15. On the plane, there are a (generally non-convex) quadrilateral and a pentagon, and no vertex of one lies on the side of the other. What is the maximum possible number of intersection points of their sides?
15. Note that no line can intersect the sides of a polygon in an odd number of points, since otherwise, moving along this line in a certain direction - after entering the polygon for the last time, we would not be able to leave its boundaries. Therefore, each side of the quadrilateral can have no more than four intersection points with the contour of the pentagon. Hence, the total number of intersection points between the sides of the quadrilateral and the pentagon cannot exceed $4 \times 4=16$. The fact that this number can equal 16 follows from Fig. 52.