# 18.A34 Mathematical Problem Solving (Putnam Seminar)

**Fall 2019, MIT** (Link to the most current version of the course)

**Class meetings:** Mondays and Wednesdays 1–2pm in 4-149

**Instructor:** Yufei Zhao (see link for contact info)

Please include “18.A34” in the subject line of your emails

## Course description

18.A34 is a first-year undergraduate seminar on mathematical problem solving. One of the goals of the seminar is to prepare students for the Putnam Mathematical Competition in December. Each week, one meeting will be a lecture (often by a guest speaker) on some topic/technique, and the other meeting will be student presentations of homework problems, where there will be emphasis on developing good classroom presentation skills. This seminar is most suitable for students with some previous exposure to Mathematical Olympiads.

**William Lowell Putnam Mathematics Competition:** The Putnam Competition is an annual mathematics
contest for undergraduates in the USA and Canada. This year it will be held **Saturday, December 7, 2019**.

**All students officially registered in the class are required to participate in the Putnam competition.**MIT students should sign up here.

**Grading.** Based on homework and in-class presentations.

- Homework will be graded on correctness and presentation. Illegible or extremely sloppy write-ups are unacceptable.
- Students are expected to regularly present solutions in class. Class presentations will be critiqued on correctness and clarity.
**Class attendance is required.**Please notify me in advance if you cannot make it to class. Too many unexcused absences is cause for concern.

**Listeners.** MIT students who are not officially enrolled in the subject are welcome to sit in and participate, but should not hand in homework.

Student Support Services (S^{3}) and Student Disability Services

## Schedule

*Subject to change. See homework policy below*

Lecture date | Topic | Lecturer | HW due & presentations | Supplementary set |
---|---|---|---|---|

W 9/4 | Probability | Evan Chen | M 9/9 | #1 |

W 9/11 | Independence and uniformity | Yufei Zhao | M 9/16 | #2 |

W 9/18 | Abstract algebra | Zilin Jiang | M 9/23 | #3 |

W 9/25 | Inequalities | Mehtaab Sawhney | M 9/30 | #4 |

W 10/2 | Analysis | Ashwin Sah | M 10/7 | #5 |

W 10/9 | Generating functions (notes) | Benjamin Gunby | W 10/16 (HW due & new lec.) |
#6 |

W 10/16 | Congruences and divisibility | Junyao Peng | M 10/21 | #7 |

W 10/23 | Sums and integrals | Shengtong Zhang | M 10/28 | #8 |

W 10/30 | Recurrences | Hung-Hsun Yu | M 11/4 | #9 |

W 11/6 | Combinatorial configurations | Yuan Yao | W 11/13 |
#10 |

M 11/18 (extra presentations) |
||||

W 11/20 | Linear algebra | Allen Liu | M 11/25 | #11 |

W 11/27 (extra presentations) |

M 12/2: Students present solutions to Putnam 2018 exam

W 12/4: Students present solutions to Putnam 2017 exam

**Saturday 12/7: Putnam Competition**

M 12/9: Students present solutions to part A of Putnam 2019 exam

W 12/11: Students present solutions to part B of Putnam 2019 exam

## Homework

Each problem set contains a (sometimes long) list of problems. You are encouraged to work on as many as you like, but only hand in **your six best solutions** (please do not submit more than six), at least four from the topics problem set based on the lecture. Do not hand in supplementary problems rated strictly less than [2]; these are too easy. For multi-part problems, you may decide what counts as “one solution”, as long as it is reasonable (i.e., not too trivial).

Homework submissions are due at the end of class. Please type or write legibly. They will be graded similarly to the Putnam competition. (Non-registered students should not hand in solutions.)

*Sources.* It is **required** to write, at the top of each assignment, a list of collaborators and sources (people, books, websites, etc.). If no additional sources are consulted, you must write `Sources consulted: none`

. Failure to do this will result in an automatic 10% deduction. **Do not look up solutions to homework problems online (or offline)**.

*Collaborations.* You are strongly encouraged to work on the problems on your own, though reasonable collaboration is permitted. Everyone must write their solutions individually and acknowledge their collaborators as noted above.

**Stellar/Gradebook** (primarily used for recording grades)

## Additional resources

You may find the following optional resources helpful for additional preparation. The books can be found at Hayden Library.

**Previous Putnam problems and solutions**

- Putnam archive by Kedlaya
- The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions and Commentary by Kedlaya, Poonen and Vakil

**Additional books helpful for preparation**

- Problem-Solving Through Problems by Larson
- Putnam and Beyond by Gelca and Andreescu