Web page for the Northwestern University Math Problem Solving Group.

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

- 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. - 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: **October 26, 2021**

- Announcements
- MathPSG Emailing List
- How the group works
- Proposed problems
- Problems from AMM
- Lists of problems
- Mathematical Results
- Solved problems
- Problem Solving Groups
- Other Links
- Bibliography
- Putnam Training

subscribe MathPSGwhereYour Name

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

Clicking on one of those links will fire up the mail program associated to your browser with the appropriate command for the listserver already typed in.
**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.

- Pick a problem not yet solved by the group, and for which the deadline for submission has not arrived yet.
- 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. - 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.

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

**Note**: If you are member of the university you can get
links to the proposed problems as follows:

- Access this page
**from inside the university network, or using the university VPN**. - Then, when requested, use the group authentication username and password. 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 (*) | |||
---|---|---|---|---|---|

- 2020 - | |||||

Mathematics Magazine, Vol. 92, No.4, Oct 2019 | March 1, 2020 | 2078 | |||

College Mathematics Journal, Vol. 50, No. 4, Sep 2019 | March 15, 2020 | 1157, 1159 | |||

Mathematics Magazine, Vol. 92, No.5, Dec 2019 | May 1, 2020 | 2082 | |||

College Mathematics Journal, Vol. 50, No. 5, Nov 2019 | May 15, 2020 | 1161 | |||

Mathematics Magazine, Vol. 93, No.1, Feb 2020 | July 1, 2020 | 2086 | |||

College Mathematics Journal, Vol. 51, No. 1, Jan 2020 | July 15, 2020 | 1167 | |||

College Mathematics Journal, Vol. 51, No. 2, Mar 2020 | July 15, 2020 | 1175 | |||

Mathematics Magazine, Vol. 93, No.2, Apr 2020 | September 1, 2020 | 2094 | |||

Mathematics Magazine, Vol. 93, No.3, May 2020 | November 1, 2020 | 2098 | |||

College Mathematics Journal, Vol. 51, No. 3, May 2020 | November 15, 2020 | 1176 | |||

- 2021 - | |||||

Mathematics Magazine, Vol. 93, No.4, Oct 2020 | March 1, 2021 | 2103 | |||

College Mathematics Journal, Vol. 51, No. 4, Sep 2020 | March 15, 2021 | 1183 | |||

Mathematics Magazine, Vol. 93, No.5, Dec 2020 | May 1, 2021 | 2106, 2107 | |||

College Mathematics Journal, Vol. 51, No. 5, Dec 2020 | May 15, 2021 | 1186 | |||

Mathematics Magazine, Vol. 94, No.1, Feb 2021 | July 1, 2021 | 2111 | |||

College Mathematics Journal, Vol. 52, No. 1, Jan 2021 | July 15, 2021 | 1192, 1194 | |||

Mathematics Magazine, Vol. 94, No.2, Apr 2021 | September 1, 2021 | 2118 | |||

College Mathematics Journal, Vol. 52, No. 2, Mar 2021 | September 15, 2021 | 1200 | |||

Mathematics Magazine, Vol. 94, No.3, Apr 2021 | November 1, 2021 | 2123 | |||

College Mathematics Journal, Vol. 52, No. 3, May 2021 | November 15, 2021 | 1203 | |||

College Mathematics Journal, Vol. 52, No. 3, May 2021 (corrections) | |||||

- 2022 - | |||||

Mathematics Magazine, Vol. 94, No.4, Oct 2021 | March 1, 2022 | 2128 | |||

College Mathematics Journal, Vol. 52, No. 4, Sep 2021 | March 15, 2022 | ||||

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

source | deadline | submitted solutions | |||
---|---|---|---|---|---|

- 2020 - | |||||

American Mathematical Monthly, Vol. 126, No. 08, Oct 2019 | February 29, 2020 | ||||

American Mathematical Monthly, Vol. 126, No. 09, Nov 2019 | March 31, 2020 | ||||

American Mathematical Monthly, Vol. 126, No. 10, Dec 2019 | April 30, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 1, Jan 2020 | May 31, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 2, Feb 2020 | June 30, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 3, Mar 2020 | July 31, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 4, Apr 2020 | August 31, 2020 | 12178 | |||

American Mathematical Monthly, Vol. 127, No. 5, May 2020 | September 30, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 6, Jun-Jul 2020 | October 31, 2020 | ||||

American Mathematical Monthly, Vol. 127, No. 7, Aug-Sep 2020 | December 31, 2020 | ||||

- 2021 - | |||||

American Mathematical Monthly, Vol. 127, No. 8, Oct 2020 | February 28, 2021 | ||||

American Mathematical Monthly, Vol. 127, No. 9, Nov 2020 | March 31, 2021 | ||||

American Mathematical Monthly, Vol. 127, No. 10, Dec 2020 | April 30, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 1, Jan 2021 | May 31, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 2, Feb 2021 | June 30, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 3, Mar 2021 | July 31, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 4, Apr 2021 | August 31, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 5, May 2021 | September 30, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 6, Jun-Jul 2021 | October 31, 2021 | ||||

American Mathematical Monthly, Vol. 128, No. 7, Aug-Sep 2021 | December 31, 2021 | ||||

- 2022 - | |||||

American Mathematical Monthly, Vol. 128, No. 8, Oct 2021 | February 28, 2022 |

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

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

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

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

- 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

- The Art of Problem Solving
- Wolfram MathWorld PlanetMath
- William Lowell Putnam Mathematical Competition (MAA site)
- William Lowell Putnam Mathematical Competition (SCU site)
- IBM Research - Ponder this

- 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

WWW Address: "https://www.math.northwestern.edu/~mlerma/problem_solving/"