Question 1
Let $A$, $B$, $C$, and $D$ be distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. $O$ is an arbitrary point on the line $XY$ but not on $AD$. $CO$ intersects the circle with diameter $AC$ again at $M$, and $BO$ intersects the other circle again at $N$. Prove that the lines $AM$, $DN$, and $XY$ are concurrent.