VN-TST 2013 Problem 1

The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$ . Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively . a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$ . b) Denote $min \,\{AC, BD\} = m$ . Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$ .

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