21-295: Putnam Seminar (Fall 2012)

Po-Shen Loh

Last updated 1 December 2012.

Locations

We have doubled the number of official sections from 2 to 4, in order to maintain optimal class sizes. Meeting times and locations are:

Mon (introductory 1) meets at 4:30 in Wean 5403, for 1 hour. This is officially Section C.
Tue (introductory 2) meets at 4:30 in Wean 5409, for 1 hour. This is officially Section A.
Wed (problems 1-3) meets at 4:30 in Wean 8220 for 1 hour. This is not a substitute for any of the four main sections of the course.
Thu (intermediate) meets at 4:30 in Wean 8220, for 1 hour. This is officially Section B.
Fri (advanced) meets at 4:30 in Wean 5310, for 1 hour. This is officially Section D.
Sun (problems 4-6) meets at 3:30 in Porter A19C, for 3 hours. The first meeting is Sun Sep 2.
Office hours: Any time by appointment, or MWF 5:30pm in Skibo gym weight room.

Additional activities considered to be part of the course

The Virginia Tech Regional Mathematics Competition (VTRMC) will be held in Scaife 125 on Saturday, October 27, from 9:00am - 11:30am.
The William Lowell Putnam Mathematical Competition will be held in Scaife 125 on Saturday, December 1, 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.

This year, we will offer 4 levels of the Putnam seminar, so that students can personalize their experience by selecting the one(s) that they benefit the most from. No prior math competition experience is required, and beginners are welcome. Indeed, this seminar seeks to complement the standard math curriculum by concentrating on the raw creative problem-solving skills which are essential for original work in almost every field. The levels are roughly divided as follows.

Mon or Tue: 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.
Thu or Fri: 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 the other sections, but the style will be student-driven. Specifically, instead of following a traditional lecture-based format, these groups will advance by having students collectively brainstorm solutions with the instructor's guidance. Consequently, active participation is required.
Learning goals: to increase ability to solve Putnam problems, and to communicate solutions effectively.
Wed: Independent problem-solving session, with personalized hints. The instructor constantly moves around the room, observing students' progress, and essentially conducts a number of independent, parallel tutoring sessions at once. (This is most similar to the style he uses at the Math Olympiad Summer Program.) Students do not write up solutions during the hour-long meeting, focusing instead on raw problem solving.
Learning goals: to master questions at the Putnam 1-3 level.
Sun: Independent problem-solving session, without hints. Students will write up solutions during the session.
Learning goals: to master questions at the Putnam 4-6 level.

Grading

This year, we are very fortunate to have two graders for this course, who will provide feedback to students on weekly assignments. The main purpose of these assignments is to allow students to practice the art of writing mathematics efficiently. Since it can be difficult to solve problems on one's own, grading will be based on effort. Attendance at the weekly section meeting, together with submission of the assignment from the previous week (with either a full solution or ideas), will count for one weekly point. The final grade will be calculated through the following formula:

Score = (number of weekly points) + (3 for the VTRMC) + (6 for the Putnam).

Ratio = Score / MaximumPossible.

MaximumPossible is 23 for Tue, 22 for Mon (due to Labor Day), and 22 for Thu/Fri (due to Thanksgiving).

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

Detailed syllabus

Week 1	Proof by contradiction Lecture / Wed
Week 2	Number theory Sun / Lecture / Wed
Week 3	Polynomials Sun / Lecture / Wed
Week 4	Calculus Sun / Lecture / Wed
Week 5	Continuous functional equations Sun / Lecture / Wed
Week 6	Inequalities Sun / Lecture / Wed
Week 7	Convergence Sun / Lecture / Wed
Week 8	Recursions Sun / Lecture / Wed
Week 9	Linear algebra Sun / Lecture / Wed
VTRMC (Sat Oct 27)	Competition from 9:00am - 11:30am, in Scaife 125.
Week 10	Combinatorics Lecture / Wed
Week 11	Geometry Sun / Lecture / Wed
Week 12	Integer polynomials Sun / Lecture / Wed
Week 13	Basic arguments Sun / Lecture / Wed Wed/Sun meetings will still run, with optional attendance, for students who stay at CMU for Thanksgiving.
Week 14	Higher math Sun / Lecture / Wed
Putnam (Sat Dec 1)	Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125.
Week 15	Beyond Putnam (optional, and Monday only)

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