Prove-disprove whether the claim is feasible or not

Question

Problem:

(a) Let  A€R ^mxn . Prove that one of the following systems has a solution but not both:

                     Ax = 0                                  A^Ty≤0

                     x≥0                                      A^Ty≠0

(b) Prove or disprove the following claim:

     Assume that both the linear program

                      min  c^Tx

                      s.t.  Ax=b

                            x ≥0               

and its dual

              min  b^Ty

              s.t    A^Ty≤c

are feasible. Then at least one of them has an unbounded feasible region.