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I teach at a small liberal-arts college and advise our Math Club. In past years, I have talked a few students into taking the Putnam Exam. I've found that I pretty much have to "coerce" them into taking it at all, let alone prepare for it adequately. In the month or so leading up to the exam, I hold evening sessions after classes where I go over past "easy" problems and remind them to really focus on problems 1 and 2 in each session since those are more doable. This proves helpful for some, but other students just don't bother to attend.

I'd also like to encourage our students to work on problems posed in journals, like the Pi Mu Epsilon journal, as well as the MAA's Mathematics Magazine and College Math Journal. Working in groups on these problems can really foster the students' "ownership" of their mathematical expertise. This can even lead to a mention in a journal, which can boost their confidence and make a nice addition to their CV. Even Project Euler problems would be great, as well.

In short, I want to encourage our math majors to work on problems outside of their course work, for the reasons mentioned above as well as just to strengthen their education. What are some effective methods for fostering students to work on challenging problems, like those mentioned above, in groups outside of their standard course work?

I'm interested in hearing about your personal experiences with this, and would love to hear about methods that have/haven't worked. I'm also interested in any research about intrinsic student motivation and how to encourage that, because I imagine that's germane here. (If students want to solve problems outside of their courses, then they'll make time for it.)