CMO 1987

Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$ .

We are given an equilateral triangle ABC with the length of its side equal to $1$ . There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$ . All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called nodes. We assign a real number for each node such that the following conditions are satisfied: (I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively; (II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal.

1. Find the minimum distance between the node with the maximal number to the node with the minimal number;
2. Denote by $S$ the sum of the numbers of all nodes, find $S$ .

Some players participate in a competition. Suppose that each player plays one game against every other player and there is no draw game in the competition. Player $A$ is regarded as an excellent player if the following condition is satisfied: for any other player $B$ , either $A$ beats $B$ or there exists another player $C$ such that $C$ beats $B$ and $A$ beats $C$ . It is known that there is only one excellent player in the end, prove that this player beats all other players.

Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$ . Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied: i) the five points are covered by the three equilateral triangles; ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$ ; iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$ .

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions: i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$ ; ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$ . Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$ . Find, with proof, the maximum value of $3m+4n$ .