The William Lowell Putnam Mathematical Competition
If there is enough interest, we will organize one or more preparation sessions. We can start these at any time, no need to wait until the contest is too close.
Here is the pdf file of the introductory handout.
Below are some sample materials for such sessions (based on what I
have done in previous
years):
| Topic 1 Problems |
Each Putnam test contains a couple of problems (usually A1 and B1) that are easier than the others. In some cases they are so run-of-the mill that one can only wonder why they appear on the Putnam. This session will introduce you to oodles of easier problems like this, with the intent to convince you that solving two or even three problems on the contest is quite real. |
| Topic 2 Problems |
In this session we will consider three techniques that prove useful. The pigeonhole principle is a favorite among authors of math contest problems. It requires no advanced math, yet sometimes it has far-reaching consequences. The principle of math induction is a useful brute-force machine when proving statements of the form For all positive integers n..., yet it is considered too simple to have a whole problem on the contest that can be solved by induction alone. The corresponding practice problems will be arguably simpler than a regular Putnam problem. The principle of the minimal counterexample supersedes the principle of induction, and works in more complicated situations. |
| Topic 3 Problems |
In this session we will consider some problems whose solution, although not immediately obvious, becomes natural if we introduce convenient notation, re-interpret the problem, or make use of symmetry. |
| Topic 4 Problems |
Linear recursions and Linear Algebra. |
| Topic 5 Problems |
Sequences and series, including some non-linear recursions (a.k.a. Dynamical Systems). |
| Topic 6 Problems |
Combinatorics and counting. Power series and generating functions. Probability. |
| Topic 7 Problems |
Complex numbers. Geometry. |
| Topic 8 Problems |
Inequalities. |
| Topic 9 Problems |
Number theory. Polynomials. Abstract Algebra. |
| Topic 10 Problems |
Real Analysis, Calculus. Differential Equations. Also, some functional equations. |
| Topic 11 Problems |
Games of strategy. I will bring some printouts and copies of articles describing and discussing two-person games. We will look at several games played with matchsticks, some played on special boards, and the old favorite, Tic-Tac-Toe. I will bring also some materials so we can play and discover optimal strategies. An optimal strategy for chess will not be disclosed at this meeting. |
Questions? Suggestions? Contact me.
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