On the maximal number of solutions of a problem in linear inequalities

The problem y=Ax+c, x≧0, y≧0, (x, y)=0 is considered, where the square real matrix A and the real vector c are the data and a solution is a pair of vectors x, y. Under certain conditions on the matrix A there exists a solution for every vector c, but it cannot be unique for every c. We prove that under these conditions the maximal number of solutions is 2 ⁿ - 1.