2006 OIM Problems/Problem 1

Problem

In the scalene triangle $ABC$ , with $\angle BAC = 90^{\circ}$ , the inscribed and circumscribed circles are considered. The tangent line at $A$ to the circumcircle intersects the line $BC$ at $M$ . Let $S$ and $R$ be the points of tangency of the circle inscribed with the lines $AC$ and $AB$ , respectively. Line $RS$ intersects line $BC$ at $N$ . Lines $AM$ and $SR$ meet at $U$ . Show that the triangle $UMN$ is isosceles.