China Girls Math Olympiad 2002

1. Find all positive integers $ n$ such $ 20n+2$ can divide $ 2003n + 2002.$
2. There are $ 3n$ $(n \in \mathbb{Z}^+)$ girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $ 3n$ students had just one time to be on duty on the same day.
   a) When $ n=3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion.
   b) Prove that $ n$ is an odd number.
3. Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality \[ k(ab + bc + ca) > 5(a^2 + b^2 + c^2),\] there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.
4. Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that \[ \frac{AH}{HF} = \frac{AC}{CD}.\]
5. There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n-1}_{i=1} \frac{1}{P_i + P_{i+1}} > \frac{n-1}{n+2}.\]
6. Find all pairs of positive integers $ (x,y)$ such that \[ x^y = y^{x - y}. \]
7. An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
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.$