pairwise non-collinear vectors of the plane
Source: Russian Olympiad 2004, problem 11.6
May 4, 2004
vectorgeometry unsolvedgeometry
Problem Statement
Prove that there is no finite set which contains more than with pairwise non-collinear vectors of the plane, and to which the following two characteristics apply:
1) for arbitrary vectors from this set there are always further N\minus{}1 vectors from this set so that the sum of these is 2N\minus{}1 vectors is equal to the zero-vector;
2) for arbitrary vectors from this set there are always further vectors from this set so that the sum of these is vectors is equal to the zero-vector.