View source for 2022 SSMO Team Round Problems/Problem 12

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

==Solution==