21-295: Putnam Seminar (Fall 2015)

Po-Shen Loh

Last updated 29 October 2015.

Course Assistants

• David Altizio
• Ziang Chu
• Cody Johnson - nice discussion around CMU Putnam course
• William Xiao

Mission

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 CMU 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. This year, we are happy to include a spectacular new University of Pittsburgh student in our meetings.

Locations

• Mon (introductory 1) meets at 3:30 in Porter 125C, for 1 hour. This is officially Section A.
• Tue (introductory 2) meets at 4:30 in Baker 136A, for 1 hour. This is officially Section B.
• Wed (problems 1-3) meets at 4:30 in Wean 5310 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 5409, for 1 hour. This is officially Section C.
• 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 6.
• Office hours: Any time by appointment.

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 24, from 9:00am - 11:30am.
• The William Lowell Putnam Mathematical Competition will be held in Scaife 125 on Saturday, December 5, from 10:00am - 1:00pm and 3:00pm - 6:00pm.

Course description

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, 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

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.

• 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: 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.

• Fri: Similar to Thursday, but at ludicrous speed.

• 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

   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.

Detailed syllabus