Putnam Training: “Crunch Time” Topic Selection

There is about a month left before the Putnam Exam, and it will be the last one I could take. I have looked over several problems from previous exams, and done several dozen problems from Paul Zeitz's The Art and Craft of Problem Solving. I still believe I need more training, and so, I found a large collection of exercises at Northwestern University's math department website.

The authors of this problem set segregated the different exercises based on strategies, which I will list below. Due to schoolwork, I don't have the time to work through each one, so I thought about picking $4$ or $5$ topics and working through as many problems as I could within those problem sets.

I'm curious to hear what my fellow MSE users would choose if they were in my position today. Which 4-5 topics would you devote your time to, and how would you rank them?

In order to minimize the "opinion-based" nature of this question, I ask that people give some sort of justification.

If you'd like to see the link to the problem sets, it's here: http://www.math.northwestern.edu/putnam/training_problems-2014.pdf

Otherwise, here is the list:

1. Induction
2. Inequalities
3. Number theory
4. Polynomials
5. Complex numbers
6. Generating functions
7. Recurrences
8. Calculus
9. Pigeonhole principle
10. Telescoping
11. Symmetries
12. Inclusion-Exclusion
13. Combinatorics and probability
14. Miscellaneous

I'll start. Here is my ranking:

1. Pigeonhole principle
2. Generating Functions
3. Number theory
4. Telescoping
5. Inequalities

Here's my justification:

Most Putnam exams that I have seen have some sort of Pigeonhole problem in some capacity. For whatever reason, I suspect that the test makers just love them. Furthermore, the strategy is pretty unique, and isn't the byproduct of clever algebraic manipulation as other strategies (i.e., telescoping series) are.

At the college I went to, generating functions were largely ignored, which I always thought was a bit unfair. I don't have raw data (though I'd love to have some) but after talking to enough people at MAA and SIAM conferences, I've found that generating functions are not as stressed upon in college curricula as they should. Generating functions teach a lot of unique properties, and using them wisely could mean the difference of solving a Putnam problem in minutes rather than wasting an entire hour to not find a solution. In fact, if pigeonhole problems weren't so prevelant, I'd really place this one at #$1$. Be sure to check out Putnam and Beyond's chapter on generating functions. You'll learn a lot.

Number theory: another choice that's often seen on the Putnam, though to largely varying degrees. My guess is that depending on the curriculum you took in college, you probably know enough number theory to get by (slightly)... but a refresher into some advanced theorems could mean the difference in solving some problems.

I included telescoping sums at #$4$ because I think the strategy can be pretty useful, and not too hard to master. I used to use this principle to solve several contest-style problems at a time, and I think all arsenals are better off having mastered it. Furthermore, the problem set you posted only has a handful of telescoping sums, so you could do them in a relatively short amount of time (1/week?) until the exam comes.

I have inequalities at 5 because the identities pop up in questions all the time. Even when you're solving problems that require other strategies (i.e., number theory, pigeonhole, etc) inequalities can help you place upper and lower bounds on a solution, so if you get stuck doing whatever you started with, inequalities could narrow down your solution set dramatically to where you could just test whatever solution possibilities you have. Not the most rigorous of strategies, but hey... it's a contest, and you're timed.

I too would be interested in hearing what others have to suggest.

• That makes sense. What about inclusion-exclusion? You don't feel as strongly about those? – daOnlyBG Nov 5 '14 at 21:45