47. (USS 1) A square $A B C D$ is given. A line passing through $A$ intersects $C D$ at $Q$. Draw a line parallel to $A Q$ that intersects the boundary of the square at points $M$ and $N$ such that the area of the quadrilateral $A M N Q$ is maximal.
47. $M$ lies on $A B$ and $N$ lies on $B C$. If $C Q \leq 2 C D / 3$, then $B M=C Q / 2$. If $C Q>2 C D / 3$, then $N$ coincides with $\mathrm{C}$.