# 18.A34 Mathematical Problem Solving (Putnam Seminar)

**Fall 2022, MIT**

**Class meetings:** Mondays and Wednesdays 1–2pm, room 2-132

**Instructor:** Prof. Yufei Zhao

**Undergraduate Assistants (UA):**
Dain Kim
and
Tomasz Slusarczyk

**Emails and Slack:**

- For quick questions, ask me after class
- Include both UAs in all class related communication, including everything homework related (submission, extensions, grading, etc.)
- Use the class Slack for any discussions of general interest (e.g., discussing solutions after due date, math related to a lecture)
- Begin your email subject line with “[18.A34]”

## Course description and policies

- A first-year undergraduate seminar
- Intended for students with previous math competition experience
- Lectures highlight problem solving techniques as well as connections to further mathematics
- Emphasis on developing mathematical communication skills, including blackboard presentation and proof writing
- Discussions of academic and career topics relevant for students from a math competition background

**William Lowell Putnam Mathematics Competition:**
The Putnam Competition is an annual mathematics contest for undergraduates in the USA and Canada.
This year it will be held **Saturday, December 3, 2022**.

All registered students will be required to participate in the Putnam competition. Students should self-register on the official Putnam website.

Seminar participants are selected through the First-year Advising Selection process. Unfortunately I cannot add additional students. See below for policies on lecture attendance.

### Class format

**Lectures**are open to all MIT students.**Presentations/Discussions**(i.e., non-lectures) are only open to registered class members. These classes will begin with a brief discussion, followed by student blackboard presentations of homework solutions. Active participation is expected.- Students can volunteer at the beginning of each presentation class by writing on the blackboard their name and the solution they wish to present.
- Aim for no more than 10 minutes per presentation. You should skip routine or uninteresting calculations. I may need to end a presentation if it runs over time.

**UA office hours**will give opportunities for students to practice presentations with the UAs in a small group setting, and also to discuss writing and problem solving techniques. Every student will be**required**to attend a certain number of UA office hours, roughly once every two weeks. More information will be announced later.

**Class attendance is required** for registered students.
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.

**Non-registered MIT students** are welcome to attend the lectures but not the discussion sessions and may not turn in homework.

### Grading

Pass/Fail. Based on homework and participation. Homework will be graded on correctness and presentation. Illegible or sloppy write-ups are unacceptable.

Students needing support should consider reaching out to Student Support Services (S^{3}) or Student Disability Services.

## Schedule and due dates

Lectures are open to all MIT students.

All other sessions are restricted to official seminar participants.

W 9/7 Class introduction

M 9/12 Discussion & Presentations. Due: Probability & SS1

W 9/14 Lecture by Yufei Zhao

M 9/19 Discussion & Presentations. Due: Hidden independence and uniformity & SS2

W 9/21 Lecture by Ashwin Sah

M 9/26 Discussion & Presentations. Due: Analysis & SS3

W 9/28 Lecture by Carl Schildkraut

M 10/3 Discussion & Presentations. Due: Combinatorial configuration & SS4

W 10/5 Lecture by Daniel Zhu

M 10/10 No class & Indigenous Peoples Day

W 10/12 Discussion & Presentations. Due: Generating functions & & SS5

M 10/17 Discussion & Presentations. Due: Congruences and divisibility & SS6

W 10/19 Lecture by Dain Kim

M 10/24 Discussion & Presentations. Due: Polynomials & SS7

W 10/26 Lecture by Mihir Singhal

M 10/31 Discussion & Presentations. Due: Abstract algebra & SS8

W 11/2 Lecture by Tomasz Slusarczyk

M 11/7 Discussion & Presentations. Due: Inequalities & SS9

W 11/9 Lecture by Allen Liu

M 11/14 Discussion & Presentations. Due: Linear algebra & SS10

W 11/16 Lecture by Edward Wan

M 11/21 Discussion & Presentations. Due: Sums and integrals & SS11

W 11/23 Discussion & Presentations (on any previously assigned problems)

M 11/28 Discussion & Presentations: Putnam 2020

W 11/30 Discussion & Presentations: Putnam 2021

Saturday 12/3 Putnam Competition

M 12/5 Discussion & Presentations: Putnam 2022

W 12/7 Discussion & Presentations: Putnam 2022

M 12/12 Discussion & Presentations: Putnam 2022

W 12/14 Discussion & Presentations: Putnam 2022

For past Putnam problems see the Putnam Archive

## Homework

Each problem set contains a long list of problems. You are encouraged to try many problems, but please only hand in **your three best solutions** (do not submit more than three):

- At least two problems should be from the topic set, i.e., at most one problem can come from the supplementary problem 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).
- After each homework is due, you are allowed and encouraged to discuss your solutions on Slack

If you wish to get a head start on later problem sets, you can check out the material from previous semesters (see links at the bottom). This year’s problem sets will likely be mostly the same, although there could be minor changes and re-numbering.

### Submission

- Begin each solution on a new page.
- Homework must be submitted on Gradescope (accessible from Canvas) by 1pm, before the beginning of the class meeting, preferably earlier.
- A certain minimum number of solutions (out of the three problems) should be typed in LaTeX and submitted as PDF (the remaining solutions may be typed or handwritten):

Homework # | # solutions required to be typed in LaTeX |
---|---|

1 — 2 | ≥ 0 |

3 — 5 | ≥ 1 |

6 — 8 | ≥ 2 |

9 — 11 | 3 |

- If you are new to LaTeX, I recommend checking out the Overleaf tutorial and using the Overleaf online editor (which does not require any LaTeX installation). You can also reach out to the UAs for LaTeX help.
- Submission should be carefully handwritten or typed (illegible submissions are unacceptable).
- Homework will be graded similarly to the Putnam competition.
- Problems range widely in difficulty. You are encouraged to challenge yourself and submit your best solutions.
- Do not worry if a problem set covers an area of mathematics you have not yet formally learned (e.g., algebra, analysis). Try your best.
- 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 the UAs and me in advance or have S
^{3}send us a message. Let us know how many days extension you need. - My policy is to not grant extension based on forseeable circumstances including other academic workload, extracurriculars, and poor study habits.

### Collaborations

- You are encouraged to first work on the homework problems yourself before seeking collaboration.
- Meaningful collaboration is allowed if it helps with your learning (e.g., solving a problem together)
- Unacceptable practices include: “dividing up” the problems among a group and then distributing the solutions; asking for a solution from a friend.
- You must write up your own solutions.

### Acknowledging collaborators and sources

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

- At the beginning of the submission for
**each problem**, write`Collaborators and sources:`

followed by a list of collaborators and sources consulted (people, books, papers, websites, software, etc.), or write`none`

if you did not use any such resources. - Failure to acknowledge will result in an automatic 1pt penalty per problem.
- Acceptable uses of resources include: looking up a standard theorem/formula/technique; using Wolfram Alpha/Mathematica/Python for a calculation
- You may NOT intentionally look up (or ask from others) solutions to homework problems prior to solving the problems yourselves. Once you have solved a problem, it is fine to seek and learn alternate solutions.

Intentional violations of the above policies may be considered academic dishonesty/misconduct.

## 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 2017, 2018, (also MIT OCW), 2019, 2020, 2021.