Problem of the Day #37

POTD January 11, 2024

In a circle \Omega of radius 9, draw a diameter, and two parallel and congruent chords of the circle that intersect the diameter such that there exist circles of radius 4 tangent to both chords, the diameter, and internally tangent to \omega. Let \omega be the acute angle formed by the intersection of a chord with the diameter. If \tan(\omega) can be expressed as \frac{a}{b}, where a and b are positive, relatively prime, integers, find a+b.