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## Welcome

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.

## 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.

## 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

## 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).

## September 12, 2019

Today’s strategy is to look for appropriate extremal objects: those that maximize or minimize some function. A common technique is to use extremality to show that an object must have some desired property. The following two problems can be solved using this technique (the solutions to the sample problems appear at the end of the worksheet).

Sample Problem 1. Let $$\Omega$$ be a set of points in the plane. Each point in $$\Omega$$ is a midpoint of two points in $$\Omega$$. Show that $$\Omega$$ is an infinite set.

Sample Problem 2. 2n points are given in the plane, no three collinear. Exactly n of these points are farms, the remaining n points are wells. It is intended to build a straight line road from each farm to one well. Show that the wells can be assigned bijectively to the farms, so that none of the roads intersect.

## September 5, 2019

Today’s theme is problems that involve colorings. A coloring is a way of visualizing the partition of a set into a finite number of subsets. The prototypical example is the following problem (its solution appears at the end of the worksheet).

Sample Problem. Consider an 8 × 8 chessboard, and cut two diagonally opposite corners of the board. In how many ways can you cover the 62 squares with dominoes?

## August 29, 2019

Today was our first session for the Fall 2019 semester!

The theme of today was the invariance principle. This idea applies when we have a situation that is changing over time, and the point is to find an invariant, that is, a quantity that stays the same.

A variation of the invariance principle is to find a quantity that perhaps does not always stay the same, but we know that it always increases (or decreases).

You can find today’s handout here.

## Fall 2019

For the Fall 2019 semester, the Putnam Problem-Solving Seminar meetings will start on August 29th and will be held:

• Thursdays,  5:00-6:30pm
• PHSC 321

As usual, please do come even if you need to show up a bit later or leave early. Everyone is welcome, so tell your friends!

## Fall 2018

For the Fall 2018 semester, the Putnam Problem-Solving Seminar meetings will start on August 30th and will be held

• Thursdays,  5:00-6:30pm
• PHSC 313

As usual, please do come even if you need to show up a bit later or leave early. Everyone is welcome, so tell your friends!

## Fall 2017: First meeting

It’s the fall semester again, which means it’s Putnam time! Our first meeting of the semester will be held:

• Thursday, August 31, 5:00-6:30pm
• PHSC 1025

As usual, please do come even if you need to show up a bit later or leave early. This is not a class and the format easily allows for people to come and go as they need.