CGMO 2002 Problem 8

Assume that $A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $l$ taken arbitrarily on the plane, assume that projections of $A_1, A_2, \ldots, A_8$ on the line are $P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $P_{i_1}, P_{i_2}, \ldots, P_{i_8}$ according to the direction of line $l.$ Thus we get one permutation for $1, 2, \ldots, 8,$ namely, $i_1, i_2, \ldots, i_8.$ In the figure, this permutation is $2, 1, 8, 3, 7, 4, 6, 5.$ Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $N_8 = N(A_1, A_2, \ldots, A_8).$ Find the maximal value of $N_8.$ Attachments: