# Math 191

## Advanced problem solving

Classes: MWF, 1-2 pm, Room 5 Evans Hall
Office hours: Tu 11:00 am - 12:00 noon, Th 12:00 noon -- 1:00 pm at the Student Learning Center, and by appt.

### Topics covered

• 08/26: Mathematical induction.
• 08/29: Generalization, specialization, analogy.
• 08/31: Euler's solution of the Basel problem.
• 09/02: Discussion of the 1st homework: counting integer points.
• 09/05: No class (Labor Day).
• 09/07: Discussion of the 1st homework: parameter-dependent problems.
• 09/09: Discussion of the 2nd homework: divisibility and congruences.
• 09/12: Discussion of the 2nd homework: generating functions.
• 09/14: Discussion of the 2nd homework: recurrence relations.
• 09/16: Discussion of the 3rd homework: infinite series, majorization.
• 09/19: Discussion of the 3rd homework: around Wilson's theorem.
• 09/21: Discussion of the 3rd homework: infinite products.
• 09/23: Discussion of the 3rd homework: parameters, small perturbations.
• 09/26: Discussion of the 4th homework: Fibonacci numbers.
• 09/28: Discussion of the 4th homework: combinatorial averaging.
• 09/30: Discussion of the 5th homework: integration by parts.
• 10/03: Discussion of the 5th homework: pigeonhole principle.
• 10/05: Discussion of the 5th homework: quadratic forms.
• 10/07: Mock Putnam test.
• 10/10: Discussion of the mock Putnam test: counting solutions to Diophantine equations.
• 10/12: Discussion of the 6th homework: yet another series for pi.
• 10/14: Discussion of the 7th homework: divided differences.
• 10/17: Discussion of the 7th homework: Riemann sums and integrals.
• 10/19: Discussion of the 7th homework: degree of approximation.
• 10/21: Discussion of the 7th homework: approximation and smoothness.
• 10/24: Discussion of the 8th homework: trigonometric polynomials.
• 10/26: Discussion of the 8th homework: the winding number.
• 10/28: Discussion of the 8th homework: zeros of entire functions.
• 10/31: Discussion of the 9th homework: double sums / integrals.
• 11/02: Discussion of the 9th homework: complex numbers as geometric tools.
• 11/04: Discussion of the 9th homework: inclusion-exclusion principle.

### Resources

• Books.
• Gleason, A. M.; Greenwood, R. E.; Kelly, L. M.; The William Lowell Putnam Mathematical Competition. Problems and solutions: 1938--1964. Mathematical Association of America, Washington, D.C., 1980. ISBN 0-88385-428-7
• The William Lowell Putnam mathematical competition. Problems and solutions: 1965--1984. Edited by Gerald L. Alexanderson, Leonard F. Klosinski and Loren C. Larson. Mathematical Association of America, Washington, DC, 1985. ISBN 0-88385-441-4
• Kedlaya, Kiran S.; Poonen, Bjorn; Vakil, Ravi; The William Lowell Putnam Mathematical Competition, 1985--2000. Problems, solutions, and commentary. MAA Problem Books Series. ISBN 0-88385-807-X
• Contests in higher mathematics. Miklos Schweitzer Competitions 1962--1991. Edited by Gabor J. Szekely. Problem Books in Mathematics. Springer-Verlag, New York, 1996. ISBN 0-387-94588-1
• Honsberger, Ross; Mathematical morsels. The Dolciani mathematical expositions, no.3. The Mathematical Association of America, 1978. ISBN 0-88385-303-5
• Shklarsky D. O.; Chentzov N. N.; Yaglom I. M.; The USSR Olympiad problem book. W. H. Freeman and Company, San Francisco and London, 1962.
• Journals.
• William Kahan, who has been teaching a similar class H90, has put together a large collection of notes and problems with solutions at his Math H90 website.
• International Mathematics Olympiads official website and an unofficial one with additional info and references.
• William Lowell Putnam Mathematics Competition Archive.

The instructor welcomes cooperation among students and the use of books. However, handing in homework that makes use of other people's work (be it from a fellow student, a book or paper, or whatever) without explicit acknowledgement is considered academic misconduct. Make sure to include proper acknowledgement when quoting anything directly or paraphrasing.

### Assignments

Homework is due at the beginning of class on the day specified. No late homework. If you cannot make it to class, find a way to submit your work before the deadline.

• Homework #1 due September 2nd, in PS and in PDF.
• Homework #2 due September 9th, in PS and in PDF.
• Homework #3 due September 16th, in PS and in PDF.
• Homework #4 due September 23rd, in PS and in PDF.
• Homework #5 due September 30th, in PS and in PDF.
• Homework #6 due October 7th, in PS and in PDF.
• Homework #7 due October 14th, in PS and in PDF.
• Homework #8 due October 21st, in PS and in PDF.
• Homework #9 due November 28th, in PS and in PDF.
• Homework #10 due November 4th, in PS and in PDF.
• Homework #11 due November 14th, in PS and in PDF.
• Homework #12 due November 28th, in PS and in PDF.

### Misc. handouts

• Mathematical induction, in PS and in PDF.
• Integers and primes, in PS and in PDF.
• Congruence, in PS and in PDF.
• Rational and irrational numbers, in PS and in PDF.
• Complex numbers, in PS and in PDF.
• Progressions and sums, in PS and in PDF.
• Diophantine equations, in PS and in PDF.
• Quadratic reciprocity, in PS and in PDF.
• Polynomials, in PS and in PDF.
• Rolle's theorem and Descartes' rule of signs, in PS and in PDF.
• Polynomial equations and symmetric functions, in PS and in PDF.
• Classical inequalities, in PS and in PDF.
• Complex analysis in a nutshell, in PS and in PDF.
• Linear algebra in a nutshell, in PS and in PDF.
• Inclusion-exclusion principle, in PS and in PDF.
• Pigeonhole principle, in PS and in PDF.
• Generating functions, in PS and in PDF.