# 18.A34 Mathematical Problem Solving (Putnam Seminar)

**Fall 2020, MIT**

**Class meetings:** Mondays and Wednesdays 10–11am (Note time change)

**Instructor:** Yufei Zhao (see link for contact info)

**Assistants:** Dain Kim and Daniel Zhu

Please include “18.A34” in the subject line of your emails

## Course description and policies

This is a first-year seminar that prepares students for the Putnam Mathematical Competition, and with a focus on developing mathematical communication and presentation skills. This seminar is fast-paced and is targeted for students with previous experience in mathematical Olympiads, though others may benefit as well. (Units: 2-0-4)

The **William Lowell Putnam Mathematics Competition** is an annual mathematics contest for undergraduates in USA and Canada.
Due to COVID-19, the 2020 Putnam has been postponed until February 20, 2021; see the official Putnam website for updates and announcements.

All students receiving credit for this class will be required to participate in the Putnam competition (in whichever format it will be held, either officially or unofficially).

*Class format.*
On most weeks, one meeting will be a lecture (often a guest lecture by an upperclassman) on a specific topic, and the other meeting will be student presentations of homework problems.
We intend to record the lectures (recordings accessible to registered participants) but not the presentation classes.

During the student presentation classes, students will be called on to present their solutions. The presenting student should pick a problem, and screen-share their solution, and explain their solution verbally (not simply reading from the written solution). I, as well as other members of the class, will give critical feedback on both the mathematical content as well as the presentation.

This is a new format as we are adapting to remote-teaching. We may experiment with different formats as the semester progresses.

*Attendance.*
Class attendance is **required** for registered students.
Active participation is an important aspect of this seminar.
Please notify me in advance if you cannot make it to class.
Too many unexcused absences is cause for concern and may lead to a non-passing grade.

You are asked to **turn on your video camera** during class (please let me know if there is some reason you might not want to do this, e.g., poor internet connection, inconvenient environment).

*Non-registered students.*
MIT students not registered in the class are welcome to attend the lectures but not the presentation sessions (see schedule below) and may not turn in homework.

Seminar participants were selected through the first-year advising selection process, and unfortunately I will not be able to add additional students. *Listeners are not permitted in 18.A34.*

*Grading.* Based on homework and in-class participation.
Homework will be graded on correctness and presentation. Illegible or extremely sloppy write-ups are unacceptable.

Student Support Services (S^{3}) and Student Disability Services

## Schedule

*Subject to change. See homework policy below.* (SS = Supplementary set)

Lectures are open to all MIT students (Zoom link; MIT Touchstone authentication required).

All other sessions are restricted to official seminar participants (link in Canvas).

**W 9/2**Introductions (only official participants attend)~~M 9/7~~*No class (Labor Day)***W 9/9**Lecture by Yufei Zhao**M 9/14**Presentations. Due: Hidden independence and uniformity and SS1**W 9/16**Lecture by Mehtaab Sawhney**M 9/21**Presentations. Due: Inequalities and SS2**W 9/23**Lecture by Shengtong Zhang**M 9/28**Presentations. Due: Sums and integrals and SS3**W 9/30**Lecture by Yuan Yao**M 10/5**Presentations. Due: Combinatorial configurations and SS4**W 10/7**Lecture by Hung-Hsun Hans Yu (notes)Presentations. Due: Recurrence and SS5*T*10/13**W 10/14**Lecture by Ashwin Sah**M 10/19**Presentations. Due: Analysis and SS6**W 10/21**Lecture by Daniel Zhu**M 10/26**Presentations. Due: Generating functions and SS7**W 10/28**Lecture by Zilin Jiang**M 11/2**Presentations. Due: Abstract algebra and SS8**W 11/4**Lecture by Dain Kim**M 11/9**Presentations. Due: Polynomials and SS9~~W 11/11~~*No class (Veterans Day)***M 11/16**Presentations.**W 11/18**Lecture by Junyao Peng (notes)*(Thanksgiving break)***M 11/30**Presentations. Due: Congruences and divisibility and SS10**W 12/2**Lecture by Allen Liu**M 12/7**Presentations. Due: Linear algebra and SS11**W 12/9**Final class wrap-up and discussions

## Homework

Each problem set contains a (sometimes long) list of problems on a specific topic. You are encouraged to work on as many as you like, but only hand in **your six best solutions** (please do not submit more than six):

- At least 4 problems should be from the topic set
- Do not hand in supplementary problems rated strictly less than [2]; these are too easy.
- For multi-part problems, you may decide what counts as “one solution”, as long as it is reasonable (i.e., not too trivial).

*Submitting homework:*

- Homework must be submitted on Gradescope (accessible from Canvas) before the beginning of the class meeting.
- Submission may be carefully handwritten or typed (illegible submissions are unacceptable).
- Begin each solution on a new page.
- Homework will be graded similarly to the Putnam competition.
- Non-registered students may not hand in solutions.

*Late policy*

- Late submissions will not be accepted without a valid excuse.
- If you need an extension for valid excuses (e.g., unanticipated health or family issues), please email me in advance or have S3 send me a message.
- My policy is to not grant extension based on forseeable circumstances including other academic workload, extracurriculars, and poor study habits.

*Sources:* It is required to acknowledge your sources (even if you worked independently)

- At the top of
**each problem**, write a list of collaborators and sources (people, books, websites, etc.). - If no additional sources are consulted, you must write
`Sources consulted: none`

or equivalent. - Failure to do this will result in an automatic 1pt penalty.
- You may NOT look up solutions to homework problems online (or offline).

*Collaborations.*

- You are strongly encouraged to work on the problems on your own, though reasonable collaboration is permitted.
- Everyone must write their solutions individually and acknowledge their collaborators as noted above.
- Pset partners — tool for finding problem set collaborators (MIT Touchstone required)

## Additional resources

You may find the following optional resources helpful for additional preparation. Some resources may be available electronically from MIT Library.

**Previous Putnam problems and solutions**

- Putnam archive by Kedlaya
- The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions and Commentary by Kedlaya, Poonen, and Vakil

**Additional books helpful for preparation**

- Problem-Solving Through Problems by Larson
- Putnam and Beyond by Gelca and Andreescu

Previous course homepage from Fall 2017, Fall 2018 (also MIT OCW), and Fall 2019.