The Putnam Mathematical Competition is a math contest held every year on the first Saturday of December for undergraduate students in the U.S. and Canada. The problems range across the undergraduate mathematics curriculum, but most do not require specialized knowledge of mathematics beyond calculus.

Over the years the exam has built up a good reputation for containing nice, interesting problems - the kind you find it hard to stop thinking about, whether you've managed to solve it or not. The best problems reveal attractive and intricate bits of mathematics lurking in seemingly simple situations.

The best way to learn to solve Putnam problems is to start trying to solve some. Although no two Putnam problems are alike, as you start to see more of them, you gradually build up a store of intuition and ideas to draw upon in solving them. Even if you don't succeed in getting very far on many problems, you will find that the mere process of grappling with them strengthens your ability. For this reason, we hold a problem seminar at OU each fall to give students the chance to attempt to solve Putnam problems.

Everybody is welcome at the seminar, whether they're thinking of participating in the competition or not. Working on difficult, but interesting, mathematical problems is a good way to improve your mathematical abilities and have fun at the same time. You'll not only be better at mathematics proper, you'll be able to work better in any field of science or engineering that uses mathematics!

Below is information about this year's seminar, as well as links to some of the problems considered in this year and in past 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.

This fall the seminar is being run by Profs. Alejandro Chávez-Domínguez and Roi Docampo, and meets on Thursdays in Physical Science Center room 1025 from 5 to 6:30 (come and go as you please).

Alejandro and Roi have set up a very nice web page about the seminar.

They have also arranged for OU to participate in the Virginia Regional Math Competition, a kind of "mini-Putnam", on Saturday Oct. 22. For information on how to register, follow the link above to their webpage.

The problem-solving seminar will be held this fall in Physical Sciences 1025 (the 10th floor classroom) on Wednesdays from 5:00 to 6:00 pm. Murad Özaydın and I are taking turns bringing problems each week for students to work on.

The seminar will not meet on Nov. 25. The last seminar of the semester will be on Dec. 2. The Putnam exam will be held this year on Saturday, Dec. 5 at 9:00 am in the Physical Sciences 8th floor classroom. Contestants should show up about 10 minutes before the exam.

Nov. 18: In addition to the problems from the 2004 Putnam exam from last week, we also looked at the 2007 Putnam exam. We also handed out a copy of this sheet of advice from a Stanford Putnam contestant. (You can also find this and other interesting material by following the links below under "Previous Years".)

Nov. 11: We solved problem 2002-A2 from the sheet of "easy" Putnam problems from Oct. 7, and problems 3 and 4 on the handout of problems from Larson's book from last week. We also got a new handout of problems from Larson's book, and decided to take a look at all the problems from the 2004 Putnam exam, to see what some of the harder Putnam problems look like. We have already solved 2004-A3 (see Sept. 30). We started thinking about 2004-A5 a bit, but haven't got very far yet. Problems 2004-A1 and 2004-B2 are supposedly pretty easy.

Oct. 28: We solved problem 2007-A1 from the Oct. 7 handout of "easy" Putnam problems. I also handed out a few fun and relatively easy problems from Loren C. Larson's book "Problem-solving through problems", and a new batch of problems from the NU selection test.

Oct. 21: We solved all the problems on the "Putnam selection test" handout from last week, except problem A6.

Oct. 14: We knocked off a bunch of the "easy" Putnam problems on the sheet from last week. We've done problems 2012-A1, 2012-B1, 2011-A1, 2009-B1, 2008-A1, and 2008-A2. We also talked about a couple of problems on Northwestern University's ``Putnam selection and training test" from 2015, but we haven't completely solved any of them yet.

Oct. 7: We handed out this sheet of "easy" Putnam problems, taken from the nice Putnam problem-solving website maintained at Northwestern University. We've done the first four problems on this sheet so far in our seminar.

Sept. 30: We spent most of the time on the first problem on this sheet, which turns out to be problem A3 from the 2004 Putnam. We made some significant progress, but haven't solved the problem yet. In the last few minutes, we solved the second problem on the sheet. Or at least we sketched the solution, though it would still be a good idea for someone to write it out carefully.

On Sept. 23, we looked at the problems on this sheet. We solved (2014 B1), (1986 A1), and (1986 A2). Actually we gave two different solutions to (1986 A2). We haven't solved (1986 A3) yet.

Here are the problems from the first meeting, Sept. 16. We solved (A1 2014). For (A3 2014), we conjectured an explicit formula for a_k and verified the formula by induction; next we were thinking of trying substituting this formula into the infinite product and seeing if we can find what number the partial products approach. We're going to think about (B4 1995) for next week.

In the fourth meeting, on Nov. 3, students presented the solution of problems 14 and 15 from last week's set. We also looked at the problem set for Week 4, and solved problem 19.

Here's the problem set for Week 3. We presented the solution to problem 9 and 12 from last week's set, and we worked for a while on number 14 from this week's set, but haven't got it yet.

At the second meeting on Oct. 20, we solved problems 4 and 6 from the sheet from Week 1, and looked at a couple of the problems from the problem set for Week 2.

The first meeting was Monday, Oct. 13. We solved the first three problems from this problem set. (Note: the sheet I handed out at the seminar contained a typo on problem 5 which has been corrected here.)

Here is the home page of the Putnam Seminar held at Carnegie Mellon University by Po-Shen Loh in 2012.

The book "Putnam and Beyond" by R. Gelca and T. Andreescu is available on the Internet to those of you who can log in to Bizzell Library with an OU ID.

Have you ever wondered what it's like to grade the Putnam exam?.

Have you ever wondered what it's like to write the Putnam exam?.

Some people like to "study" for the Putnam by doing lots of problems from the problem section of the American Mathematical Monthly. I'm not sure this method is for everybody, but it is an interesting experience to try to do one of these problems once in a while. Here is a problem from the November 2012 edition of the Monthly. If you find a solution and send it in to the Monthly before March 31, 2013, and it is judged correct by the problem editors, then your name will get published in a future issue along with your solution.

*Proposed by Jose Luis Palacios, Universidad Simon Bolivar, Caracas, Venezuela.*
A random walk starts at the origin and moves up-right or down-right with equal
probability. What is the expected value of the first time that the walk is k steps below its
then-current all time high? (Thus, for instance, with the walk UDDUUUUDDUDD...,
the walk is three steps below its maximum-so-far on step 12.)

We started out the semester by looking at Problem 2 from the 2012 International Math Olympiad and Problem B1 from last year's Putnam Exam.

Elias Wegert kindly sent along this paper concerning some results which are an outgrowth of the 1986 IMO problem
described below on this page under the **Fall 2004** heading.

- On Sept. 30, we looked at this sheet of (relatively) easy Putnam problems. We solved 2004-A1 and 2001-A1 and partly solved 2007-B1. One of us was working on 1997-A5 and seemed to be getting somewhere.
- On Sept. 16, we solved problem A1 from this handout and (mostly) solved A2.

Here are some practical tips on taking the Putnam Exam, and an outline of George Polya's general problem-solving strategy.

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

This year we looked at problem sets from Ravi Vakil's website: the problems involve recurrences, complex numbers, invariants and monovariants, and probability. Most weeks we have concentrated on the "sample problems" on these problem sets, and not gone much into the rest of the problems. Some notable exceptions, though, are that Jacob came up with a solution to Problem 10 on the "number theory" set, Huy solved Problem 2 on the "complex number" set (without using complex numbers), and Adam and an anonymous friend solved Problem 2 on the "recurrence relations" set.

Besides the solution we found to Sample Problem 4 on the "probability" set, there are a number of other interesting ways to solve this problem and related problems: see Alexander Bogomolny's article on this problem at http://www.cut-the-knot.org/Curriculum/Probability/TriProbCartesian.shtml. Follow the links given on that page; they are well worth looking at.

At the second meeting, Sept. 18, we looked at some problems in number theory, again taken from the Stanford problem-solving seminar website. Before looking at the problems, we briefly went over this handout on some famous and useful theorems of number theory. For a comprehensive, book-length treatment of how to solve Putnam-type problems in number theory, see "104 Number Theory Problems from the training of the USA IMO Team" by Titu Andreescu et al. For those of you looking at this page on a machine in the .ou domain, you can read an electronic copy of this text online by clicking here.

At the first meeting this semester, on Sept. 11, we again started with problems on the pigeonhole principle and induction, from the Stanford Polya problem-solving seminar website.

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.)

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?"