About 2500 mathematics majors from colleges in the United States and Canada spend six hours on the first Saturday in December wrestling with 12 hard problems in the William Lowell Putnam Mathematical Competition (http://math.scu.edu/putnam/). A measure of their difficulty is that the best score in the examination is usually about 100/120. Half the students score less than 10 (meaning half get less than one problem right).
Each year since I came to UMass Boston in 1972 I coach (if that's the right word) undergraduates who choose to inflict this frustration on themselves. They know that just taking the Putnam exam puts them in the top 2500 math majors in the country, even if they get no problems right. I always find at least one volunteer and have occasionally found four. Once our three person team managed a significant national ranking because one student got four problems right.
During the Fall semester we meet weekly to go over old exams, learn general problem solving techiques and some specific Putnam problem tips and explore interesting mathematics we encounter along the way. Sometimes we read in George Polya's Induction and Analogy in Mathematics.
On the day of the exam we convene at UMass along with my old friend Paul Mason, who became my friend in 1972 when as a Freshman he was the first UMass student to take the Putnam exam. (When he took it for the fourth time several years later he ranked 28th in the country!) No one from UMass has done nearly as well since, but the students always come away with a sense of time well spent.