Each year, in early December, the Mathematical Association of America sponsors a problem-solving competition for undergraduates known as the William Lowell Putnam Competition.
Over 
5000 students from 500 schools across the US and Canada take the Putnam exam, which consists of 12 extremely challenging problems. A single completely correct solution is already a significant achievement, likely to result in a score well above the median. (The participants are self-selected from among the best in the continent.) The cash prizes and fame that the top winners receive are enviable, but the real thrill of taking the Putnam exam, or ofattending the practice sessions, is the pleasure of tackling and solving interesting and unusual problems. The emphasis is on ingenuity: first-year students can do as well as seniors.
Practice problem sessions for the Putnam competition will be held throughout the Fall semester. All are invited to attend the practice sessions. It is a great opportunity to tangle with challenging problems and hone your problem solving skills.
A sample problem: Players 1, 2, 3, ... , n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.
If you are intersted in participating in the Putnam exam, be on the lookout for an email from the departmental Putnam coordinator, or contact the department chair to ask who is organizing the exam.