Iran NMO 2002/3
In a convex quadrilateral $ABCD$ with $\angle ABC=\angle ADC=135^ \circ$,points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD=\angle NCB=90^\circ$.The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$.Prove that $AK \perp KC$.
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