AIME I 2026 (Problem 3)

Solid GeometryMedium

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

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

T

divided by the area of the disk is

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers. Find

p+q