18.A34 Mathematical Problem Solving (Putnam Seminar)

Fall 2017, MIT

Meetings: Tuesdays and Thursdays 11am–noon in 2-135

Instructor: Yufei Zhao (see website for contact info)

Course description

18.A34 is a freshman advising seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Much (but not all) of the seminar will be geared toward preparing for the Putnam Competition.

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, 2017. All students officially registered in the class are required to participate in the competition.

MIT students should sign up here by noon, Friday, September 29.

Grading. Based on homework and in-class presentations. Beyond grading for correctness, the grader may deduct points or grant bonus points based on the elegance of the solution and clarity of the writing. Class attendance is required. Please email me in advance if you cannot make it to class. Too many unexcused absences is cause for concern.

MIT students who are not officially enrolled are welcome to participate, but should not hand in homework.

Schedule

(Subject to change)

  T   R  
Sept 5 Registration day 7 Congruences
  12 Student presentations (HW1 due) 14 Recursive sequences
  19 Student presentations (HW2 due) 21 Probability
  26 Student presentations (HW3 due) 28 Linear algebra (guest lecturer: Alexey Balitskiy)
Oct 3 Student presentations (HW4 due) 5 Polynomials
  10 Columbus Day (no class) 12 Student presentations (HW5 due)
  17 Student presentations (HW6 due) 19 Generating functions
  24 Student presentations (HW7 due) 26 Abstract algebra
  31 Student presentations (HW8 due)    
Nov     2 Analysis
  7 Student presentations (HW9 due) 9 Inequalities
  14 Student presentations (HW10 due) 16 Independence and uniformity
  21 Student presentations (HW11 due) 23 Thanksgiving (no class)
  28 Putnam practice    
Dec     1 Putnam practice
        Putnam Competition: Sat Dec 2
  5 Putnam solutions 7 Putnam solutions
  12 Presentations of student-created problems    

Homework

Instructions. You should hand in carefully written solutions to exactly six problems each week, at least four from the topics problem set based on the lecture. Do not hand in supplementary problems rated strictly less than [2]; these are too easy.

You are encouraged to work on more problems for your own enjoyment, but they will not be graded. (Non-registered students should not hand in solutions.)

Late policy. Late homework will not be accepted.

Sources. At the top of each assignment, you must write either “Sources consulted: none” or a list of all sources consulted other official course material. Examples include: names of people you discussed solutions with (whether or not they are taking the class), books, Wikipedia and other websites. You should avoid looking up solutions to homework problems online (or offline); if you do, you must mention the precise source on your assignment.

Collaborations. You are encouraged to collaborate, but you must write up your homework independently. You should include names of your collaborators in your sources consulted.

Proper write-ups. You are expected to write your solutions at a level of detail and rigor appropriate for the Putnam Competition. Write in complete sentences. For model write-ups, see the Additional resources section below.

Please ask me if you have questions about homework.

(The ordering of the topics may be subject to change.)

# Due date Topic problem set Supplementary problems
1 T 9/12 Congruences Supplementary set #1
2 T 9/19 Recursive sequences Supplementary set #2
3 T 9/26 Probability Supplementary set #3
4 T 10/3 Linear algebra Supplementary set #4
5 R 10/12 Polynomials Supplementary set #5
6 T 10/17 Pigeonhole principle Supplementary set #6
7 T 10/24 Generating functions Supplementary set #7
8 T 10/31 Abstract algebra Supplementary set #8
9 T 11/7 Analysis Supplementary set #9
10 T 11/14 Inequalities Supplementary set #10
11 T 11/21 Independence and uniformity Supplementary set #11
12 Su 12/10 Create your own problem  

Stellar/Gradebook

Student Support Services (S3) and Student Disability Services

Additional resources

You may find the following optional resources helpful for additional preparation. The books have been placed on reserve at Hayden Library.

Previous Putnam problems and solutions

Additional books helpful for preparation