Putnam Club Archive: Difference between revisions
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'''Join us on Wednesdays 5:00-6:30pm (from September 21) at Van Vleck 9th floor lounge for some challenging questions, pizzas and drinks!!!''' | '''Join us on Wednesdays 5:00-6:30pm (from September 21) at Van Vleck 9th floor lounge for some challenging questions, pizzas and drinks!!!''' | ||
<div style="text-align: center;"><span style="color:blue"><font size="3">[https://www.maa.org/math-competitions/putnam-competition The 83nd Putnam competition] will take place on Saturday, December 3, | <div style="text-align: center;"><span style="color:blue"><font size="3">[https://www.maa.org/math-competitions/putnam-competition The 83nd Putnam competition] will take place on Saturday, December 3, at Van Vleck B107 (9AM-Noon, 2PM-5PM). Make sure to register for the competition!!!</font></span> | ||
The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems. | The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems. | ||
Revision as of 17:29, 21 November 2022
Fall 2022
Join us on Wednesdays 5:00-6:30pm (from September 21) at Van Vleck 9th floor lounge for some challenging questions, pizzas and drinks!!!
The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems.
To sign up for Putnam club email announcements, please send an email to putnam-club+join@g-groups.wisc.edu (empty subject and body works).
| Meeting September 21, 2022
Botong Wang |
We will go over some recent Putnam problems and the Undergraduate math competition problems. |
| Meeting September 28, 2022
Botong Wang |
We will continue to discuss the problems in the sample set and the latest undergrad math competition. |
| Meeting October 5, 2022
Brian Lawrence |
We will discuss polynomials. |
| Meeting October 12, 2022
Brian Lawrence |
We will continue to work on the worksheet from last week. |
| Meeting October 19, 2022
Botong Wang |
We will compute integrals! The file is updated to include a very nice proof. |
| Meeting October 26, 2022
Botong Wang |
We continue to work on the above worksheet. |
| Meeting November 2, 2022
Paul Apisa |
We will discuss linear algebra techniques. |
| Meeting November 9, 2022
Paul Apisa |
We continue to work on the above worksheet. |
| Meeting November 16, 2022
Dima Arinkin |
Number theory! |
| Meeting November 30, 2022
Dima Arinkin |
|
| Meeting December 7, 2022
Brian Lawrence |
|
| Meeting December 14, 2022
Brian Lawrence |
Spring 2022
Join us on Wednesdays 5:00-6:30pm (from February 2) at Van Vleck 9th floor lounge for some challenging questions, pizzas and drinks!!!
If you would like to participate in the undergraduate competition, please register here. Registration is not required - it is only to help us with organization.
The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems.
To sign up for Putnam club email announcements, please send an email to putnam-club+join@g-groups.wisc.edu (empty subject and body works).
| Meeting February 2, 2022
Dima Arinkin |
Meeting at the 9th floor lounge Van Vleck. When: February 2, 2022 05:00 PM. |
| Meeting February 9, 2022
Dima Arinkin |
Functions and calculus |
| Meeting February 16, 2022
Tatyana Shcherbina |
We are going to discuss Analysis problems |
| Meeting February 23, 2022
Tatyana Shcherbina |
We are going to continue to discuss Analysis problems 2 |
| Meeting March 2, 2022
Botong Wang |
We are going to discuss Graph theory |
| Meeting March 9, 2022
Botong Wang |
We are going to discuss Geometry |
| Meeting March 23, 2022
Dima Arinkin |
Games |
| Meeting March 30, 2022
Dima Arinkin |
Generating functions and telescoping series |
| Meeting April 6, 2022
Mihaela Ifrim |
Competition problems |
| Meeting April 13, 2022
Tatyana Shcherbina |
We are going to discuss Complex numbers |
| Meeting April 20, 2022
Tatyana Shcherbina |
We are going to discuss Probability problems |
| Meeting April 27, 2022
Mihaela Ifrim |
Fall 2021
Join us on Wednesdays 5:00-6:30pm (from September 22) at Van Vleck 9th floor lounge for some challenging questions, pizzas and drinks!!!
The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems.
To sign up for Putnam club email announcements, please send an email to putnam-club+join@g-groups.wisc.edu (empty subject and body works).
| Meeting September 22, 2021
Botong Wang |
Meeting at the 9th floor lounge Van Vleck. When: Sep 22, 2021 05:00 PM.
In the first two lectures, we will take a look at some of the Putnam competition problems from 2020 and 2019. We plan to start with algebraic problems such as 2019 A1, A2, B1, B3, 2020 A1, A2, B1. Then continue to the analytic ones such as 2019 A3, A4, B2, 2020 A3. | |
| Meeting September 29th, 2021
Botong Wang
|
Continue the two sets of Putnam problems from last time. | |
| Meeting October 6th, 2021
Dima Arinkin
|
Polynomials. We'll look at the following problems (the last few problems are quite hard, and we probably won't get to them during the meeting). | |
| Meeting October 13th, 2021
Dima Arinkin
|
Continuing with the problems from last time. If you have the time, please look at the problems in advance, and then if you have ideas or questions, we would talk about it at the start. | |
| Meeting October 20th, 2021
Tatyana |
We are going to discuss number theory problems. The plan is to look on some of the following problems (we probably will not cover them all)
| |
| Meeting October 27th, 2021
Tatyana |
We will continue to discuss some number theory problems from the list | |
| Meeting November 3rd, 2021
Botong |
We are going to discuss Inequalities! | |
| Meeting November 10th, 2021
Botong |
We continue the discussion of inequalities | |
| Meeting November 17th, 2021
Mihaela Ifrim |
We started discussing Various problems from past competitions | |
| Meeting November 24th, 2021
|
Thanksgiving break | |
| Meeting December 1st, 2021
Mihaela Ifrim |
Working in the extra problems added to the Nov 17th pdf. Solutions to all the problems: Various problems from past competitions | |
| Meeting December 8th, 2021
Mihaela Ifrim |
Worked on the actual Putnam problems. Covered half of them.
| |
| Meeting December 15th, 2021
Mihaela Ifrim |
Worked on the remaining problems; we still have some left: A6 and B5. See you all after the Winter break!
|
Spring 2021
The Putnam Exam, offered by the Mathematical Association of America, is the premier American math competition for undergraduate students. In a regular year is given on the first Saturday in December. The exam consists of 12 problems, 6 in the 3 hour morning session and 6 in the 3-hour afternoon session. Each problem is worth 10 points, so the maximum score is 120. National winners usually get around 100 points. The median score is generally around 0-2 points. This is a difficult exam with many interesting and fun problems.
Due to the coronavirus crisis, the 81st Putnam Competition, originally scheduled for Fall 2020, has been postponed until Feb. 20, 2021. The competition was in an unofficial mode, with no proctors, no prizes, no awards, and no national recognition of high-scoring individuals or teams. The solution papers are uploaded by participants for grading. Scores will be reported back privately to the individual participants and the local supervisor of each institution will receive a report of the scores of the students for that institution.
We will not continue our Putnam club meetings after March 24. See you all in the fall!
| Meeting Feb 10, 2021
Botong Wang |
ZOOM MEETING: Hi there, You are invited to a Zoom meeting.
When: Feb 10, 2021 05:00 PM Central Time (US and Canada) Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJMrd-yhpjstHt34UK8YJ9_mZJ7NT_G3NLcM After registering, you will receive a confirmation email containing information about joining the meeting. The first two meetings will be presented by Botong. We will look at some of the most recent Putnam exams to help you prepare for the coming one. On Feb 10, we will go over some of the 2019 Putnam problems. | |
| Meeting Feb 17, 2021
Botong Wang
|
Zoom: Please join using the above link!
We will continue to discuss problems in the 2019 Putnam competition. We will also look at some of the 2018 Putnam problems.
| |
| Meeting Feb 24, 2021
Dima Arinkin
|
Zoom: Please join using the above link!
We will look at some of the problems from this year's Putnam (which took place last weekend). | |
| Meeting Mar 3, 2021
Dima Arinkin
|
Zoom: Please join using the above link!
Let's keep looking at the Putnam problems - some interesting ones left! | |
| Meeting Mar 10 and Mar 17, 2021
Botong Wang |
Zoom: Please join using the above link!
This week we will discuss several apects of combinatorics: graphs, set theory and geometric combinatorics. The worksheet is here. |
|
| Meeting Mar 24, 2021
Mihaela Ifrim |
Zoom: Please join using the above link!
This week we will go over problems in linear algebra and analysis. The worksheet is here. |
Fall 2020
Due to the coronavirus crisis, most students in the US and Canada are unable to return to campuses this fall. Therefore, the 81st Putnam Competition, originally scheduled for Fall 2020, will be postponed until February 20, 2021. If most students can return to campuses in the spring, the competition on that date can go forward in much the same form as in previous years. On the other hand, if most students are unable to return to campuses in the spring, the competition on that date will proceed in an unofficial mode, with no proctors, no prizes, no awards, but with solution papers submitted for grading by participants themselves and scores reported back privately to the individual participants.
| Meeting 16 SEPT 2020
Mihaela Ifrim |
ZOOM MEETING: Hi there, You are invited to a Zoom meeting.
When: Sep 16, 2020 05:00 PM Central Time (US and Canada) Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJApcumhrz4pEtJRe_o0WTGM26u2zTM8T6J5 After registering, you will receive a confirmation email containing information about joining the meeting. You will meet us all at some point in time:) First two meetings will be presented by Mihaela. We will each teach two consecutive meetings. UPDATE (09/17/2020): e met and all the documents (problems discussed and solutions given by you in class, are now shared with you via onedrive (app related to your outlook wisc account!)). I have also sent to you an invitation to use the witheboard attached to the same wisc account! But, do not feel discouraged in case you want to join later! You are always welcomed! Please check out the following book: Putnam and Beyond! We will use it throughout the meetings! The first list of problems is posted! Please email me if you want access to it! File:Putnam Problem Set 1.pdf
|
| Meeting 23 SEPT 2020
Mihaela Ifrim
|
Zoom: Please join using the above link!
We discussed the problems given in the Putnam_Problem_Set_1 and their solutions are now posted on the shared folder. In addition I have also gave you the following problems to think at. File:Putnam September 23 2020.pdf
|
| Meeting 30 SEPT 2020
Tatyana Shcherbyna |
ZOOM: Please consider joining !
The next meeting will cover NUMBER THEORY! Please see the attached list of problems!! Enjoy!!! File:Putnam nt1(1).pdf
|
| Meeting 7 OCT 2020
Tatyana Shcherbyna |
ZOOM: Please join! Here is the list of problems for the next meeting!!! File:Putnam nt2.pdf If you have solutions for the previous proposed set of problems, please email them so that we can compile a set of solutions. We will post them in our onedrive folder. If you are new, please email us and we will give access to it! |
| Meeting 14 OCT 2020
Mihaela Ifrim |
ZOOM: Please join! Here is the list of problems for the next meeting!!! File:Putnam-Problems-and-Theory-form-Oct-14th-2020-short.pdf A more detailed list is posted on onedrive! |
| Meeting 21 OCT 2020
Mihaela Ifrim |
ZOOM: Please join! We worked out some of the problem proposed on the previous meeting!!! File:Putnam-Problems-and-Theory-form-Oct-14th-2020-short.pdf Some solutions will be posted on onedrive! Reach out to us if you would like access to the solutions and you did not register yet! |
| Meeting 28 OCT 2020
Tatyana Shcherbina |
ZOOM: Please join! The next two meeting will cover Polynomials! Please see the attached list of problems!! Enjoy!!! File:Putnam pol1.pdf |
| Meeting 4 Nov 2020
Tatyana Shcherbina |
Here are the theory discussed on Oct 28 and hints to the problems File:Putnam pol1 hints.pdf and File:Polynomials theory.pdf |
| Meeting 4 Nov 2020
Tatyana Shcherbina |
Here are the solutions of Polynomials problems File:Putnam Oct28 sol.pdf |
| Meeting 11 Nov 2020
Dima Arinkin |
On ZOOM (the link is above). The topic for the next two meetings is Linear algebra. Here are the problems I plan to discuss, and here is a short summary (from NWU) of some common linear algebra techniques for math contests. |
| Meeting 18 Nov 2020
Dima Arinkin |
On ZOOM (the link is above). Continuing with the linear algebra: here are the problems (including some left-overs from the last time). |
| Meeting 2 Dec 2020
Botong Wang |
On ZOOM (the link is above). The topic for the next two meetings is limits of sequences (sections 3.1.3, 3.1.4, 3.1.5 in Putnam and Beyond). We will go over some basic theorems and discuss how to apply them. Here is the worksheet. |
Spring 2020
The Putnam Club will help you prepare for the exam by practicing on problems from previous years and other olympiad-style problems. The meeting time is 5pm on Wednesdays in VV B139.
The book we will mainly use as a guide in preparing our meetings is: "Putnam and Beyond" by Razvan Gelca and Titu Andreescu.
- February 5: Binomial coefficients and generating functions (Answers and hints) Botong
- February 19: Number theory (Answers and hints) Botong
- March 4 and 11: Inequalities ( Note comming up!) Kim
Fall 2019
The Putnam Club will help you prepare for the exam by practicing on problems from previous years and other olympiad-style problems. The meeting time is 5pm on Wednesdays in VV B139. The first meeting will be on the 25th of September in Van Vleck hall, room B139.
We will continue using the Piazza page from last semester for discussions. The book we will mainly use as a guide in preparing our meetings is: "Putnam and Beyond" by Razvan Gelca and Titu Andreescu.
- September 25: Introductory meeting Botong
- October 2: Integral inequalities Mihaela
- October 9: More about Integral inequalities (I will post notes on Wednesday morning and we will discuss more in class!) Mihaela
- October 16: ODE of the first order Chanwoo
- November 6: Number theory Dima
- November 13: ??
- November 20: Geometry ( Note comming up!) Mihaela
- November 27: No meeting! Thanksgiving!
- December 4: Last meeting of the semester! Please come and bring your friends too!! It will be fun! Mihaela
Spring 2019
The Putnam Club will help you prepare for the exam by practicing on problems from previous years and other Olympiad-style problems. The meeting time is 5pm on Wednesdays in VV B139. The first meeting will be on the 6th of February in Van Vleck hall, room B139.
! Important announcement: We will begin preparing you for the Putnam exam earlier this year. The material covered will be presented gradually. More details will be explained to you during your first meeting of this semester (Feb 6th). We put together a Piazza account that will help the participants to discuss and collaborate with their pairs. Here is the link you need to access in order to register for this "class": piazza.com/wisc/fall2018/putnam2018 . Our intervention on Piazza will be minimal (some of the instructors will, from time to time, visit the piazza questions and provide some help). Also, based on your requests, we have decided to structure our meetings in a way that will provide more insight on methods and certain tricks that are very often used in this type of math competitions. The book we will mainly use as a guide in preparing our meetings is: "Putnam and Beyond" by Razvan Gelca and Titu Andreescu.
- February 6: The basics by Botong
- February 13: Botong
- February 20: Alex Ordered Sets
- March 6th: Mihaela Algebra
- March 13: Mihaela
- March 27: Botong Matrices
If this material is completely new to you then read through the definitions in the first section and try the interspersed exercises which are direct applications of the definitions. If you are familiar with the basic material then review the problem solving strategies and the example problems which directly utilize the techniques. Finally, if you are a veteran, go ahead and jump right to the exercises!
- February 27: Alex: Review results from 2/20. Bring written solutions and/or be prepared to present your
- March 6th: Mihaela
- March 13: Mihaela
etc.
Fall 2018
The Putnam Club will help you prepare for the exam by practicing on problems from previous years and other olympiad-style problems. The meeting time is 5pm on Wednesdays in VV B139. The first meeting will be on the 26th of September in Van Vleck hall, room B139.
! Important announcement: We put together a Piazza account that will help the participants to discuss and collaborate with their pairs. Here is the link you need to access in order to register for this "class": piazza.com/wisc/fall2018/putnam2018 . Our intervention on Piazza will be minimal (some of the instructors will, from time to time, visit the piazza questions and provide some help). Also, based on your requests, we have decided to structure our meetings in a way that will provide more insight on methods and certain tricks that are very often used in this type of math competitions. The book we will mainly use as a guide in preparing our meetings is: "Putnam and Beyond" by Razvan Gelca and Titu Andreescu.
- September 26: topic Introductory meeting by Mihaela Ifrim. We covered only the first 3 problems. I encourage you to work out all the problems!
- October 3: Linear Algebra by George Craciun.
- October 10: Polynomials by Botong Wang.
- October 17: Sequences by Alex Hanhart.
- October 24: Convergence and Continuity by Mihaela Ifrim.
- October 27: Virginia Tech Math Contest: 9-11:30am in VV B115.
- October 31: Geometry: cartesian coordinates, complex coordinates, circles and conics by George Craciun.
- November 7: Combinatorics: Set theory and geometric combinatorics by Botong Wang.
- November 14: Techniques from Group Theory by Alex Hanhart.
- November 21: No meeting: Happy Thanksgiving!
- November 28: Number Theory by Mihaela Ifrim.
- December 1: Putnam Competition! Starts at 9am!!!! The competition will take place December 1st 2018 (Saturday December 1st). The competition is administered in two sessions (A and B) on the same day, December 1st! Session A will start at 9 am and it will end at 12pm, and Session B will start at 2pm and it will end at 5pm. You should arrive at least 10 minutes prior to each session. You should bring your own pencils and pens (blue or black ink are permitted). Number 2 pencils with erasers will be needed to complete the identification forms. Erasers are also permitted, but nothing else will not be allowed in the exam room. I plan on bringing 20 such no 2 pencils. The exam room is B239 which is a class room located in Van Vleck Hall, at the level B2. Thank you all for participating and see you all there! If you have friends that would like to take the exam please encourage them to do so.
Spring 2018
The Putnam Club does not meet in the spring, but we had the fourth annual UW Undergraduate Math Competition on April 24th, 2018, 5:30-8pm in VV B239.
Fall 2017
The Putnam Club will help you prepare for the exam by practicing on problems from previous years and other olympiad-style problems. The meeting time is 5pm on Wednesdays in VV B139.
- September 20: Introductory meeting by D.Arinkin
- September 27: Equations with functions as unknowns by M.Ifrim (by request: here is a solution to problem 6; problem 7 is problem B5 of 2016 Putnam exam; you can see the solution here).
- October 4: Inequalities by G.Craciun.
- October 11: Polynomials by D.Arinkin.
- October 18: Equations by M. Ifrim
- October 21: Virginia Tech Math Contest: 9-11:30am in VV B203.
- October 25: Review of this year's Virginia Tech Contest by G.Craciun.
- November 1: Functions and calculus by D.Arinkin.
- November 8: Past Competitions by M.Ifrim
- November 15: Recurrences by G.Craciun.
- November 22: No meeting: Happy Thanksgiving!
- November 29: Complex numbers by D.Arinkin.
- December 2: Putnam Exam in VVB115. Morning session: 9-12pm; Afternoon session: 2-5pm.
Spring 2016
The Putnam Club does not meet in the spring, but we had the third annual UW Undergraduate Math Competition on April 19th, 2017.
Fall 2016
- September 20: Introductory meeting
- September 27: Calculus and analysis
- October 4: Generating functions (by Vlad Matei)
- October 11: Review of last year's UW Math competition
- October 18: Functional equations
- October 22: Virginia Tech Math Contest
- October 25: Review of this year's VT contest
- November 1: Matrices (by Vlad Matei)
- November 15: Two algebra problems
- November 22: No meeting: Happy Thanksgiving!
- November 29: Assorted problems
- December 3: Putnam Exam: Morning session: 9am-noon, Afternoon session: 2-5pm in VV B135.
Spring 2016
The Putnam Club does not meet in the spring, but we had the second annual UW Undergraduate Math Competition on April 12th, 2016.
Fall 2015
.
- September 23rd: Introductory meeting
- September 30th: Pigeonhole principle
- October 7th: Review of 2015 UW math competition
- October 14th: Matrices and determinants
- October 21st: Virginia Tech practice
- October 24th: Virginia Tech Regional Mathematics Contest: 9-11:30 am
- October 28th: Review of the 2015 Virginia Tech contest.
- November 4th: Polynomials
- November 11th: Assorted problems
- November 18th: Assorted problems
- No meeting on November 25th
- December 2nd: TBA
- December 5th: Putnam competition: Morning session: 9am-12pm, afternoon session: 2-5pm in VV B115.
Spring 2015
The Putnam Club does not meet in the spring, but we had our first UW Undergraduate Math Competition!
Fall 2014
- September 17: Introductory meeting
- September 22: Coloring and pigeonhole principle
- October 1st: Went through HW problems from last time
- October 8th: Number theory
- October 15th: Games
- October 22nd: Problems from last year's Virginia Tech contest
- October 25th: Virginia Tech Regional Mathematics Contest
- October 29th: Review of this year's Virginia Tech contest
- November 5th: Inequalities and functional equations
- November 12th: Polynomials
- November 19th: Combinatorics
- December 3rd: Recursions
- December 6th: Putnam competition: Morning session: 9am-12pm, Afternoon session: 2pm-5pm in Van Vleck B119
- December 10th: Review of this year's Putnam
Fall 2013
- September 11: Introductory Meeting
- September 18: Assorted Problems (by Yihe Dong)
- September 25: Combinatorics
- October 2: Matrices and Linear Algebra
- October 9: Number Theory
- October 16: Functions and Calculus
- October 23: Polynomials
- October 26: Virginia Tech Regional Mathematics Contest
- October 30: Problems from this year's Virginia Tech contest
- November 6: Games
- November 13: Pigeonhole Principle
- November 20: Extreme Principle
- November 27: No meeting (Thanksgiving)
- December 4: TBA
- December 7: Putnam competition Morning session: 9am-12pm, afternoon session: 2-5pm in VV B239.
Fall 2012
- September 11: Introduction Problems
- September 18: Some Basic Techniques Problems
- September 25: Polynomials and Algebra Problems
- October 2: Number Theory Problems
- October 9: Calculus Problems
- October 16: Games and Algorithms Problems
- October 23: Combinatorics Problems
- October 30: Probability Problems
- November 6: Linear Algebra Problems
- November 13: Grab Bag Problems
- November 27: Grab Bag 2 Problems
Fall 2011
- September 21: Pigeonhole Principle (Brian Rice) Problems
- September 28: Introduction to Counting (Brian Rice) Problems
- October 5: Elementary Number Theory (Brian Rice) Problems, Problems (Hardcore)
- October 12: Polynomials (Brian Rice) Problems, Problems (Hardcore)
- October 19: A Grab Bag of Discrete Math (Brian Rice) Problems
- October 26: Calculus, Week 1 (Brian Rice) Problems
- November 2: Calculus, Week 2 (Brian Rice) Problems
- November 9: Linear and Abstract Algebra (Brian Rice) Problems
- November 16: Mock Putnam Problems, Solutions