__Undergraduate Mathematics Problem
Solving Seminar__

__Fall 2013__

(640:491)

__Seminar Info__

Organizers: Lev Borisov (borisov@math.rutgers.edu), Swastik Kopparty (swastik.kopparty@gmail.com **(Note:
swastik.kopparty@rutgers.edu has been unreliable lately)**)

Meeting time and place: Thursday 1:40 – 3:00, Allison Road Classroom Building 212

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

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.

A secondary goal 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.)

CLICK HERE FOR INSTRUCTIONS ON HOW TO REGISTER FOR THE PUTNAM.

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

The seminar qualifies as an honors seminar for the honors track . Students who have taken the seminar previously may not register for it, but are very welcome to attend.

All students taking the seminar are expected to:

·
Attend regularly.

·
Participate actively in group
problem solving.

·
Take their turn presenting a problem
solution to the class.

·
Read text book chapter to be covered
in advance of the seminar meeting.

·
Work on some of the assigned
problems and turn in a carefully written solution for at least one problem per
week.

Some
appetizers:

·
When you multiply the numbers 1, 2,
3, …, 400, how many trailing 0’s does the answer have?

·
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?

·
Determine (without using a
calculator) which is larger: or ?

·
Find the remainder when 123^{456789}
is divided by 91 (without using a supercomputer).

·
Show that in any sequence of n^{2}+1
distinct integers, there is either an increasing
subsequence or a decreasing subsequence of size n+1.

·
Suppose you have a
n red points and n blue points in the plane. Can you pair up the red points
with the blue points (each red point is paired with one blue point) so that all
the line segments between the pairs are nonintersecting?

__Seminar Problem Sets__

Every week you should turn in a
complete formally-written solution (or clear explanation of your good faith
attempt) for at least ONE problem

·
September 5: miscellaneous
problems

·
September 12: the
pigeon-hole principle

·
September 19: polynomials

·
September 26: 2-way
counting and inclusion-exclusion

·
October 3: analysis
and geometry

·
October 10: inequalities

·
October 17: number
theory

·
October 24: miscellaneous

·
October 31: miscellaneous

·
November 7: miscellaneous

·
November 14: some old
Putnam problems