2008 AIME II Problem 11

In triangle ABC, AB = AC = 100, and BC = 56. Circle P has radius 16 and is tangent to \overline{AC} and \overline{BC}. Circle Q is externally tangent to circle P and is tangent to \overline{AB} and \overline{BC}. No point of circle Q lies outside of \bigtriangleup\overline{ABC}. The radius of circle Q can be expressed in the form m - n\sqrt{k},where m, n, and k are positive integers and k is the product of distinct primes. Find m +nk.

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.