2008 AIME I Problem 14

Let \overline{AB} be a diameter of circle \omega. Extend \overline{AB} through A to C. Point T lies on \omega so that line CT is tangent to \omega. Point P is the foot of the perpendicular from A to line CT. Suppose \overline{AB} = 18, and let m denote the maximum possible length of segment BP. Find m^{2}.

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.