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Olympiad Geometry on AIME yay

Source:

February 2, 2024

geometryAMCAIME

Problem Statement

Let

\triangle ABC

have side lengths

C

and

CA = 10.

The tangents to the circumcircle of

\triangle ABC

at

B

and

C

intersect at point

D,

and

\overline{AD}

intersects the circumcircle at

P \ne A.

The length of

\overline{AP}

is equal to

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

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