Team Selection Test for IMO

It is easy to observe that the point $F$ is unique when $A,B,C,D$ are fixed. Hence it is enough to show that the intersection of the circumcircle of $ABD$ and $[CB]$ satisfies the properties of the point $F$. Let the intersection be $F^{\prime }$. $\angle DB{F}^{\prime }=\angle DA{F}^{\prime }$. Then $\angle BA{F}^{\prime }+\angle EBX=\angle DAB$. On the other hand we have $\angle DEC=\angle DAB=\angle D{F}^{\prime }C$. Thus, $D,E,F^{\prime },C$ are concyclic. Hence $\angle E{F}^{\prime }B=\angle EDC=\angle DAB=\angle BA{F}^{\prime }+\angle EB{F}^{\prime }$ and we are done.