China Girls' Mathematical Olympiad

(1) For two parallel and directed lines with the same direction, the order of projections of $A_1,A_2,\dots ,A_8$ must be the same. So, we need only to discuss all directed lines passing through a fixed point $O$.

(2) If a directed line taken is perpendicular to a line joining two given points, then the projections of these two points must coincide, and a corresponding permutation will not be produced. When the directed lines taken are not perpendicular to any line joining two given points, any two projections of $A_1,A_2,\dots ,A_8$ must not coincide. Hence there is a corresponding permutation.

(3) Suppose that the number of lines through point $O$ and perpendicular to a line joining two given points is $k$. Then $k\le C_8^2=28$. Then there arise $2k$ directed lines placed anticlockwise. Assume that they are in order of $l_1,l_2,\dots ,l_{2k}$. For an arbitrary directed line $l$ (different to $l_1,\dots ,l_{2k}$), there must be two consecutive directed lines $l_j$ and $l_{j+1}$ such that $l_j,l,l_{j+1}$ are placed anticlockwise. It is obvious that for given $j$, the corresponding permutations obtained from such $l$ must be the same.

(4) For any two directed lines $l$ and $l^{\prime }$ different from $l_1,\dots ,l_{2k}$, if we cannot find $j$ such that both $l_j,l,l_{j+1}$ and $l_j,l^{\prime },l_{j+1}$ satisfy (3). Then there must be $j$ so that $l^{\prime },l_j,l$ are placed anticlockwise. Assume that $l_j$ is perpendicular to the line joining $A_{j_1}$ and $A_{j_2}$, it is obvious that the orders of the projections of points $A_{j_1}$ and $A_{j_2}$ on the directed lines $l$ and $l^{\prime }$ must be different, so the corresponding permutations must also be different.

(5) It follows from (3) and (4) that the number of different permutations is $2k$. Note that $k=C_8^2$ is obtainable, so $N_8=56$.