The Mathematical Association of America's William Lowell Putnam Mathematical Competition provides the excuse to run a problem-guided tour of mathematics, while also developing core problem-solving skills that enhance the ability to learn and use higher mathematics. The questions which appear on the handouts are specially selected to spark discussions about famous mathematical results.
Although this course is named after a contest, its purpose is not to breed competition. Rather, we seek to develop a collaborative team spirit among the community of sharp and motivated students who self-identify themselves by joining this course. The Carnegie Mellon University Putnam seminar currently involves about 3% of the undergraduate student body, and provides a venue for this fellowship of scholars to gather on a weekly basis, with topics of discussion ranging from mathematics to career advice.
Finally, our scope is not limited to CMU. Long-term, we aim to develop the talent base in the Greater Pittsburgh Area through partnerships with other institutions.
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 carefully-selected problems which simultaneously develop sophisticated problem-solving techniques and inspire discussions about more advanced mathematics. The instructor's aim is to use the competition problems to provide a tour through many interesting topics, and to expose a bridge to higher mathematics.
There is no official reference for this course. 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. Much of the material will come from the instructor's experience in coaching the United States Math Olympiad team (also organized by the Mathematical Association of America).
There will be 6 meeting days per week, to provide optimal class sizes for the many CMU students who are on our extended Putnam Team. Meeting times and locations are:
The aim of this class is to provide a comprehensive introduction to the various branches of mathematics which happen to 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 5 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, although familiarity with 21-127 Concepts of Mathematics is assumed. 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.
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.
Learning goals: to increase ability to solve Putnam problems, and to communicate solutions effectively.
Learning goals: to master questions at the Putnam 1-3 level.
Learning goals: to master questions at the Putnam 4-6 level.
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 counts for 1/3 of a weekly point, and the other 2/3 comes from the assignment turned in that week (gaining credit from either full solutions or ideas). 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), 22 for Thu (due to Thanksgiving), and 21 for Fri (due to Mid-Semester Break and Thanksgiving).
Grades are then based on The Ratio, using the standard scale A = 90%+, B = 80%+, etc.
Week 1 | Introduction
Lecture / Wed |
Week 2 | Polynomials
Sun / Lecture / Wed |
Week 3 | Number theory
Sun / Lecture / Wed Thu (C) and Fri (D) meet together at 4:30 Thu Sep 15 in Hammerschlag B131. |
Week 4 | Calculus
Sun / Lecture / Wed |
Week 5 | Functional equations
Sun / Lecture / Wed |
Week 6 | Inequalities
Sun / Lecture / Wed |
Week 7 | Convergence
Sun / Lecture / Wed |
Week 8 | Recursions
Sun / Lecture / Wed |
VTRMC
(Sat Oct 22) |
Competition from 9:00am - 11:30am, in Scaife 125. |
Week 9 | Linear algebra
Lecture / Wed |
Week 10 | Combinatorics
Sun / Lecture / Wed |
Week 11 | Integer polynomials
Sun / Lecture / Wed |
Week 12 | Probability
Sun / Lecture / Wed |
Week 13 | Bare hands
Sun / Lecture |
Week 14 | Geometry
Sun / Lecture / Wed |
Putnam
(Sat Dec 3) |
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125. |
Week 15 | No class |
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