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 C.