__Undergraduate Mathematics Problem
Solving Seminar__

__Fall 2016__

(640:491)

__Seminar Info__

Organizer: Swastik Kopparty (swastik.kopparty@gmail.com,
swastik@math.rutgers.edu)

Meeting time and place: Thursday 1:40 – 3:00, Hill 525

**Applying and Registering:** To apply for admission to the
seminar, submit an 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 you have a finite collection
of points on the plane, such that whenever you draw a line through any 2 of
them, that line passes through a 3^{rd} point. Must all the points be
collinear?

·
Is the 50000^{th} Fibonacci
number odd or even?

·
How many vectors in
3-dimensionsional space can you find so that the angle between any two is 90 degrees?

·
Can the product of 2 consecutive
integers be a perfect square?

·
Suppose you have 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 attempt) for at
least ONE problem

·
September 8: assorted
topics

·
September 15: the
pigeonhole principle

·
September 22: polynomials

·
September 29: two-way
counting and inclusion-exclusion

·
October 6: analysis
and geometry

·
October 13: inequalities

·
October 20: miscellaneous problems

·
October 27: miscellaneous problems

·
November 3: miscellaneous problems

·
November 10: tougher problems

·
November 17: tougher problems

·
TUESDAY November 22 (note
nonstandard day): tougher problems

·
December 1: putnam practice problems

·
December 8: putnam post mortem