RU-ORAL-GEOMETRY 2014

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$ . Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$ .

Let $ABCD$ be a parallelogram. On side $AB$ , point $M$ is taken so that $AD = DM$ . On side $AD$ point $N$ is taken so that $AB = BN$ . Prove that $CM = CN$ .

Is there a convex pentagon in which each diagonal is equal to a side?