China Mathematical Olympiad 1987 problem2
Source: China Mathematical Olympiad 1987 problem2
January 6, 2014
geometryrhombuscombinatorics unsolvedcombinatorics
Problem Statement
We are given an equilateral triangle ABC with the length of its side equal to . There are points on each side of the triangle that equally divide the side into segments. We draw all possible lines that pass through any two of all those points such that they are parallel to one of three sides of triangle . All such lines divide triangle 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 are assigned to 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 the sum of the numbers of all nodes, find .