Is Hamlet mad?
Yes; although Hamlet claims his madness to be an act, he is delusional to believe that this is true, as all of his actions are driven by a force that has overtaken his capacity for deliberate action.
No; although Hamlet becomes increasingly violent and unreasonable, his mental faculties are still intact, but tragedy has stretched him to new lengths of desperation.
Yes; although Hamlet feigns madness to amuse himself with Polonius, his obsession with revenge is a form of madness, because he increasingly loses sight of his own morality as the play continues.
No; although Hamlet is a complicated character and often seems on the brink of mental deterioration, he never manifests true madness because he retains the ability to interact with people.
No and yes; although tragedy has stretched Hamlet to new lengths of desperation, his preoccupation with revenge manifests a particular form of derangement and madness.