You should have a full-time house parent, and a senior adult on hand at all times. The academic programme. First of all, there should be lots of structured social time for the students to meet each other. It's difficult to come up with really good problems, but there are now incredible resources where you can find problems. If you have any questions at all, just ask.
Homework is a bad idea.
But a fun challenge problem for them to think about over the evening is not. It's hard to give a good talk to this age group, especially if you're used to an undergraduate, didactic environment. The best talks tend to be interactive, without much jargon. There are lots of fun things to talk about in geometry, combinatorics, logic, and number theory. There's no need to be too ambitious.