I am organizing extracurricular problem-solving sessions for enthusiastic undergraduate and master's students at the School of Mathematics, University of Bristol, focused on competitive math problems. One the aims of the sessions is to select and train a team of students to represent the University of Bristol at the International Mathematics Competition for University Students IMC 2024 which will take place in the summer.
The training sessions will be ran by me, Safoura Zadeh, Oleksiy Klurman, Joseph Najnudel. The ultimate aim of the sessions is to enhance the students' problem-solving skills and mathematical thinking. We will mainly focus on presenting material and solving competition problems in the fields of Algebra (Linear and Abstract), Analysis (Real and Complex), Number Theory, Geometry and Combinatorics.
Should you have any questions, please contact me at besfort.shala@bristol.ac.uk.
Textbooks are a good place to start for competitions for university students. A solid understanding of the basic undergraduate material is often helpful and enough to solve the first one or two problems. One good book focused on competitive problems is Putnam and Beyond by Rǎzvan Gelca and Titu Andreescu.
Problems from previous competitions: see Putnam, IMC, IMO, and more generally the Art of Problem Solving forum (which should contain the preceding ones as well as much more).
Journals: see AMM, Mathematics Magazine, MathProblems, Mathematical Reflections (this last one tends to have easier problems in the undergraduate section, so it's a good place to start).
Session 1: Saturday, October 21st, 10:00-12:00, LG.02, Fry Building. Ran by Besfort Shala.
We started by discussing general tips related to problem-solving, including the following main four points: (i) getting comfortable with not being able to solve problems and spending a considerable amount of time attempting them anyway; (ii) reading solutions appropriately -- a line-by-line understanding of the arguments does not suffice, it is important to ask why all the time but understand that answering may sometimes be difficult; (iii) taking time to think about the big picture of what's going on, try summarizing solutions in a few sentences; (iv) making sure to spend a reasonable amount of time and effort on writing solutions up, do not stop once you are convinced that you've solved the problem.
In terms of math, we solved the first problems of each of day from IMC 2023 (see below) in order to get a feel for the flavor of problems in competitions. The first one was a functional equation solved by differentiating twice (as the problem statement is begging you to do) and then using a "descent" and continuity argument. A good book to learn analysis from is "Understanding Analysis" by Stephen Abbott, whereas a good reference book to refresh your memory is "Principles of Mathematical Analysis" by Walter Rudin.
The second problem was a linear algebra problem. The basic idea was to find an invariant, that is, a property of the starting matrix that does not change throughout the game and that the final matrix does not have. A natural candidate was the determinant, but due to the multiplicative nature of the game, we observed this is not immediately the right choice. However, after transforming multiplication to addition via the logarithm in order to have a "linear algebra-friendly" structure, we observed that the determinant indeed is an invariant which helped us solve the problem. You can read more about invariants and monovariants in this Brilliant.org brief article, or in the book "Problem-Solving Methods in Combinatorics: An Approach to Olympiad Problems" by Pablo Soberon. A good source to learn linear algebra is the book "Linear Algebra Done Right" by Sheldon Axler.
Session 2: Saturday, October 28th, 10:00-13:00, LG.02, Fry Building. Ran by Safoura Zadeh.
Problem Sheet 2 (to be used in the future as well).
Session 3: Saturday, November 4th, 10:00-13:00, LG.02, Fry Building. Ran by Joseph Najnudel.
The solutions of problems 1, 2 and 3 from IMC 2021 were covered during this session.
Session 4: Saturday, November 11th, 10:00-13:00, LG.02, Fry Building. Ran by Oleksiy Klurman.
In this session we did combinatorial problems, eventually ending with techniques on how to show that certain infinite series (such as the sum over primes p of 2^(-p)) converge to irrational numbers.
Session 5: Saturday, November 18th, 10:00-13:00, LG.02, Fry Building. Ran by Safoura Zadeh.
Session 6: Saturday, November 25th, 10:00-13:00, LG.02, Fry Building. Ran by Besfort Shala.
We did a rather quick overview of linear algebra, namely why would one care about diagonalizability and the relationship of the latter with eigenvalues. Then we covered the (real and complex) spectral and spectral mapping theorems. Using these, we gave two different solutions to the first problem from the problem sheet below. In the first solution, we had to work quite a bit to get ourselves in a situation to use the real spectral theorem. In the second solution, although we observed we can use the complex spectral theorem at once, there was some tension between real and complex which we were able to handle using the given condition that all eigenvalues are real.
A good source to learn linear algebra is the book "Linear Algebra Done Right" by Sheldon Axler.
There are three more problems you can have a go at using similar ideas in the problem sheet below.
Session 7: Saturday, December 2nd, 10:00-13:00, LG.02, Fry Building. Ran by Jeremy Rickard.
We looked at some old IMC problems that involved ideas from abstract algebra (specifically groups, rings and fields). Although not everybody had come across this material before, these problems mostly just involved basic definitions rather than any advanced theory, and so could be tackled after a brief introduction to the relevant concepts.
Session 8: Saturday, December 9th, 10:00-13:00, LG.02, Fry Building. Ran by Joseph Najnudel.
In this session we solved problems 1, 3 and 4 from IMC 2017.
Session 9: Saturday, January 27th, 10:00-13:00, LG.02, Fry Building. Ran by Joseph Najnudel.
In this session we solved problems 2 and 7 from IMC 2017.
Session 10: Saturday, February 3rd, 10:00-13:00, LG.02, Fry Building. Ran by Balint Toth.
In this session we solved and/or discussed problems 1, 3, 4, 6, 8 and 12 from the problem sheet below.
Session 11: Saturday, February 10th, 10:00-13:00, LG.02, Fry Building. Ran by Edward Crane.
In this session we solved and/or discussed problems from the problem sheet below.
Session 12: Saturday, February 17th, 10:00-13:00, LG.02, Fry Building. Ran by Safoura Zadeh.
In this session we solved and/or discussed problems from the first page of the problem sheet below.
Session 13: Saturday, February 24th, 10:00-13:00, LG.02, Fry Building. Ran by Besfort Shala.
In this session we discussed rank inequalities. In particular we proved Frobenius' and Sylvester's rank inequalities. These were easy corollaries of the rank-nullity theorem, but with a clever application of the "view an object in two different ways" idea. Afterwards we solved the third problem from the problem sheet below by considering when equality is achieved in Frobenius' inequality. Lastly we discussed about the fourth problem, recalling the spectral theorems, geometric thinking in linear algebra problems, and identifying a hidden group structure in the problem (which turns out to be a semidirect product).
TST: Saturday, March 2nd, 10:00-14:00, LG.02, Fry Building.