2021 April MIMC 10 Problems/Problem 24

One semicircle is constructed with diameter $AH=4$ and let the midpoint of $AH$ be $M$ . Construct a point $O$ on the side of segment $AH$ (closer to segment $AH$ than arc $AH$ ) such that the distance from $A$ to $O$ is $2\sqrt{5}$ , and that $OM$ is perpendicular to the diameter $AH$ . Three more such congruent semicircles are formed through multiple $90^{\circ}$ rotations around the point $O$ . Name the $6$ endpoints of the diameters $B$ , $C$ , $D$ , $E$ , $F$ , $G$ in a circular direction from $A$ to $H$ . Another four congruent semicircles are constructed with diameters $AB, CD, EF, GH$ , and that the distance from the diameters to the point $O$ are less than the distance from the arcs to the point $O$ . Connect $AC$ , $CD$ , $DO$ , $OG$ , and $GA$ . Find the ratio of the area of the pentagon $ACDOG$ to the total area of the shape formed by arcs $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , $FG$ , $GH$ , $HA$ .

$\textbf{(A)} ~\frac{14+10\pi}{17} \qquad\textbf{(B)} ~\frac{13+\sqrt{2}}{28} \qquad\textbf{(C)} ~\frac{4+\sqrt{2}}{7+3\pi} \qquad\textbf{(D)} ~\frac{13}{28+6\pi} \qquad\textbf{(E)} ~\frac{13}{30\pi}\qquad$

Solution

To be Released on April 26th.

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2021 April MIMC 10 Problems/Problem 24

Solution