21-295: Putnam Seminar (Fall 2011)

Po-Shen Loh

Last updated 9 December 2011.


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!

Levels and expectations

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 week 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 official sections of this course, and two optional sections.


As this course is primarily for interest, grades will only be a distraction. Therefore, grading will be based on effort. In the beginning, signing an attendance sheet will suffice, but by the middle of the course, students will turn in ungraded exercises to count for their weekly attendance. The final grade will be calculated through the following simple formula, which allows for 2 missed weekly meetings:

   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.

Detailed syllabus

Week 1
(Aug 30 / Sep 1)
C: Wed Aug 31
Proof by contradiction
Handouts: A/B, C
Week 2
(Sep 6 / 8)
C: Mon Sep 5
Mathematical induction
Handouts: A/B, C, D
Week 3
(Sep 13 / 15)
C: Wed Sep 14
The pigeonhole principle
Handouts: A/B, C, D
Week 4
(Sep 20 / 22)
C: Mon Sep 19
Handouts: A/B, C, D
Week 5
(Sep 27 / 29)
C: Wed Sep 28
Handouts: A/B, C
Week 6
(Oct 4 / 6)
C: Mon Oct 3
Number theory
Handouts: A/B, C, D
Week 7
(Oct 11 / 13)
C: Wed Oct 12
Handouts: A/B, C, D
Week 8
(Oct 18 / 20)
C: Mon Oct 17
Handouts: A/B, C, D
Week 9
(Oct 25 / 27)
C: Wed Oct 26
Handouts: A/B, C, D
(Sat Oct 29)
Competition from 9:00am - 11:30am, in Gates 4307
Week 10
(Nov 1 / 3)
C: Mon Oct 31
Linear Algebra Advanced class (Thu Nov 3) meets in Wean 5302.
Handouts by Misha Lavrov: A, B, C
Week 11
(Nov 8 / 11)
C: Wed Nov 9
Geometry Advanced class meets on Fri Nov 11, not Thu Nov 10.
Handouts: A/B, C, D
Week 12
(Nov 15 / 17)
C: Wed Nov 16
Functional equations
Handouts: A/B, C, D
Week 13
(Nov 22)
C: Mon Nov 21
Elementary methods.
Handouts: A/B, C, D No advanced class (Thu Nov 24)
Week 14
(Nov 29 / Dec 1)
C: Wed Nov 30
General strategy and wrap-up
Handouts: A/B, C, D
(Sat Dec 3)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125
Week 15
(Dec 6 / 8)
No class; Putnam solutions will be discussed between 1-3pm and after 6pm on Sat Dec 3.

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