Web page for the Northwestern University Math Problem Solving Group.

Goals

1. Acquire basic math problem solving skills.
2. Help in training future candidates for the yearly Putnam Competition.

Activities

1. Solve math problems proposed in magazines like Math Horizons, College Mathematics Journal and the Mathematics Magazine. Ideally we should solve at least one problem from each issue.
2. Discuss solutions to problems and study mathematical subjects related to problem solving.

Unless otherwise stated, the level of the problems and results shown here should be undergraduate.

Last Modified:

Index

• Announcements
• MathPSG Emailing List
  • Posting Guidelines
• How the group works
• Proposed problems
  • Past Proposed problems
• Lists of problems
• Mathematical Results
• Solved problems
• Problem Solving Groups
• Other Links
• Bibliography

Announcements

MathPSG Emailing List

subscribe MathPSG Your Name

Alternatively, please click in one of the following links, depending on whether you want to

• Subscribe MathPSG, or

• Unsubscribe MathPSG.

Note: Problems from magazines discussed and solved in the MathPSG emailing list may be submitted to the magazines in the name of the Northwestern University Math Problem Solving Group.

The MathPSG emailing list is moderated. Postings sent to the list will not be automatically forwarded to its members, they must await approval by the moderator.

• Posting Guidelines

How the group works

1. Pick a problem not yet solved by the group, and for which the deadline for submission has not arrived yet.
2. If you solve the problem, you can post your solution sending it by email to

   MathPSG@listserv.it.northwestern.edu

   or mailing it directly to the group coordinator. If you want to give a chance the other members of the group to think the problem too, instead of a complete solution you may post only some hints first.
3. Some discussion of the solution may follow. After the solution is considered correct, and if nobody has anything to add, the coordinator will write a final version and submit it to the magazine in behalf of the group. If there is some discussion, or more that one solution posted, the coordinator will try to compose a "combined solution" with the different contributions.

Proposed Problems

The problems can be found in the magazines (available by subscription), or in electronic access sites such as JSTOR.

(*) Color red means that the solution submitted by us was selected for publication.

• Past Proposed Problems

Lists of problems

Local links

• A few "old" problems (sporadically updated).
• "Easy" Problems from the Putnam Competition (periodically updated)

External links

• Putnam Archive (includes solutions from years 1995 and later.)

Mathematical Results

• Generating Functions
• Inequalities
• Mathematical Induction
• Modular Arithmetic
• Recurrence Relations
• Various Techniques - Summation by Parts, Euler-Maclaurin Summation Formula, Poisson Summation Formula.

Solved problems

• A binary operation on rationals
• Eventually constant modulo m
• John's PIN
• Josephus problem
• Sorted random numbers
• Split squares
• Sums and products of tangents squared (MathJax version)
• Ternary Addition

Other Links

• The Art of Problem Solving - AoPS Putnam Forum
• Wolfram MathWorld PlanetMath
• William Lowell Putnam Mathematical Competition (MAA site)
• William Lowell Putnam Mathematical Competition (SCU site)
• IBM Research - Ponder this
• Problems of the Miklós Schweitzer Memorial Competition

Bibliography

• G. Alexanderson et al.: The William Lowell Putnam Mathematical Competition problems and solutions: 1965-1984, The Mathematical Association of America, Washington, D.C., 1985.
• W. W. Rouse Ball and H.S.M. Coxeter: Mathematical Recreations and Essays (13th edition), Dover Publications, 1987.
• Benjamin Bold: Famous Problems of Geometry and How to Solve Them, Dover Publications, 1969.
• Heinrich Dorrie: 100 Great Problems of Elementary Mathematics, Dover Publications, 1989.
• Arthur Engel: Problem-Solving Strategies, Springer, 1998.
• Gelfand, Gerver, Kirillov, Konstantinov and Kushnirenko: Sequences, Combinations, Limits, Dover Publications, 1969.
• A.M. Gleason et al.: The William Lowell Putnam Mathematical Competition problems and solutions: 1938-1964, The Mathematical Association of America, Washington, D.C., 1980.
• Kedlaya, Poonen and Vakil: The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, The Mathematical Association of America, Washington, D.C., 2002.
• Kenneth Hardy and Kenneth S. Williams: The Green Book of Mathematical Problems, Dover Publications, Inc., 1997.
• Kenneth Hardy and Kenneth S. Williams: The Red Book of Mathematical Problems, Dover Publications, Inc., 1996.
• Frederick Mosteller: Fifty Challenging Problems in Probability with Solutions, Dover Publications, 1965.
• Donald J. Newman: A Problem Seminar, Springer, problem books in mathematics, 1982.
• G. Polya: How to Solve It, 2nd ed., Princeton University Press, 1957.
• Alfred S. Posamentier, Charles T. Salkind: Challenging Problems in Algebra, Dover Publications, 1988.
• Alfred S. Posamentier, Charles T. Salkind: Challenging Problems in Geometry, Dover Publications, 1988.
• D. O. Shklarsky, N. N. Chentzov, I. M. Yaglom: The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics, Dover Publications, 1993.
• Hugo Steinhaus: One Hundred Problems in Elementary Mathematics, Dover Publications, 1979.
• Wayne A. Wickelgren: How to Solve Mathematical Problems, Dover Publications, 1995.
• Herbert S. Wilf: Generatingfunctionology, online book, (download at http://www.math.upenn.edu/~wilf/DownldGF.html).
• Peter Winkler: Mathematical Puzzles: A Connoisseur's Collection, A K Peters, Ltd., 2004.
• A. M. Yaglom, I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions Vol. 1 & 2, Dover Publications, 1987.
• Paul Zeitz: The Art and Craft of Problem Solving, John Wiley & Sons, Inc., 1999.

Comments, suggestions...? email Miguel A. Lerma at: mlerma at math dot northwestern dot edu