IR-TST 2007 Problem 11

Triangle $ABC$ is isosceles ( $AB=AC$ ). From $A$ , we draw a line $\ell$ parallel to $BC$ . $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$ . $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$ . Prove that $$\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}$$