In this paper we investigate normal and nondegenerate forms of the maximum principle for optimal control problems with state con-straints. We propose new constraint qualifications guaranteeing non-degeneracy and normality that have to be checked on smaller sets of points of an optimal trajectory, than those in known sufficient con-ditions. In fact, the constraint qualifications proposed impose the existence of an inward pointing velocity just on the instants of time for which the optimal trajectory has an outward pointing velocity.