2019 team (left to right): Maria Monks Gillespie, Andre Leautaud, Luke Askew, Taylor Rogers, Sophia Ressler, Jenna Sickau, Henry Adams, Nikita Laurenov. Not pictured: Juanita Duque Rosero, Jingya Li, Naeem Moin, Naser Moin, James Rudd.

## What is the Putnam?

The Putnam is an annual mathematics competition for college students, and a fun way to expose yourself to new challenging math problems. The exam is typically held on the first Saturday of December, but this year it will be held on Feb 20, 2021, online. For more information, see the paragraph titled "81st Putnam Competition Update". Please sign up to take the Putnam exam and make sure to choose Colorado State University as your institution! Even though the Putnam is being run in "unofficial" mode (meaning no prizes), it will still be graded as usual and so we can see how many positive points we can earn as a team on this fun and challenging exam.Here is the Putnam Mathematical Competition website, and here is the Wikipedia page.

## What is the Putnam seminar?

We will be holding weekly practice sessions, during which we will try to solve interesting problems together. The emphasis will be on learning from each other (not on competition), so please stop by and share your ideas. It's definitely okay if you can't make some of the practice sessions.You are welcome to attend the practice sessions even if you can't make the exam, and you are welcome to take the exam even if you can't make the practice sessions.

In Fall 2020 we will be meeting approximately every-other-week on Mondays at 4:00pm, on Zoom. Our first meeting will be on Monday, August 31.

## How do I get involved?

Stop by one of the practice sessions! Alternatively, feel free to contact Henry Adams (henry [dot] adams [at] colostate [dot] edu) with questions, or to get added to our email list.

## Resources

Here are Putnam problems 1985-present and solutions 1995-present.Here are Putnam problems and solutions 1994-present.

Here are Putnam problems and solutions 1938-2003.

Here are Putnam problems and solutions 1980-2005.

Here are Putnam problems 1980-2010.

The book *Putnam and Beyond* by Gelca and Andreescu organizes problems by topic, presents examples, and contains practice problems. It is freely available to CSU students as a pdf.

## Coaches

- Henry Adams, 2015-present.
- Maria Monks Gillespie, 2019-present.
- Jae Hwang Lee, 2021-present.
- Juanita Duque Rosero, 2019-2020.
- Sean Walters, 2017-2018.
- Peter Muller, 2016-2017.
- Nabeel Moin, 2016-2017.

## Fall 2020 Schedule

Date |
Topic, problems discussed | |

Aug 31 | Putnam 2000 exam, A1 and A2 | |

Sep 14 | Putnam 2000 exam, A3 and B2 | |

Sep 28 | Putnam 2000 exam, A5 and B4 | |

Oct 12 | Proof writing: Putnam 2000 exam, A2, A5, B1, B2 | |

Nov 2 | Putnam 2003 exam, A1 (using bijections) and A2 (using AM-GM) | |

Nov 16 | Putnam 2003 exam, A3 | |

Dec 7 | Brainteasers and SET | |

Saturday, Feb 20, 2020 | Putnam exam! |

## Spring 2020 Schedule

Date |
Topic, problems discussed | |

Jan 27 | Putnam 2019 exam: B1 | |

Feb 3 | Putnam 2019 exam: A1, A2, A4 | |

Feb 10 | Andre Leautaud, Section 2.2 on polynomials | |

Feb 17 | Taylor Rogers, Section 1.5 on invariants and semi-invariants | |

Feb 24 | Colloquium | |

Mar 2 | Colloquium | |

Mar 9 | Jack Pfaffinger, Multivariable optimization and Lagrange multipliers | |

Apr 27 | Maria Monks Gillespie, Section 2.1.5 on the Arithmetic Mean - Geometric Mean Inequality.
The meeting will be on zoom. |

## Fall 2019 Schedule

Date |
Topic, problems discussed | |

Sep 9 | First meeting: Pigeonhole principle, and Putnam 2010 exam | |

Sep 23 | Putnam 2010 exam, A1 and B1 | |

Sep 30 | Putnam 1984 exam, A1 and B1 | |

Oct 7 | Generating functions | |

Oct 14 | Generating functions 2014 A1, 2015 A2 | |

Oct 21 | Generating functions plus more 2015 A2, 2008 A2 | |

Oct 28 | Problems sheet, 2001 A1, 2002 A1 | |

Nov 4 | Proof writing / sharing! | |

Nov 11 | Problem sheet, 1988 A1, 1988 B1, 2009 A1 | |

Nov 18 | Problem sheet, 2010 A2, 1991 A1, 1993 B2 | |

Dec 5 | Dinner at Henry's! | |

Saturday, Dec 7 | Putnam exam! |

## Spring 2019

In Spring 2019, instead of studying for the Putnam exam, we did a joint research project exploring the conjecture that the Vietoris-Rips simplicial complex of any set of points in the plane, at any scale parameter, is always homotopy equivalent to a wedge sum of spheres!## Fall 2018 Schedule

Date |
Topic, problems discussed | |

Aug 27 | Pigeonhole principle, and Putnam 2015 exam, B1 | |

Sep 10 | Sean Walters: Putnam 2015 exam, and pigeonhole principle | |

Sep 24 | Putnam 1997 exam, B2 | |

Oct 1 | Putnam 1997 exam, A4 and A5 | |

Oct 8 | Putnam 1994 exam, A2 | |

Oct 15 | Putnam 1994 exam, A1, B1, B2, B3 | |

Oct 29 | Putnam 1983 exam, B1 and part of A2 | |

Nov 12 | Putnam 1983 exam, A1 and A2 | |

Nov 26 | Sean Walters: Putnam 1990 and 2006 exams, A1 from 2006 | |

Dec 1 | Putnam exam! |

## Spring 2018 Schedule

Date |
Topic, problems discussed | |

Jan 29 | Putnam 2017 exam, B2 | |

Feb 5 | Putnam 2017 exam, B1 and part of A1 | |

Feb 12 | Will van Noordt, Section 3.4 on equations with functions as unknowns, #536 | |

Feb 19 | Jingya Li, Combined integral method | |

Feb 26 | Sheldon Deeny, Section 2.2.5 on irreducible polynomials, #182, 183 | |

Mar 5 | Sean Walters, Section 2.3.3 on the inverse of a matrix, #229 | |

Mar 26 | Henry Adams, Section 6.1.1 on set theory and combinatorics of sets, #823 | |

Apr 2 | Ethan Coldren, Section 6.1.3 on combinatorial geometry, #837, 841 | |

Apr 9 | No meeting - Henry has a visitor in town | |

Apr 16 | Dinner at Henry's! |

## Fall 2017 Schedule

Date |
Topic, problems discussed | |

Aug 28 | Putnam 1999 A1 and pigeonhole principle: "Prove that among any 5 integers there are always three with sum divisible by 3" | |

Sep 11 | Putnam 1999 A3 and pigeonhole principle: "At least 6 Coloradans with the same number of hairs on their head", and "At least two arrows within 1 meter" | |

Sep 18 | Putnam 1999 B1, "Probability two random triangles inscribed on the circle intersect", and "Probability two integers are relatively prime" | |

Oct 2 | Putnam 2011 B1 and "Given three points sampled uniformly at random from a circle centered at the origin, what is the probability that the origin is contained in the resulting triangle?" | |

Oct 9 | Putnam 2011 A4 and "Prove that (n-1)/(n+1)<cos(π/n) for all integers n≥4" | |

Oct 30 | Putnam 1986 B1, Putnam 1988 A1, Putnam 2007 B1 | |

Nov 6 | Putnam 1991 B1, Putnam 2009 A1, Putnam 2001 A1 | |

Nov 18 | Putnam 1988 B1, Putnam 1991 A1, Putnam 2008 A1, Putnam 2010 A1 | |

Nov 27 | Putnam 1992 B1, Putnam 2001 A2, Putnam 1997 A4 | |

Dec 2 | Putnam exam! |

## Spring 2017 Schedule

Date |
Topic, problems discussed | |

Jan 23 | Putnam 2016 exam, B1 and A4 | |

Jan 30 | Nabeel Moin, Section 3.2.1 on Taylor series | |

Feb 6 | Ethan Coldren, Section 2.4 on abstract algebra | |

Feb 13 | Sean Walters, Section 4.1.2 on the coordinate geometry of lines and circles | |

Feb 27 | Will van Noordt, Section 4.1.2 on factorization and divisibility, and Sylvester's theorem | |

Mar 6 | Dinner at Henry's! | |

Mar 20 | Bowen Li, sequences and series, including Section 3.1.4, and 2016 B6, and #360 on page 120 | |

Mar 27 | Bowen Li, sequences and series, #353 and #354 on page 119 | |

Apr 3 | Ethan Coldren, Section 6.2.2 on generating functions | |

Apr 17 | Section 6.2.2 on generating functions, #867, #870, #872 | |

Apr 17 | Putnam 2005 exam, B1 |

## Fall 2016 Schedule

Date |
Topic, problems discussed | |

Aug 24 | Putnam 2013, A1 and B1 | |

Aug 31 | The pigeonhole principle | |

Sep 7 | The pigeonhole principle, part 2 | |

Sep 14 | Ethan Coldren, Section 4.1.3 on conics & other planar curves, #604 | |

Sep 21 | Section 4.1.3 on conics & other planar curves, #606, #616 | |

Sep 28 | Peter Muller, sheet of problems, 2009 A1 | |

Oct 5 | Sheet of problems, 2015 A1, 2014 A1, 2012-A1, 2010 B2, 2012 A1 | |

Oct 12 | Putnam 1998, A2 and B1, and from prior weeks | |

Oct 19 | Putnam 1995 exam, A1 and B2 | |

Oct 26 | Putnam 2007 exam, B1 | |

Nov 2 | Nabeel Moin, Section 2.3.1 and 2.3.2 on matrices and determinants, #203, #205, #207 | |

Sunday Nov 6 | Practice exam! 2003 side A, and 2000 side A and B1 | |

Nov 16 | Discussed practice exam | |

Nov 16 | Peter Muller: Analysis section from Putnam and Beyond | |

Dec 6 | Putnam exam! |

## Spring 2016 Schedule

Date |
Topic, problems discussed | |

Feb 22 | Putnam 1996 exam, A1 | |

Feb 29 | Gavin Stewart: Section 1.1 on proofs by contradiction, #4, #8 | |

Mar 7 | Brooks Adams: Section 1.2 on mathematical induction | |

Mar 21 | Ethan Coldren: Section 6.1.5 on Ramsey theory, #854 | |

Mar 28 | #855, #24 | |

Apr 11 | #25, both combinatorially and by induction | |

Apr 18 | Putnam 2015 exam, A1, B1, B4, A6 | |

Apr 24 | Dinner at Henry's! |

## Fall 2015 Schedule

Date |
Topic, problems discussed | |

Oct 13 | Putnam 1984, A1, A2, B1, B2 | |

Oct 20 | Putnam 2010, A1, A2, B1 | |

Oct 27 | Putnam 2002, A1, A2, A4 | |

Nov 3 | Putnam 2008, A1, A2, B1, B2 | |

Nov 10 | Putnam 2009, B1, A2, A3, A5 | |

Nov 17 | Putnam 2004, A1, A2 | |

Nov 24 | Putnam 2001, A1, A2, B3 | |

Dec 5 | Putnam exam! | |

Dec 6 | Recap of 2015 exam |

2018 team (left to right): Henry Adams, Yuling Wu, Jingya Li, Yujia Chen, William van Noordt, Ethan Coldren, Andre Leautaud, Matt Beirne, Luke Askew, George Laird. Not pictured: Sean Walters.

2017 team (left to right): Zhi Li, Bowen Li, Sheldon Deeny, Nathan Larson, Will van Noordt, Amber Moin, Ethan Coldren. Not pictured: Henry Adams, Sean Walters.

2016 team (left to right): Zhi Li, Henry Adams, Bowen Li, Will van Noordt, Ethan Coldren, Brooks Adams, Joseph Gelfand, Colin Roberts, Doran Wood, Chase Ashby. Not pictured: Sean Walters, Amber Moin, Sid Katz, Peter Muller, Nabeel Moin.

2015 team (left to right): Henry Adams, Ethan Coldren, Gavin Stewart, Nabeel Moin, Amber Moin, Katy Sieviec. Not pictured: Sean Walters.