2022 SSMO Team Round Problems/Problem 12

Problem

Regular pentagon $ABCDE$ is inscribed in circle $\omega_1$ with radius $5\sqrt{5}$ . Circle $\omega_2$ is the reflection of $\omega_1$ across $\overline{AB}$ . Let $I$ be the intersection of $\overline{AD}$ and $\overline{BE}$ , let $P$ be an intersection of $\overline{DO}$ and $\omega_2$ , and let line $\ell$ be the tangent to $\omega_2$ at $P$ . The sum of the possible distances from point $I$ to line $\ell$ can be expressed as $m\sqrt{n}$ , where $n$ is a squarefree positive integer. Find $m+n$ .

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2022 SSMO Team Round Problems/Problem 12

Problem

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