Both Hamlet and Ophelia are described as being ‘mad.’ Compare and contrast the madness in each character. Are both characters truly ‘mad’? Explain your answer. How does Shakespeare use this theme to intensify the idea that something is rotten in the state of Denmark? Explain.

In Hamlet, Hamlet decides to pretend to be mad in order to gather information about what is going on in the Danish court, especially about Claudius's guilt or innocence. As Hamlet says to Horatio in act 1, scene 5, Horatio shouldn't react, no matter:

How strange or odd soe'er I bear myself

Hamlet goes on to say he will be acting as if he is crazy.
Ophelia, on the other hand, genuinely does go mad. While Hamlet contemplates suicide and sometimes wishes to be dead, he never acts on the impulse. But just as Ophelia's madness is real, so is her suicidal wish: she does actually take her life. To the extent that she does really become insane and really kills herself, she acts a doppelgänger or twin to Hamlet, truly enacting what he pretends to be or want.
To the extent that Hamlet does have a fit of near madness, it is during the frenzied scene when he visits his mother to impress upon her the wrong she has done in marrying Claudius. In this scene, his borderline madness expresses itself as violence: his mother cries out in fear that he will hurt or kill her, and Hamlet does murder Polonius, mistaking him for Claudius. Hamlet fakes madness to decide whether he should engage in the violent act of killing Claudius. Ophelia's madness, however, expresses itself gently: she tears up flowers while reciting sayings about them.
That Ophelia, by and large a sweet, innocent character, becomes so impacted by the atmosphere of the court and by Hamlet's behavior that she commits suicide does show that something is wrong or rotten in Denmark. Her young life should not have been wasted. That Hamlet has to pretend to be mad to find out what is going on also shows that the court has become corrupt. In a saner environment, he wouldn't have had to act as if he were crazy.