In Hamlet, what are the three problems that Claudius faces at the beginning of act 4?
Claudius is obviously in a very tricky position at the beginning of Act IV. He has just tried to find out the cause of Hamlet's madness, thinking that it was Ophelia. He has also been made to sit through a re-enactment of the way that he killed his brother which shocked him so much that he got up and left the play. Now he walks in to the Queen's chamber to find out that Hamlet has just exited, having killed Polonius. His one major problem therefore is how to get rid of Hamlet - both to save himself and to stop him killing any other people in his "madness". However, he has to go carefully because he is dealing with the son of his new wife, and therefore cannot dispose of Hamlet openly in the way he would like. Note what he resolves to do:
The sun no sooner shall the mountains touch,
But we will ship him hence, and this vile deed,
We must with all our majesty and skill
Both countenance, and excuse.
Of course, what he is really planning is the death of Hamlet and the end of his inconvenient presence as it is clear that Claudius knows that now Hamlet knows the real facts of the old King's death.