The William Lowell Putnam Mathematical Competition 2005

The William Lowell Putnam Mathematical Competition will take place on Saturday, December 3, 2005. The Putnam will take place in the Vidalakis room at the Schwab Residential Center, on Serra Street right by the corner with Campus Drive East. Here is a map. We'll meet at 7:40 am. The first session will be 8-11 am, and the second session will be 1-4 pm. We'll meet again for the second session at 12:50.

It is a challenging opportunity for you to test your mathematical mettle. In the 2004 competition, 3733 individuals participated, representing 515 colleges and universities across Canada and the U.S. The Stanford team placed eighth, and many individuals did very well. (I have a text file of results that I can forward to anyone interested.) The measure I look to is the number of competitors on the list of top performers sent out with the results; we had 31, making us one of the top three.

The competition emphasizes ingenuity rather than knowledge, so freshmen are not at much of a disadvantage compared to seniors. Interest in or experience with problem solving is a plus. Not just math majors have done well; many recent winners have come from nearby disciplines, including physics, computer science, and engineering.

Completely solving even one of the twelve problems is a significant achievement, and in almost all years would place you well above the median. (Keep in mind that the particpants are self-selected from among the best in the continent.)

Links:

• The introductory poster.
• The introductory meeting will take place Monday Oct. 3, 5:15-5:45 pm, in 380-383N (in the corner of the third floor of the math department). Here is the introductory handout.
• Meetings: We will have informal dinner-time problem solving practice sessions this quarter, on Mondays from 5:30 to 7. (Tentative schedule: beforehand 5:15-5:30 starting week 2, people can drop by to discuss "left-overs" from the previous week. 5:30-5:45: short discussion of new technique. 5:45-6:30: work on problems. 6:30-7: eat pizza, watch "volunteers" present solutions and explain how they thought about the problems.) Some smart faculty and graduate students always turn up with wise words, including Ben Brubaker, Brian Munson, Alok Aggarwal, Henry Segerman, Paul-Oliver Dehaye, and Mark Lucianovic. Practice problems will include some on the topics discussed, and some others. Handouts will appear here.
  • Monday, October 10: general problem-solving strategy, induction, and the pigeonhole principle. (We start with this every year.) Ravi Vakil is away, and Ben Brubaker will be running this session, with help from Paul-Oliver Dehaye and Mark Lucianovic.
    Website of the week: Benford's Law.
    Problems: ps pdf (typos in 3 and 7 corrected Oct. 18)
  • Monday, October 17: number theory.
    Website of the week: animated proofs at the Art of Problem Solving website. To see them start again, refresh your browser, or just wait.
    Problems: ps pdf
  • Monday, October 24: combinatorics.
    Problem of the week: I think this is originally from St. Petersburg. Let a(n) be the number of digits (in base 10) of 2^n, and let b(n) be the number of digits of 5^n. What is a(2005)+b(2005)? (This is strongly reminiscent of the problem from last week about powers of ten written in base 2 and base 5. If anyone can make that connection precise, I'd be happy to hear about it!)
    Problems: ps pdf
  • Monday, October 31: functional equations, by Mark Lucianovic.
    Rather than a problem-of-the-week, or a link-of-the-week, I have a "lecture-of-the-week": "Number theory: Partitions and the legacy of Dyson and Ramanujan", by Prof. Ken Ono from the University of Wisconsin. You can see his abstract on posters around the department, which you can also see here.
    Problems: ps pdf
  • Monday, November 7: inequalities. (Thanks to Mark Lucianovic and Soren Galatius for many interesting problems!)
    Problems: ps pdf
  • Monday, November 14: calculus.
    Problems: ps pdf
  • Monday, November 28: old Putnam problems.
    Problems: ps pdf
  For those of you who would like to discuss the problems with each other during the week, Paul-Oliver Dehaye has setting up a discussion group on Coursework. Click here, and sign up as follows. Go to your "My courses page". Click "Add course". Click item Z in the index page (!). Look for the section marked "RESOURCE: Z - Resource Sites". Add yourself to Resource 200. Ignore Coursework's comment that you need to register through Axess. (Thanks to Aman Kumar for figuring this out!) Unfortunately, only Stanford affiliates will be able to access this page. This isn't because we prefer it this way; it's just the way Coursework works. If you have any trouble signing up, let me know, and we'll try to figure it out!
• There will also be a Masterclass for a few experts (everyone is welcome), on Mondays 7:15-8:15 pm.
  • Monday, October 17. Problems: ps, pdf.
  • Monday, October 24. Problems: ps, pdf.
  • Monday, October 31. Problems: ps, pdf.
  • Monday, November 7. Problems: ps, pdf.
  • Monday, November 14. Problems: ps, pdf.
  • Monday, November 28. We're ending with an all-star session of seven problems! Problems: ps, pdf.
• The official Putnam website.
• Recommended reading:
  • I really like Loren Larson's "Problem Solving Through Problems" --- definitely worth owning. It's great preparation for the Putnam.
  • It is also great help (more than you would think) trying many old problems. I'd especially recommend the collection of recent Putnams "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary", by Kiran Kedlaya, Bjorn Poonen, and myself.
  • There are more good problems in the previous Putnam book, "The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965-1984", by Alexanderson, Klosinski, and Larson (look especially at those in the 1980's).
  All of these books are in the math library (2-day reserve, under the course "VAKIL-SEM"), in Building 380 (fourth floor). Also, on the web you can find
  • a compilation of old Putnam problems and solutions on the web, collected by Kiran Kedlaya;
  • to find solutions to older Putnams, you can read the official solutions in the American Mathematical Monthly, which you can see electronically (from any Stanford computer) here. At that page, 'Search this journal' for "William Lowell Putnam" in the title.
  • Some more links are available through this good UCSD site, by Patrick Fitzsimmons.
• Last year's Stanford Putnam website.

Want more? Additional suggestions for others? Please let me know.