Putnam Seminar

Each year on the first Saturday of December, several thousand students from all over the United States and Canada participate in the Putnam Mathematical Competition for undergraduate students. At O.U., we usually hold a weekly seminar in the fall semester for students who are interested in practicing solving the type of problems which appear in the competition. This semester (Fall, 2006) we are meeting every Wednesday at 4 pm in room 1025 of the Physical Sciences Center. Everyone interested in problem-solving is welcome, even if they don't plan on entering the competition in December. Below are links to some of the problems considered in the past couple of years. Some of the problems on these handouts get solved during the seminars, some get partly solved, and some remain untouched. Even when we have solved one, we might not have used the best or most elegant method. So if you find a solution to any of these problems, including the ones from previous years, it will be worth presenting it in the seminar even if another solution has already been given earlier.

You can find the problems from all Putnam competitions back to 1985 online at http://www.unl.edu/amc/a-activities/a7-problems/putnamindex.shtml. (You can also find solutions there, which you may be sometimes tempted to look at when you are stuck on a problem. Resist the temptation!)

Here are some interesting interviews with students who did well on the 2004 Putnam exam.

Fall 2007

  • On Oct. 17, we again looked at a problem sheet from the Stanford Putnam seminar, this one on algebraic tools and techniques. We solved problem W3, and spent a fair amount of time on problem W2, without solving it however.
  • On Oct. 3, we looked at some of the "warm-up" problems on this problem sheet on mathematical induction and the pigeonhole principle, taken from the website of the Putnam seminar held at Stanford last year. We did problems W2 and W4.

    Fall 2006

  • Here are the handouts from Sept. 13 and Sept. 20. So far we've done problem 1 from page 1 and problems 1 and 2 from page 2 of the Sept. 13 handout; and problems 1 and 3 from the Sept. 20 handout. These problem sets were both compiled by four-time Putnam fellow Reid Barton and are taken from this website.

    Fall 2005

    The Putnam competition will be held this year on Saturday, December 3. At O.U., the test will be given in Physical Sciences Room 1105. The first session begins at 9:00 am; anyone who wants to take it should plan to arrive 5 to 10 minutes before 9 o'clock.

  • Here is the Nov. 9 handout. We solved problems 1 and 2 from this handout.
  • Oct. 12: we did part (i) and most of part (ii) of the following problem (taken from "Mathematical Miniatures"): Let ABC be any triangle, and X any point on or in the interior of ABC. Show that the sum of lengths AX + BX + CX is (i) greater than the semiperimeter of the triangle, (ii) less than the perimeter of the triangle, (iii) less than or equal to the sum of the two longest sides of the triangle.
  • Oct. 5: we discussed q-analogues, and solved a problem chosen from "Berkeley Problems in Mathematics" (the problem handout will be here later).
  • Sept. 28: many Putnam problems involve the use of classical inequalities.
  • On Sept. 21 we talked a bit about Minkowski's theorem (no handout)
  • On Sept. 14 a couple of students presented their solutions to problems A4 and B2 from last year's competition.
  • The Sept. 7 handout was on symmetric functions.
  • On Aug. 31 we went over the solution of problems A3 and B4 from last year's competition. We also looked at the first problem on this handout.

    Fall 2004

    Here are the handouts and problem sets from last year's Putnam Seminar. Especially recommended are the handouts on Pell's equation, Diophantine equations, progressions and sums, and rational and irrational numbers, which were all written by Lucy Lifschitz.

  • The Nov. 17 handout was on Pell's equation and related problems.
  • In the Nov. 10 handout, despite appearances, problem 2 is perhaps more closely related to problem 3 than it is to problem 1!
  • The Oct. 27 handout is on Diophantine equations.
  • The Oct. 20 handout concerns cellular automata, like the so-called ``Game of Life''. You'll want to use the "Rotate Counterclockwise" button in your .pdf viewer to read it. We solved the first problem on this handout easily, but I don't think anyone in the seminar has come up with a solution to the second one yet. (Wonder where A. Gomilko is today?)
  • On Oct. 13, besides solving the Sept. 29 problem, we looked at the following one (from the 1986 International Math Olympiad):

    To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively, and y < 0, then the following operation is allowed: x, y, z are replaced by x + y, -y, z + y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

    Besides the question of whether the procedure necessarily comes to an end, one could ask further questions: (1) Does the answer change if one replaces the pentagon by a polygon with a different number of vertices? (2) What final configuration(s) are possible? (3) Is there a fastest way to arrive at the final configuration? (4) How many different ways are there to arrive at the final configuration? (The answers to questions (1), (2), and (3) are known, but (4) is to this day an open question.)

  • The Oct. 6 handout is titled "Progressions and Sums".
  • On Sept. 29 we spent an hour looking at (but not yet solving) the following problem, which is problem B-6 from the 1993 Putnam exam:

    Three nonnegative integers are given. We may choose two of them, say x and y, and if x is less than or equal to y, replace them by 2x and y-x. Prove that, after a finite number of such operations, it is possible to obtain 0.

    In "Mathematical Miniatures" this problem is attributed to the Russian algebraist Alexei Shirshov. According to Savchev and Andreescu, "The late Shirshov was a nontraditional mathematician with a highly nontraditional career. What else can you say about a man who graduated from the university with a degree in Russian language and literature, then taught them both in some remote rural area; then survived the entire Second World War fighting in the trenches from the very first to the very last day, succeeding in getting hooked by mathematics in between; then got back to the university at the age of almost thirty, graduated again with a degree in mathematics, and finally became one of the top mathematicians in his field?"

  • Here is the Sept. 22 handout.
  • On Sept. 15 we went over some problems from this handout on Helly's theorem.
  • Here is a handout on rational and irrational numbers from the Sept. 8 seminar.
  • On Sept. 1, we went over the solutions of problems A6 and B1 from the 2003 Putnam competition.

    Fall 2003

    Here are problem handouts from the Fall 2003 seminar. You can look at the handouts from Sept. 10, Sept. 17, Sept. 24, Oct. 1, Oct. 15, Oct. 22, Nov. 5 and Nov. 12.

    Other stuff

    Here is a list of a few other problems we've looked at in past years. I think these are taken either from old Putnam competitions, or from the "Hungarian Problem Books" published by the Mathematical Association of America.