IMO 1977 LL USS47

Origin: USS

Problem

A square ABCD is given. A line passing through A intersects CD at Q. Draw a line parallel to AQ that intersects the boundary of the square at points M and N such that the area of the quadrilateral AMNQ is maximal.

Solution

M lies on AB and N lies on BC. If CQ \leq2CD/3, then BM = CQ/2. If CQ > 2CD/3, then N coincides with C.