2026 AIME I Problem 3

A hemisphere with radius 200 sits on top of a horizontal circular disk with radius 200, and the hemisphere and disk have the same center. Let \mathcal{T} be the region of points P in the disk such that a sphere of radius 42 can be placed on top of the disk at P and lie completely inside the hemisphere. The area of \mathcal{T} divided by the area of the disk is \frac{p}{q}, where p and q are relatively prime positive integers. Find p+q.

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.