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

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