Fall 2012

(640:491)

Seminar Info

Organizer: Swastik Kopparty (swastik.kopparty@rutgers.edu)

Office Hours: Wednesday 3:30-4:30 (Hill Center 432)

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.

One of the main 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.)

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 . 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 123456789 is divided by 91 (without using a supercomputer).

·         Show that in any sequence of n2+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?

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 6: assorted topics

·         September 13: the pigeonhole principle

·         September 20: 2-way counting & inclusions-exclusion

·         September 27: polynomials

·         October 4: number theory

·         October 11: inequalities

·         October 18: no class (makeup class on Tuesday, October 30)

·         October 25: analysis and geometry

·         October 30: makeup class (CANCELLED due to hurricane)

·         November 1: CANCELLED due to hurricane

·         November 8: power series

·         November 15: no class (makeup class after Thanksgiving)

·         Tuesday, November 20: (Thursday Schedule)

·         November 22: NO CLASS (Thanksgiving)

·         November 29: more practice problems

·        December 1: PUTNAM EXAM (10am – 6pm in Hill 705)

·         December 6: Putnam exam post-mortem