A variational problem with inequality-type constraints satisfying a differential equation with delay is discussed. This constrained optimization problem is reduced (by means of exact penalization) to an unconstrained optimization problem. The problem under consideration is characterized by the presence of strict inequalities and complicated differential relations. Necessary optimality conditions are derived making use of Nonsmooth Analysis techniques. For practical considerations it is important to note that the optimality conditions are quite constructive and can be used for developing numerical algorithms.