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: September 22, 2008

Index

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

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

Note: Downloading the lists of proposed problems requires authentication by password. I will distribute periodically the necessary username and password among the members of the Math Problem Solving Group. If you are a member of the group and have forgotten the password, let me know at MATHPSG-request@listserv.it.northwestern.edu.

	source		deadline		submitted solutions (*)
			- 2007 -
	Math Horizons, Vol. 14, No. 2, Nov 2006		January 10, 2007		206
	College Mathematics Journal, Vol. 37, No. 5, Nov 2006		February 15, 2007		836, 838
	Pi Mu Epsilon Journal, Vol. 12, No. 5, Fall 2006 (external link)		March 1, 2007		1139
	Mathematics Magazine, Vol. 79, No. 4, Oct 2006		March 1, 2007		1752, 1753, 1754
	College Mathematics Journal, Vol. 38, No. 1, Jan 2007		April 15, 2007		841, 842
	Mathematics Magazine, Vol. 79, No. 5, Dec 2006		May 1, 2007		1758
	Math Horizons, Vol. 14, No. 3, Feb 2007		June 10, 2007		S117, S118, 207, 208 (wrongly labeled S114, S115, 205 and 206 resp.)
	College Mathematics Journal, Vol. 38, No. 2, Mar 2007		June 15, 2007		847
	Mathematics Magazine, Vol. 80, No. 1, Feb 2007		July 1, 2007		1761
	Math Horizons, Vol. 14, No. 4, Apr 2007		August 10, 2007		S119, S120
	College Mathematics Journal, Vol. 38, No. 3, May 2007		August 15, 2007		851
	Mathematics Magazine, Vol. 80, No. 2, Apr 2007		September 1, 2007		1766, 1767
	Pi Mu Epsilon Journal, Vol. 12, No. 6, Spring 2007 (external link)		October 1, 2007		1147, 1150
	Harvard College Mathematics Review, Vol. 1, No. 1, Spring 2007 (external link)		November 1, 2007		S07-2
	Mathematics Magazine, Vol. 80, No. 3, Jun 2007		November 1, 2007		1773
	Math Horizons, Vol. 15, No. 1, Sep 2007		November 10, 2007		210, 212
	College Mathematics Journal, Vol. 38, No. 4, Sep 2007		December 15, 2007		856
			- 2008 -
	Math Horizons, Vol. 15, No. 2, Nov 2007		January 10, 2008		S121
	College Mathematics Journal, Vol. 38, No. 5, Nov 2007		February 15, 2008		861, 864
	Pi Mu Epsilon Journal, Vol. 12, No. 7, Fall 2007 (external link)		March 1, 2008		1159, 1167, 1168
	Mathematics Magazine, Vol. 80, No. 4, Oct 2007		March 1, 2008		1780
	Harvard College Mathematics Review, Vol. 1, No. 2, Fall 2007 (external link)		March 1, 2008		F07-4
	College Mathematics Journal, Vol. 39, No. 1, Jan 2008		April 15, 2008		866, 868, 870
	Mathematics Magazine, Vol. 80, No. 5, Dec 2007		May 1, 2008		1782, 1783
	Math Horizons, Vol. 15, No. 3, Feb 2008		June 10, 2008		214, 216
	College Mathematics Journal, Vol. 39, No. 2, Mar 2008		June 15, 2008		871, 872, 874, 875
	Mathematics Magazine, Vol. 81, No. 1, Feb 2008		July 1, 2008		1786, 1790
	Math Horizons, Vol. 15, No. 4, Apr 2008		August 10, 2008		S126, S127, S128
	College Mathematics Journal, Vol. 39, No. 3, May 2008		August 15, 2008		877, 880
	Mathematics Magazine, Vol. 81, No. 2, Apr 2008		September 1, 2008		1795
	Harvard College Mathematics Review, Vol. 2, No. 1, Spring 2008 (external link)		September 15, 2008		S08-3
	Pi Mu Epsilon Journal, Vol. 12, No. 8, Spring 2008 (external link)		October 1, 2008		1174, 1180
	Mathematics Magazine, Vol. 81, No. 3, Jun 2008		November 1, 2008
	Math Horizons, Vol. 16, No. 1, Sep 2008		November 10, 2008
	College Mathematics Journal, Vol. 39, No. 4, Sep 2008		December 15, 2008

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

• Past Proposed Problems

Problems for the Brave

(See note above about requiring username and password for downloading the lists of proposed problems.)

	source		deadline		submitted solutions
			- 2007 -
	American Mathematical Monthly, Vol. 113, No. 8, Oct 2006		February 28, 2007		11244, 11245
	American Mathematical Monthly, Vol. 113, No. 9, Nov 2006		March 31, 2007		11251
	American Mathematical Monthly, Vol. 113, No. 10, Dec 2006		April 30, 2007		11262, 11263
	American Mathematical Monthly, Vol. 114, No. 1, Jan 2007		May 31, 2007
	American Mathematical Monthly, Vol. 114, No. 2, Feb 2007		May 31, 2007
	American Mathematical Monthly, Vol. 114, No. 3, Mar 2007		June 30, 2007		11278
	American Mathematical Monthly, Vol. 114, No. 4, Apr 2007		July 31, 2007		11289
	American Mathematical Monthly, Vol. 114, No. 5, May 2007		August 31, 2007
	American Mathematical Monthly, Vol. 114, No. 6, Jun-Jul 2007		September 30, 2007		11301
	American Mathematical Monthly, Vol. 114, No. 7, Aug-Sep 2007		November 30, 2007		11306
			- 2008 -
	American Mathematical Monthly, Vol. 114, No. 8, Oct 2007		February 28th, 2008
	American Mathematical Monthly, Vol. 114, No. 9, Nov 2007		March 31th, 2008
	American Mathematical Monthly, Vol. 114, No. 10, Dec 2007		April 30th, 2008
	American Mathematical Monthly, Vol. 115, No. 1, Jan 2008		May 31st, 2008
	American Mathematical Monthly, Vol. 115, No. 2, Feb 2008		June 30th, 2008
	American Mathematical Monthly, Vol. 115, No. 3, Mar 2008		July 31th, 2008
	American Mathematical Monthly, Vol. 115, No. 4, Apr 2008		August 31th, 2008		11355
	American Mathematical Monthly, Vol. 115, No. 5, May 2008		September 30th, 2008
	American Mathematical Monthly, Vol. 115, No. 6, Jun-Jul 2008		October 31th, 2008
	American Mathematical Monthly, Vol. 115, No. 7, Aug-Sep 2008		December 31th, 2008		11376

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

• Past Proposed Problems from AMM

Lists of problems

Local links

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

External links

• William Lowell Putnam Mathematics Competition Directory (includes solutions from years 1995 and later.)
• Rusin's Putnam Exams

Mathematical Results

• Generating Functions
• Inequalities
• Mathematical Induction
• Modular Arithmetic
• Recurrence Relations
• Various Techniques

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
• Ternary Addition

Problem Solving Groups

• G.R.A.20 Problems Group (Universita' di Roma)

Other links

• Harvard College Mathematics Review (HCMR)
• Pi Mu Epsilon Journal - Problem Department
• Eric Weisstein's World of Mathematics
• PlanetMath
• William Lowell Putnam Mathematical Competition
• This Is Mega-Mathematics!
• Homepage of Hojoo Lee (contains various interesting documents with math problems and related topics.)

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