USAJMO 2015 Question 5

Let $ABCD$ be a cyclic quadrilateral. Prove that there is $X\in BD$ with $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there is $Y\in AC$ with $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$. Here $X,Y$ are on the respective segments.

Proposed by Sungyoon Kim (credited in the supplied notes).