Math 410-01, Problem solving techniques, Fall 2006
Instructor: Professor Junping Shi Phone: 221-2030  email:
Office: Jones Hall 122 Office Hour: TThF 2-3:00pm or by appointment
Course Description
Techniques of solving mathematical problems will be reviewed, and the seminar will also prepare the students for college level mathematical competitions, like Putnam Exam, and Virginia Tech competition. Parallel to a systematic review of knowledge on algebra, combinatorics, number theory, calculus, geometry and other topics, students will also solve problems from selected problem sets, present and discuss their solutions in seminar.
Meeting times and location
Monday 4:30-5:50pm, Small Hall 152

Plan of the seminar
The structure of each meeting is as follows: first 20-30 minutes, the instructor will give a presentation on a mathematical topic, including main concepts and theorems, problem solving techniques, and a few typical examples to illustrate the theory; next 50-60 minutes, the class will discuss a problem set assigned prior week. Each week one problem set consiting of 7-10 problems will be assigned.  Students should think about these problems before the class, and if you can solve some problems in the set, sketch your answer in notebooks, and try to present your answer in class. For each problem set, one problem will be assigned to each student, and he/she will lead the discussion of the problem, or present his/her solution.
We may have one or two take-home midterm "tests", similar to mathematical competitions. You can think these tests as mock competitions. You will be given a period of 4-5 days try to solve the problems. But the results of the tests has no effect on your grade. Our "midterm exam" is Virginia Tech competion in October 28, Saturday, and our "final exam" is Putnam exam in December 2, Saturday. Again, the results has no effect on your grade.
Your grade will mainly be determined by participation of discussions, and your presentations.
Resource for this seminar course is from (1) several books on this subject; and (2) internet. For the books, the instructor has now a collection of 6 books, and students in the seminar are welcomed to use these books for their need. For the internet resource, plese check the course website:
Textbooks and refererence books
There is no required textbook for the class. There are many good books on the subject of problem solving, and student can choose to pruchase one or more books for their interests. Recommended general problem solving books are
Problem Solving Through Problems  By: Loren C. Larson
Problem-Solving Strategies (Problem Books in Mathematics) By: Arthur Engel
Mathematical Olympiad Challenges By: Titu Andreescu and Razvan Gelca
Mathematical Miniatures (New Mathematical Library)  By: Svetoslav Savchev, Titu Andreescu
The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary By: Kiran S. Kedlaya, et al
The William Lowell Putnam Mathematical Competition  Problems and Solutions: 1965-1984 By: Gerald L. Alaxanderson, et al

Topics and Schedule

Week 1 (9/4)
Problem set 1 Pigeon hole principle handout
Week 2 (9/11)
Problem set 2
Mathematical induction
Week 3 (9/18)
Problem set 3
Series, and sum handout
Week 4 (9/25)
Problem set 4
Congruence handout
Week 5 (10/2)
Problem set 5 Recurrence handout
Week 6 (10/9)
Problem set 6
Inequalities handout
Week 7 (10/16)
no problem set, fall break
Floor function handout
Week 8 (10/23)
Problem set 7
Polynomial handout
Va Tech Competition
Week 9 (10/30)
Problem set 8
Probability handout
Week 10 (11/6)
Problem set 9
Binomial coefficient handout
Week 11 (11/13)
Problem set 10
Games, strategy, puzzle handout
Week 12 (11/20)
Problem set 11
Linear Algebra handout
Week 13 (11/27)

Putnam Exam
10am-1pm, 3-6pm
Week 14(12/4)
Review of Putnam problems