Problem sets

In general, undergraduate math competitions are meant to reflect the material studied by a math major, so the best way to prepare for them is to take relevant classes. In some sense, taking the "gateway" math course is the initiation process for a math major; for students at UM - Dearborn this is the MATH 300 course whereas at many larger schools there is an official course or seminar specific to problem solving. After taking this intro course, a student may proceed to take all the other interesting and relevant math courses in analysis, algebra, geometry, topology, number theory, and perhaps others.

This process can take a long time, so becoming familiar with some of the most typical definitions, theorems, and concepts beforehand can help a lot with handling competitions (while also giving a taste for some things which you may not have seen before). We hope to prepare interested students by offering summaries and examples of these topics in our problem sets. A description of the sets for our workshop follow.

Set One: Induction and Probability

This set amounts to a brief overview of the material that would be seen in the Mathematical Language, Proof & Structure (MATH 300). Basically, what is required in a proof? Some emphasis is on the kinds of proofs especially commonly seen in math competitions.

Set Two: Counting and Combinatorics

The subject of combinatorics is treated a little bit in courses such as Discrete Mathematical Methods for Computer Engineers (MATH 276) or Probability (MATH 325), but it seems that many of the related topics are never seen together or only the same theorems are repeated everywhere. Here we look at some of the important ideas.

Set Three: Modular Arithmetic and Polynomials

Algebra involves more general study than just these two topics, but the notes in this set are similar to some things found in Elementary Number Theory (MATH 395) or First Course in Modern Algebra (MATH 412). We try to discuss some of the important ideas per usual but without introducing too many special definitions (like group, ring, or field which are important in abstract algebra but don't appear in competitions often).

Set Four: Probability

This set is an overview of the sorts of things which might appear in Probability (MATH 325) but weren't covered in Set Two, especially continuous distributions.

Set Five: Summation and Integration

This set improves on the tricks that occur in Calculus I, II, and III (MATH 115, 116, 215). It serves both as a summary of the wide array of important things to remember while also providing a glimpse at what isn't mentioned in those courses.

Set Six: Plane Geometry

Geometry spans a lot of material, but some ideas that aren't typically seen in High School turn out to be very useful. This set aims to survey some of the theorems that relate more to analytic geometry rather than Euclidean geometry, although some obscure theorems are mentioned to give an idea of what else one might want to know.

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