Mathematical Problem Solving (Putnam Seminar)
Course facts
- Payment: Free, no certificate
- Credits: Non Credit
- Learning style: Self-paced, start anytime
- Resources: Instructor Insights; Problem Sets
- Before you start: There are no prerequisites. It is offered as one of MIT’s First-Year Advising Seminars.
MIT OpenCourseWare details
- As taught in: Fall 2018
- Department: Mathematics
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Topic-based problem sets and supplementary problems for the Putnam Competition, on inequalities, recurrences, probability and more.
Challenging competition mathematics takes practice across many areas, and the William Lowell Putnam Mathematical Competition asks for all of them. This seminar pairs a topic with a long problem set each week, on subjects such as recurrences, inequalities, probability, congruences and generating functions. Supplementary problems range over every topic, and past Putnam exams and student-created Putnam-style problems close the term.
What You Will Achieve
- Apply techniques for inequalities, recurrences and generating functions to hard problems
- Choose among methods from analysis, algebra and combinatorics for a new problem
- Present a solution clearly and find flaws or gaps in other arguments
- Work through past Putnam Competition exams under your own timing
Program Overview
Analysis and Probability
Techniques built around sums, limits and chance.
- Hidden independence and uniformity
- Sums and integrals
- Analysis and inequalities
- Probability
Algebra and Number Patterns
Algebraic tools for competition problems.
- Recurrences
- Linear algebra
- Abstract algebra
- Congruences
Combinatorics and the Putnam Exam
Counting methods, then the real exam problems.
- Combinatorial configurations
- Generating functions
- Solutions to past Putnam exams
- Student-created Putnam-style problems
Course Completion
Job Outlook
Problem solving at this level feeds into further study in mathematics and computer science, where many of the students in the class were headed.
Good for: Students who enjoy hard mathematical problems and want organized practice for the Putnam Competition.
Keep in mind: The published materials are problem sets without lectures or solutions listed, and the weekly student presentations and feedback happened in class. Entering the competition itself is up to you.
Bottom line: A generous bank of competition problems arranged by technique, ideal for self-driven practice.
Frequently Asked Questions
How many problems should I attempt each week?
Students were encouraged to work on as many as they liked and handed in their six best solutions, at least four of them from the topic problem set based on the lecture.
What are the supplementary problems?
Each week adds a list of supplementary problems ranging over all topics, not necessarily related to that week's lecture, numbered in sets that run through the term.
Are there topics beyond the weekly list?
Yes. Additional topics from previous editions of the course are published too, with problems on the Pigeonhole principle, integer part, roots of polynomials and limits.
Who wrote the problem sets?
Most were created by anonymous guest lecturers. Some are based on files by Prof. Richard Stanley from earlier versions of the course, and others are credited to named guest lecturers.
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