O C^{\prime}}$, which contains point $B$ (since $\widehat{B}$ is the smallest angle of $A B C$), and will have an amplitude equal to $180^{\circ}-48^{\circ}=132^{\circ}$. The corresponding portion of the lake will therefore be equal to $\frac{132}{360}=\frac{11}{30}$ of the surface.

![](https://cdn.mathpix.com/cropped/2024_04_17_48746ba046504379ff31g-3.jpg?height=369&width=396&top_left_y=1000&top_left_x=886)