BULGARIAN NATIONAL MATHEMATICAL OLYMPIAD

The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ intersect at point $O$. The points $A_1$, $B_1$, $C_1$ and $D_1$ from the segments $AO$, $BO$, $CO$ and $DO$, respectively, are such that $A{A}_1=C{C}_1$ and $B{B}_1=D{D}_1$. Let $M$ be the second intersection point of the circumcircles of $\angle AOB$ and $\angle COD$; $N$ be the second intersection point of the circumcircles of $\angle AOD$ and $\angle BOC$; $P$ be the second intersection point of the circumcircles of $\angle A_{1}O{B}_1$ and $\angle C_{1}O{D}_1$, and $Q$ be the second intersection point of the circumcircles of $\angle A_{1}O{D}_1$ and $\angle B_{1}O{C}_1$. Prove that the points $M$, $N$, $P$ and $Q$ are concyclic.