18.A34 Mathematical Problem Solving (Putnam Seminar)

Fall 2023, MIT

[Dropbox (schedule & homework PDFs)] [Canvas]

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

Instructor: Prof. Yufei Zhao

Undergraduate Assistants (UA): Mark Saengrungkongka and Tomasz Slusarczyk

Emails:

• For quick questions, ask me after class
• Include both UAs in all class related communication, including everything homework related (submission, extensions, grading, etc.)
• Begin your email subject line with “[18.A34]”

Course description and policies

• A first-year undergraduate seminar. Seminar participants are selected through the First-year Advising Selection process. Unfortunately I cannot add additional students.
• 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 2, 2023.
  • All registered students will be required to participate in the Putnam competition. Sign up information will be announced.

Class format

• Lectures are open to all MIT students.
• Presentations/Discussions (i.e., non-lectures) are only open to registered class members.
  • Each class starts with a brief discussion, followed by student blackboard presentations of homework solutions
  • Active participation is expected.
  • Students should volunteer to present via a Google Form sent out in Canvas
  • 10-minute time limit per presentation. Feel free to skip routine calculations and details
• UA evening office hours
  • In small groups lead by a UA, students practice giving presentations and receive feedback from the UA
  • Attendance is required roughly once every two weeks for each student
  • See Canvas announcements
• Class attendance is required for registered students.
  • Please notify me in advance if you cannot make it to class (e.g., due to illness).
  • Too many unexcused absences is cause for concern and may lead to a non-passing grade.
• Non-registered MIT students are welcome to attend sessions below marked “lecture” but not other sessions

Grading

• Pass/Fail. Based on homework and participation.
• Homework will be graded on correctness and presentation.

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

Schedule and due dates

SS = Supplementary set

♡ = recommended problems

• W 9/6 Class introduction
• M 9/11 Discussion & Presentations. Due: Probability & SS1
• W 9/13 Lecture by Yufei Zhao
• M 9/18 Discussion & Presentations. Due: Hidden independence and uniformity & SS2
• W 9/20 Lecture by Mehtaab Sawhney
• M 9/25 Discussion & Presentations. Due: Inequalities & SS3
• W 9/27 Lecture by Mingyang Deng
• M 10/2 Discussion & Presentations. Due: Combinatorial configurations & SS4
• W 10/4 Lecture by Luke Robitaille
• M 10/9 No class: Indigenous Peoples Day
• W 10/11 Discussion & Presentations. Due: Congruences and divisibility & SS5
• M 10/16 Discussion & Presentations. Due: Generating functions & SS6
• W 10/18 Lecture by Dain Kim
• M 10/23 Discussion & Presentations. Due: Polynomials & SS7
• W 10/25 Lecture by Ashwin Sah
• M 10/30 Discussion & Presentations. Due: Analysis & SS8
• W 11/1 Lecture by Edward Wan
• M 11/6 Discussion & Presentations. Due: Sums and integrals (notes) & SS9
• W 11/8 Lecture by Papon Lapate
• M 11/13 Discussion & Presentations. Due: Abstract algebra & SS10
• W 11/15 Lecture by Allen Liu
• M 11/20 Discussion & Presentations. Due: Linear algebra & SS11
• W 11/22 Discussion & Presentations (on any assigned problem)
• M 11/27 Discussion & Presentations: Putnam 2021
• W 11/29 Discussion & Presentations: Putnam 2022
• Saturday 12/2 Putnam Competition
• M 12/4 Discussion & Presentations: Putnam 2023 A
• W 12/6 Discussion & Presentations: Putnam 2023 A
• M 12/11 Discussion & Presentations: Putnam 2023 B
• W 12/13 Discussion & Presentations: Putnam 2023 B

Past Putnam problems: 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). If you don’t know which ones to start, try the ones marked by ♡ first.
• At least two problems should come 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).
• 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
• State your sources at the top of each problem (even if you worked independently); see below
• Homework must be submitted on Gradescope (accessible from Canvas) by 1pm, before the beginning of the class meeting, preferably earlier.
• Submissions should be typed in LaTeX and submitted as PDF. Reach out to the UAs for help with LaTeX if needed.
• 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 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.

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.

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.

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 2001–-2016: Problems, Solutions, and Commentary by Kedlaya, Kane, Kane, and O’Dorney
• 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, 2019, 2020, 2021, 2022.