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The Putnam Mathematical Competition is the most prestigious math contest for undergraduate students in the U.S. and Canada. The exam is designed to test creative thinking in addition to technical competence, and is held every year on the first Saturday of December. The problems range across the undergraduate mathematics curriculum, but most do not require specialized knowledge of mathematics beyond calculus.

Unlike standard textbook problems, where many of them are quite similar, Putnam problems are challenging (and fun!) precisely because it is generally not immediately obvious how to approach them. There are some standard techniques that often can be useful to start making progress towards a solution, but a good amount of practice is needed to develop an intuition of when to apply each one. For this reason, we hold the Putnam Problem-Solving Seminar at OU each fall: we meet weekly to solve Putnam-style problems and learn about some problem-solving strategies.

Anyone interested in problem-solving 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 analytical 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 that requires analytical reasoning!

Below is information about the 2016-2019 seminars, and at this link you can find the archives for 2015 and before.

For the fall 2019 semester, the Putnam Problem-Solving Seminar is being run by Alejandro Chávez-Domínguez (jachavezd[at]ou[dot]edu) and Roi Docampo (roi[at]ou[dot]edu).

We are meeting on Thursdays 5:00-6:30pm in PHSC 321, starting on August 29th. Please do come even if you need to arrive late or leave early, this is not a class and the format easily allows for people to come and go as they need.

Putnam Exam 2019

December has arrived, which brings with it not just the holidays and the finals, but also the Putnam exam!

It will take place this Saturday, December 7, from 9:00am-12:00pm and 2:00pm-5:00pm in PHSC 1105. All you need to bring are writing instruments (and perhaps erasers, if that is something you would like to have available). Accordingly, the last session of the Problem-Solving seminar will take place this Thursday, December 5.

There are two three-hour sessions because you will get a different set of six problems for each session (the A problems in the morning, and the B problems in the afternoon). During the two hour break, we will get lunch. We will start right at 9:00am, so please make sure to arrive a few minutes early.

For planning purposes, if you intend to take the exam please email jachavezd [ at ] ou.edu to let us know (even if you are not completely sure). If you know other students who might be interested, please do share this information with them.

November 21, 2019

Today’s theme is solving geometric problems through the use of vectors (or complex numbers). The idea is to translate the geometric conditions to algebraic equations, and then solve the problem by doing algebraic calculations.

If this sounds like analytic geometry, it does share the same philosophy. The approach with vectors, however, has the advantage of not using coordinates. This makes for nicer, cleaner, calculations.

Click here for the worksheet.

November 14, 2019

Today’s theme is functional equations, that is, equations where the unknown is a function.

Some typical strategies are:

  • Plug in special values: 0, 1, etc.
  • If there are two variables: set them equal to each other, or one to be the negative of the other, etc.
  • Change of variable: from x to -x, or some other transformation suggested by the equation.
  • Take derivatives.

Click here for the worksheet.

October 17, 2019

Today’s theme is the principle of mathematical induction.

Recall that induction is a proof technique used to prove that a property P(n) holds for every natural number n, i.e. for n = 1, 2, 3,…
It consists of two parts:
(I) The base case: proving the property for n=1.
(II) The induction step: assuming that we already know the property holds for n=k, we prove it for n=k+1.

There are slight variations of this technique, the most common one being strong induction: for the step, we assume that the property holds for all n less than or equal to k, and then prove it for n=k+1.

Click here for the worksheet.

October 3, 2019

Today’s theme was Number Theory, which in the context of math competitions typically refers to properties of the integers, particularly divisibility.

Click here for the worksheet.

September 26, 2019

Today’s theme was enumerative combinatorics, which deals with counting things; typically, the number of ways that certain patterns can be formed.
Some typical basic strategies that are used to solve this type of problems are:

  • Sum rule
  • Product rule
  • Recursion
  • Counting by bijection, or counting the same objects in two different ways
  • Principle of inclusion and exclusion

Click here for the worksheet.

September 19, 2019

Today’s strategy is the well-known box (or pigeonhole) principle: if n+1 items are distributed among n boxes, at least one box must contain more than one item.

More generally, if kn+1 items are distributed among n boxes, at least one box must contain at least k+1 items.

Sample problem: Five points with integer coordinates are chosen on the plane. Prove that you can always choose two of these points so that the segment joining them passes through another point with integer coordinates. (The solution is at the end of the worksheet).

For the worksheet, click here.

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