21-295: Putnam Seminar (Fall 2010)

Po-Shen Loh

Last updated 2 December 2010.


Additional activities considered to be part of the course

Course description

Problem solving is an essential skill in every discipline, from mathematics to carpentry. This class seeks to develop that ability through challenging (but fun) problems which require some creativity to solve. These problems will generally come from mathematical competitions, and students will also have the opportunity to try their hands at two regional/national competitions, the VTRMC and the Putnam.

Yet although competitions are the title of this course, the syllabus will actually be constructed around general problem-solving techniques in mathematics. The instructor's aim is to use the competition problems to teach some mathematics which is not usually seen in ordinary classes, but is also beneficial to learn.

There is no official reference for this course, as much of the material will come from the instructor's experience in coaching the United States Math Olympiad team. Nevertheless, it should be mentioned that there will be some correlation with the book Putnam and Beyond, by Razvan Gelca and Titu Andreescu. Students are not required to purchase the book!

Rough format of class meeting, and levels

The aim of this class is to provide a comprehensive introduction to the various branches of mathematics which appear in the Putnam and related competitions. Each session will focus on a particular theme, and the teaching style will be midway between a pure lecture and a pure problem-solving session.

There are two sections of this course. Section A should be accessible to any interested student (including freshmen). Section B will be better suited for students who already have prior experience with the variety of mathematics which appears in these competitions.


As this course is primarily for interest, grades will only be a distraction. Therefore, grading will be by attendance-only, using the following simple formula:

   Score = (number of classes attended) + (3 for the VTRMC) + (6 for the Putnam).

   Ratio = Score / MaximumPossible.

   MaximumPossible is 23 for Section A and 22 for Section B (due to Thanksgiving).

Grades are then based on the Ratio, using the standard scale A = 90%+, B = 80%+, etc.

Therefore, the homework sets mentioned above will not be graded. They are meant to supply deeper problems which require more time to think about (and therefore cannot be done together in lecture). Nevertheless, students are encouraged to work on them either individually or in groups. They will be briefly discussed at the beginning of each lecture.

Detailed syllabus

Week 1
(Aug 24 / 26)
No class.
Instead, try this problem for fun, and tell us next week how much money you can manage to earn. Note: in the context of the problem, when you press button A (or B), you also choose the number i it is applied with (the number i is not random).
Week 2
(Aug 31 / Sep 2)   
Proof by contradiction
Week 3
(Sep 7 / 9)
Mathematical induction
Week 4
(Sep 14 / 16)
The pigeonhole principle
Week 5
(Sep 21 / 23)
Bonus class: Fri Sep 24, 4:30-5:30 in Wean 8220.
Week 6
(Sep 28 / 30)
Bonus class: Wed Sep 29, 4:30-5:30 in Wean 5302.
Week 7
(Oct 5 / 7)
Number theory
Bonus class: Wed Oct 6, 4:30-5:30 in Wean 5302.
Week 8
(Oct 12 / 14)
Bonus class: Wed Oct 13, 4:30-5:30 in Wean 5302.
Week 9
(Oct 19 / 21)
Bonus class: Wed Oct 20, 4:30-5:30 in Wean 5302.
Week 10
(Oct 26 / 28)
Bonus class: Wed Oct 27, 4:30-5:30 in Wean 5302.
(Sat Oct 30)
Competition from 9:00am - 11:30am, in Scaife 125
Week 11
(Nov 2 / 4)
Functional equations
No bonus class.
Week 12
(Nov 9 / 11)
Bonus class: Mon Nov 8, 4:30-5:30 in Wean 5302.
Week 13
(Nov 16 / 18)
Bonus class: Mon Nov 15, 4:30-5:30 in Wean 5302.
Week 14
(Nov 23)
Elementary methods. No advanced class (Thu 25th)
Bonus class: Mon Nov 22, 4:30-5:30 in Wean 5302.
Week 15
(Nov 30 / Dec 2)
General strategy and wrap-up
Handout A; Handout B
Bonus class: Fri Dec 3, 4:30-5:30 in Wean 8220.
(Sat Dec 4)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125

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