Following is a problem from the 2004 exam. You can find more problems at
http://www.unl.edu/amc/a-activities/a7-problems/putnam/
       Basketball star Shanille O'Keal's team statistician keeps track of the number, S(N), of successful free throws she has made in her first N attempts of the season. Early in the season, S(N) was less than 80% of N, but by the end of the season, S(N) was more than 80% of N. Was there necessarily a moment in between when S(N) was exactly 80% of N?
Answer:
        Yes. Suppose otherwise. Then there would be an N such that S(N) < 80% and S(N + 1) > 80%; that is, O'Keal's free throw percentage is under 80% at some point, and after one subsequent free throw (necessarily made), her percentage is over 80%. If she makes m of her first N free throws, then m/N < 4/5 and (m + 1)/(N + 1) > 4/5. This means that 5m < 4n <
5m + 1, which is impossible since then 4n is an integer between the consecutive integers 5m and 5m + 1.
Remark:
      This same argument works for any fraction of the form (n - 1)/n for some integer n > 1, but not for any other real number between 0 and 1.

       The William Lowell Putnam Mathematical Competition is the premier competition for undergraduate mathematics students. It is sponsored by the Mathematical Association of America and held the first Saturday in December. Putnam problems test originality as well as technical competence; they are challenging, though of varying difficulty. Students have 6 hours to work on 12 questions.
       About 3500 students from colleges and universities in Canada and the United States participated in the competition this year including six UO students. Nathan Collins scored 30. This earned him a rank of 327.5 and a 91.2 percentile. Travis Willse scored a 20 for a rank of 596.5 and a spot in the 84^th percentile. Josiah Thornton scored a 12 with a rank of 825.5 and a 88^th percentile. Aaron Henner's percentile was 75 with a score of 11. David Jordan scored a 9, which put him in the 65^th. Tom Denton also took the exam and scored a 0. The median score was about 0. The U of O team finished in 68^th place.
       Arkady Vaintrob held a seminar on Putnam problems in the fall and continued it throughout the year. Next fall the Putnam seminar will probably be taught by Arkady Vaintrob again. If you are interested, stop in to the seminar next fall and see what the problems look like.