CGMO 2002

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.

(1) When $n=3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion.

(2) Prove that $n$ is an odd number.

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.

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 $$\frac{AH}{HF} = \frac{AC}{CD}.$$

There are $n \geq 2$ permutations $P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $\{1,\ldots,n\}.$ Prove that

$$\sum^{n-1}_{i=1} \frac{1}{P_i + P_{i+1}} > \frac{n-1}{n+2}.$$

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 $ABC.$

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: