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: November 17, 2009

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 (*)
			- 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		1797
	Math Horizons, Vol. 16, No. 1, Sep 2008		November 10, 2008		220
	College Mathematics Journal, Vol. 39, No. 4, Sep 2008		December 15, 2008		883, 885
			- 2009 -
	Math Horizons, Vol. 16, No. 2, Nov 2008		January 10, 2009		223
	College Mathematics Journal, Vol. 39, No. 5, Nov 2008		February 15, 2009		887, 890
	Mathematics Magazine, Vol. 81, No. 4, Oct 2008		March 1, 2009		2002
	Pi Mu Epsilon Journal, Fall 2008 (external link)		March 1, 2009		1186, 1189
	College Mathematics Journal, Vol. 40, No. 1, Jan 2009		April 15, 2009		892
	Mathematics Magazine, Vol. 81, No. 5, Dec 2008		May 1, 2009		1810
	Math Horizons, Vol. 16, No. 3, Feb 2009 - (PNG-1) (PNG-2) - (PDF, color)		June 1, 2009		226
	College Mathematics Journal, Vol. 40, No. 2, Mar 2009		June 15, 2009		900
	Mathematics Magazine, Vol. 82, No. 1, Feb 2009		July 1, 2009		1815
	Math Horizons, Vol. 16, No. 4, Apr 2009		August 10, 2009		231
	College Mathematics Journal, Vol. 40, No. 3, May 2009		August 15, 2009		903
	Mathematics Magazine, Vol. 82, No. 2, Apr 2009		September 1, 2009		1816, 1818
	Pi Mu Epsilon Journal, Spring 2009 (external link)		October 1, 2009		1196, 1202
	Mathematics Magazine, Vol. 82, No. 3, June 2009		November 1, 2009		1822, 1825
	Math Horizons, Vol. 17, No. 1, Sep 2009		November 10, 2009		232
	College Mathematics Journal, Vol. 40, No. 4, Sep 2009		December 15, 2009		906
			- 2010 -
	Math Horizons, Vol. 17, No. 2, Nov 2009		January 10, 2009
	College Mathematics Journal, Vol. 40, No. 5, Nov 2009		February 15, 2010		912
	Mathematics Magazine, Vol. 82, No. 4, October 2009		March 1, 2010		1827

(*) 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
			- 2008 -
	American Mathematical Monthly, Vol. 114, No. 8, Oct 2007		February 28th, 2008
	American Mathematical Monthly, Vol. 114, No. 9, Nov 2007		March 31st, 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 31st, 2008
	American Mathematical Monthly, Vol. 115, No. 4, Apr 2008		August 31st, 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 31st, 2008
	American Mathematical Monthly, Vol. 115, No. 7, Aug-Sep 2008		December 31st, 2008		11376
			- 2009 -
	American Mathematical Monthly, Vol. 115, No. 8, Oct 2008		February 28th, 2009		11383
	American Mathematical Monthly, Vol. 115, No. 9, Nov 2008		March 31st, 2009		11395
	American Mathematical Monthly, Vol. 115, No. 10, Dec 2008		April 30th, 2009		11397, 11402
	American Mathematical Monthly, Vol. 116, No. 1, Jan 2009		May 31st, 2009
	American Mathematical Monthly, Vol. 116, No. 2, Feb 2009		June 30th, 2009
	American Mathematical Monthly, Vol. 116, No. 3, Mar 2009		July 31st, 2009
	American Mathematical Monthly, Vol. 116, No. 4, Apr 2009		August 31st, 2009
	American Mathematical Monthly, Vol. 116, No. 5, May 2009		September 30th, 2009
	American Mathematical Monthly, Vol. 116, No. 6, June-July 2009		October 31st, 2009
	American Mathematical Monthly, Vol. 116, No. 7, August-September 2009		December 31st, 2009
			- 2010 -
	American Mathematical Monthly, Vol. 116, No. 8, October 2009		February 28th, 2010
	American Mathematical Monthly, Vol. 116, No. 9, November 2009		March 31st, 2010

(*) 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!

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