Problem (Geometry):
A circle with center $O$ and radius 1 contains chord $\overline{A B}$ of length 1 , and point $M$ is the midpoint of $\overline{A B}$. If the perpendicular to $\overline{A O}$ through $M$ intersects $\overline{A O}$ at $P$, compute $[M A P]$.

Solution:
Draw auxiliary segment $\overline{O B}$, as shown in the diagram below.



Triangle $O A B$ is equilateral, so $\mathrm{m} \angle O A B=60^{\circ}$. Then $\triangle M A P$ is a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle with hypotenuse $A M=1 / 2$. Thus $A P=1 / 4$ and $M P=\sqrt{3} / 4$, so

$$
\begin{aligned}
{[M A P] } & =\frac{1}{2}\left(\frac{1}{4}\right)\left(\frac{\sqrt{3}}{4}\right) \\
& =\frac{\sqrt{3}}{\mathbf{3 2}} .
\end{aligned}
$$

Final answer: $\frac{\sqrt{3}}{32}$