Graduate Exam Archive - Department of Mathematics and Statistics - University of Toledo
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For links to external exam archives, see the bottom of this page.
Disclaimer: Historical accuracy is not guaranteed here. Because of the changeover from quarters to semesters or because of isolated cases of unusual scheduling, the dates on some exams may disagree by a semester or so with their location on the grid. Also, exams may be preliminary versions and may differ somewhat from the exam that was actually written.
M.A Comprehensive Exam in Algebra, Fall 1993: tex | pdf
Ph.D. Comprehensive Exam in Complex Analysis, 1994: tex | pdf
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The Putnam Archive
The William Lowell Putnam Mathematics Competition is a North American math contest for college students, organized by the Mathematical Association of America (MAA). Each year on the first Saturday in December, several thousand US and Canadian students spend 6 hours (in two sittings) trying to solve 12 problems. Individual and team winners (and their schools, in the latter case) get some money and a few minutes of fame.
This web site contains problem statements, solutions, and competition results for recent years.
Problem statements given here are verbatim from the competition (except some diagrams are missing), and are copyrighted by the MAA. They appear here with permission, but further redistribution is subject to copyright restrictions which I am not authorized to waive; contact the MAA for assistance.
Solutions given here have been compiled (in some combination) by Manjul Bhargava, Kiran Kedlaya, and Lenhard Ng based on numerous sources (see below). Copyright is held by the named authors, who request that you link to this page in lieu of reproducing these solutions elsewhere. Also, please do not refer to these as "official solutions", as this describes the solutions issued by the MAA which reflect the intent of the problem setters. Those appear in the official competition summary (see below).
New problems and solutions will be posted here no sooner than the following Monday at 8pm Eastern (5pm Pacific); this embargo was set in consultation with competition director Daniel Ullman, in order to give exam supervisors time to mail their exam papers. We suggest that others observe a similar delay before discussing the exam online.
Competition results up to 2016 were transcribed by hand from the printed summary of results, which was sent by mail. As of the 2017 competition, both the original summary of results, taken from the MAA historical Putnam results page, and old-style HTML transcriptions are provided.
New results will be posted here as soon as they are received; this generally takes place in mid-February. (Until 2016, results were mailed in late March or early April.)
Some additional resources concerning the Putnam competition include the following.
- The MAA official Putnam home page includes information about the competition, including exam administration details; an archive of problems and solutions starting from 2017; and information about three published compilations of Putnam exams (a fourth is under development):
- The William Lowell Putnam Mathematical Competition. Problems and Solutions: 1938-1964, by Gleason, Greenwood and Kelly;
- The William Lowell Putnam Mathematical Competition. Problems and Solutions: 1965-1984, edited by Alexanderson, Klosinski and Larson;
- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, by Kedlaya, Poonen and Vakil.
- The (English) Wikipedia entry on the Putnam competition is quite lengthy and includes a lot of detail on historical results.
- The Art of Problem Solving discussion forums include numerous threads concerning recent Putnam competitions. Some discussion therein is reflected in the solutions posted here. (The long-time moderator of the Putnam discussions, Kent Merryfield, died unexpectedly on November 27, 2018.)
- Joe Gallian maintains several pages of information about the Putnam, including a historical summary, a database of career trajectories of Putnam fellows, and a PDF summary of the database.
- For each competition, the MAA has published an official competition summary with problems, solutions, results, and statistics in the American Mathematical Monthly a few months after the exam date. The Monthly back catalog is available via JSTOR if you are affiliated with a subscribing institution. (Another MAA publication, Mathematics Magazine, has also published exam summaries in recent years.)
- There is a rich literature of books about math competitions and preparation for them, but books focusing on the Putnam are rare. One book specifically about the Putnam is Putnam and Beyond by Andreescu and Gelca.
- A Conversation about the Putnam Competition.
This page is maintained by Kiran Kedlaya; contact me with corrections and other comments about this material only. I cannot assist with exam administration. If you are a student, see your school's Putnam coordinator; if you are a coordinator, see the MAA's Putnam home page for instructions.
The MAA has announced that the 81st competition, originally scheduled for Saturday, December 5, 2020, will be postponed because of the COVID-19 pandemic. Further details will be reported here as I receive them.
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