2022 AIME I Problem 8

Equilateral triangle \triangle ABC is inscribed in circle \omega with radius 18. Circle \omega_A is tangent to sides \overline{AB} and \overline{AC} and is internally tangent to \omega. Circles \omega_B and \omega_C are defined analogously. Circles \omega_A, \omega_B, and \omega_C meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of \triangle ABC are the vertices of a large equilateral triangle in the interior of \triangle ABC, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of \triangle ABC. The side length of the smaller equilateral triangle can be written as \sqrt{a} - \sqrt{b}, where a and b are positive integers. Find a+b.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.