18.A34 Mathematical Problem Solving (Putnam Seminar)

Quick links: [Canvas]

Fall 2021, MIT (Link to the most current version of the course)

Class meetings: Mondays and Wednesdays 1–2pm, room 2-147

Instructor: Prof. Yufei Zhao

Undergraduate Assistants (UA): Dain Kim and Carl Schildkraut

Emails and Piazza

• For quick questions, ask me after class instead of emailing
• Include both UAs in all class related emails, including everything homework related (submission, extensions, grading, etc.)
• Use Piazza for any discussions of general interest (e.g., discussing solutions after due date)
• Begin your email subject line with “[18.A34]”

Course description and policies

This is a first-year undergraduate 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)

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 4, 2021

All registered students will be required to participate in the Putnam competition. See Putnam official website for registration information.

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.
• Discussions (i.e., non-lectures) are only open to registered class members. Most sessions will begin with a brief meta-discussion, followed by student blackboard presentations of homework solutions. Active participation is expected.
  • Students should volunteer at the beginning of each presentation class by writing on the blackboard their name and the solution they wish to present, along with “requests” for specific solutions.
  • Aim for no more than 10 minutes per presentation. You should skip routine or uninteresting calculations. I will cut you off if the presentation runs unnecessarily long.
• UA office hours will give opportunities for students to practice presentations with the UAs in 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, as will be explained during class announcements on Canvas.

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

Based on homework and participation (in class, office hours, Piazza). Homework will be graded on correctness and presentation. Illegible or extremely sloppy write-ups are unacceptable.

Students needing support should consider reaching out to Student Support Services (S³) or Student Disability Services.

Schedule and due dates

Lectures are open to all MIT students.
All other sessions are restricted to official seminar participants.

SS = Supplementary set

• W 9/8 Lecture by Mehtaab Sawhney

• M 9/13 Discussion — Presentations. Due: Inequalities & SS1

• W 9/15 Lecture by Ashwin Sah

• M 9/20 Discussion — Presentations. Due: Analysis & SS2

• W 9/22 Lecture by Yufei Zhao

• M 9/27 Discussion — Presentations. Due: Hidden independence and uniformity & SS3

• W 9/29 Lecture by Daniel Zhu

• M 10/4 Discussion — Presentations. Due: Generating functions & SS4

• W 10/6 Lecture by Evan Chen

• M 10/11 No class — Indigenous Peoples Day

• W 10/13 Discussion — Presentations. Due: Sums and integrals & SS5

• M 10/18 Discussion — Presentations. Due: Probability & SS6

• W 10/20 Lecture by Shengtong Zhang

• M 10/25 Discussion — Presentations. Due: Congruences and divisibility & SS7

• W 10/27 Lecture by Carl Schildkraut

• M 11/1 Discussion — Presentations. Due: Combinatorial configuration & SS8

• W 11/3 Lecture by Mihir Singhal

• M 11/8 Discussion — Presentations. Due: Abstract algebra & SS9

• W 11/10 Lecture by Dain Kim

• M 11/15 Discussion — Presentations. Due: Polynomials & SS10

• W 11/17 Lecture by Allen Liu

• M 11/22 Discussion — Presentations. Due: Linear algebra & SS11

• W 11/24 Discussion — More presentations (on any problem set)

• M 11/29 Discussion — Presentation of Putnam 2019 problems

• W 12/1 Discussion — Presentations of Putnam 2020 problems

• Saturday 12/4 Putnam Competition

• M 12/6 Discussion — Presentations of Putnam 2021 A problems

• W 12/8 Discussion — Presentations of Putnam 2021 B problems

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 (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).
• After each homework is due, you are allowed and encouraged to discuss your solutions on Piazza (link in Canvas).

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.
• Submission may 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 put your best foot forward.
• 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. Don’t be afraid to learn new things on the fly. “Fake it till you make it.”
• 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 S³ 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.
• Pset partners — a tool for finding problem set collaborators (MIT Touchstone required)

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