KöMaL 2005 B3831

There is a square of side 2005 units long drawn on squared paper. The sides of the square are lattice lines. Draw a closed non self-crossing polygon inside the square, such that each side of the polygon is drawn along a lattice line, and it passes through every lattice point inside or on the boundary of the square exactly once. Show that the area bounded by the polygon is more than half the area of the square.