Problem:
The perimeter of triangle APM is 152 , and angle PAM is a right angle. A circle of radius 19 with center O on AP is drawn so that it is tangent to AM and PM. Given that OP=m/n, where m and n are relatively prime positive integers, find m+n.
Solution:
Let T​ and B be the points where the circle meets PM and AP, respectively, with ABP. Triangles POT and PAM are right triangles that share angle MPA, so they are similar. Let p1​ and p2​ be their respective perimeters. Then OT/AM=p1​/p2​. Because AM=TM, it follows that p1​=p2​−(AM+TM)=152−2AM. Thus 19/AM=(152−2AM)/152, so that AM=38 and p1​=76. It is also true that OP/PM=p1​/p2​, so
21​=PMOP​=152−(38+19+OP)OP​
It follows that OP=95/3, and m+n=98​.
The problems on this page are the property of the MAA's American Mathematics Competitions
