IMO 1970 LL FRA26

Origin: FRA

Problem

Consider a finite set of vectors in space {a1, a2, . . . , an} and the set E of all vectors of the form x = \lambda1a1 +\lambda2a2 +\cdot \cdot \cdot+\lambdanan, where \lambdai are nonnegative numbers. Let F be the set consisting of all the vectors in E and vectors parallel to a given plane P. Prove that there exists a set of vectors {b1, b2, . . . , bp} such that F is the set of all vectors y of the form y = µ1b1 + µ2b2 + \cdot \cdot \cdot + µpbp, where the µj are nonnegative.