If 0 < c < 1, Then as we keep allocating more bonus to one of the professors, the other professor's utility keeps on decreasing. No matter how much of money we have left to allot, whenever we give some more money to one of the professors, the utility of the other professor has to decrease. However in this case since c is between 0 and 1, if we split the money equally between both professors, they will still have positive utility. So in this case, it is possible to have both professors with positive utility after allocating all the money