Math Problem of the Month

PROBLEM OF THE MONTH

April 2002

Let P be a regular hexagon. Show that it is impossible to inscribe a triangle in P with one side parallel to one of the sides of the hexagon, and whose area is more than 3/8 of the area of the hexagon. Are there any inscribed triangles whose area is exactly 3/8 of the area of the hexagon? What happens if we do not assume anymore that P is regular ?