21-295: Putnam Seminar (Fall 2011)

Po-Shen Loh

Last updated 9 December 2011.

Location

Section A (general lecture) meets Tue 4:30 in Doherty 2210, for 1 hour.
Section B (active seminar) meets Thu 4:30 in Wean 8220, for 1 hour.
Section C (problems 1-3) meets at 4:30 for 1 hour per week, alternating Mon (Wean 5302) / Wed (Wean 8220).
Section D (problems 4-6) meets Sun 3:30 in Porter A19C, for 3 hours. The first meeting is Sun Sep 4.
Office hours by appointment.

Additional activities considered to be part of the course

The Virginia Tech Regional Mathematics Competition (VTRMC) will be administered in Gates 4307 on Saturday, October 29, from 9:00am - 11:30am.
The William Lowell Putnam Mathematical Competition will be held in Scaife 125 on Saturday, December 3, from 10:00am - 1:00pm and 3:00pm - 6:00pm.

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.

Section A: For students who are new to problem solving. This is meant to provide a broad introduction to the area, and to inspire excitement. It will be taught in lecture format, with participation encouraged.
Learning goals: By the end of the course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Students should also be able to recognize when proofs are written with sufficient rigor, and should gather a greater appreciation for mathematics as a broad field.
Section B: For students who have prior experience, and would like to sharpen their ability to discover new multi-step arguments. The content will overlap substantially with Section A, but the style will be student-driven. Specifically, instead of following a traditional lecture-based format, Section B will advance by having students collectively brainstorm solutions with the instructor's guidance. Consequently, active participation is required. Outside of class, students will also write up one solution per week in advance of each meeting. Feedback will be provided through peer review.
Learning goals: to increase ability to solve Putnam problems, and to communicate solutions effectively.
Section C: Independent problem-solving session, with hints. Students will not write up solutions during the hour-long meeting, but may choose to write up a solution to bring for peer review at the Section B meeting.
Learning goals: to master questions at the Putnam 1-3 level.
Section D: Independent problem-solving session, without hints. Students will write up solutions during the session, for peer-review evaluation at the end.
Learning goals: to master questions at the Putnam 4-6 level.

Grading

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	Polynomials Handouts: A/B, C, D
Week 5 (Sep 27 / 29) C: Wed Sep 28	Calculus 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	Inequalities Handouts: A/B, C, D
Week 8 (Oct 18 / 20) C: Mon Oct 17	Convergence Handouts: A/B, C, D
Week 9 (Oct 25 / 27) C: Wed Oct 26	Recursions Handouts: A/B, C, D
VTRMC (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
Putnam (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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