Organizers: Professor Michael Saks and Professor Shubhangi Saraf

What is the smallest positive integers having exactly 100 divisors?

Suppose we have a matrix of distinct numbers and we sort each row in increasing order, then we sort each column in increasing order. Are the rows necessarily still in increasing order?

If we have a set S of points in n-dimensional space each having integer coordinates, but such that the midpoint between any two points of S does not have integer coordinates, what is the largest that S can be?

Determine (without using a calculator) which is larger: e to the power pi or pi to the power e?

This is a seminar in mathematical problem solving. It is aimed at undergraduate students who enjoy solving mathematical problems in a variety of areas, and want to strengthen their creative mathematical skills, and their skills at doing mathematical proofs.

One of the goals of this seminar is to help interested students prepare for the William Lowell Putnam Undergraduate Mathematics Competition , which is an annual national mathematics competition held every December. Any full-time undergraduate who does not yet have a college degree is eligible to participate in the exam. (However, you are free to participate in the seminar without taking the exam, and vice versa.)

Students who have taken the seminar previously may not register for it, but are very welcome to attend.

The meetings of the seminar will be a mixture of presentations by the instructors, group discussions of problems, and student presentations.

The seminar qualifies as an honors seminar for the
honors
track .

All students taking the seminar are expected to:

**Applying and Registering **
To apply for admission to the seminar, submit a
Special
Permission Application Form -- Honors Courses Only form
to the undergraduate office (Hill 303).

** Problem Sets **

- Problem Set 1 (miscellaneous problems)
- Problem Set 2 (miscellaneous problems)
- Problem Set 3 (pigeon-hole principle)
- Problem Set 4 (counting)
- Problem Set 5 (inequalities)
- Problem Set 6 (number theory)
- Problem Set 7 (analysis and geometry)
- Problem Set 8 (polynomials)