Undergraduate Mathematics Problem Solving Seminar

Fall 2016



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.)



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 0s 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 3rd point. Must all the points be collinear?

       Is the 50000th 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