If the goal is to show them "interesting" mathematics, there are many sources available, depending on what you think falls into the class of "interesting" as well as appropriate. Just to start, Martin Gardner's books (and if you have a nice citation index, some books which cite Gardner's books) should keep you busy for a while. Using just his books alone, you can write a short description of 10 problems, each with three or four paragaphs of setup, result, and application, and refer to one of Gardner's books for an expansion on the topic. There are other authors you could use in place of Gardner.
If the goal is to stimulate their interest in mathematics through showing them problems, I humbly suggest the following: comb through open problem lists as well as indices of books like Gardner's, pick 100 or so problems from each, and start writing 10 lists of 20 problems, half which are solved/understood, half which aren't, and all of which are accessible. On MathOverflow alone, you see at least one such problem every two or three days. Then for the next ten semesters, you have a list of twenty problems from various fields to capture the students' imaginations. Bonus points to you if you can get them to research and cite the literature on what work has been done on the problems. Keep handy a record of your sources to compare to those found by your students.
If you are asking for specific problems from me, your question needs to say so, in which case I can tell you some problems I have worked on that are accessible that could do with some more elbow and brain grease applied to them.
Gerhard "Ask Me About System Design" Paseman, 2011.03.12
