To do this problem a solid theoretical background is useful.

First of all, the strict inequalities are just there to fool you; 
since we work with integers, these are easily converted to normal
inequalities fitting to the following discussion.

So we are left with the problem:

is Ax <= b infeasible ?

It is easy to see

   Ax <= b 

is infeasible iff
             T
   max (0,-1) (x,y) < 0
 
   [A -I] (x,y) <= b
   x UIS, y>=0

Constructing the dual problem we get

    Ax <= b 

is infeasible iff
        T
   min b z < 0
      T
   | A | z  = 0
   |-I |   <= -1
   z>=0

iff
       T
  min b z < 0
   T
  A z = 0
  z>=0


Now it is time to use the special structure of the given equations.
Consider a graph with vertices {a1,...a(n+1)} with an edge ai->a(j+1)
representing the sum from i to j if i<j or the negative sum from j to i
if i>j.

Now the vector z from the above correspong to a circulation in this 
network, and we have reduced the problem to determining if the network
has a circulation of value < 0.

Thinking about the cycle removal algorithm will be helpful, note that we
immediately have a feasible solution in the 0-flow.

questions can be directed to arnsfelt@daimi.au.dk