How to Solve It

Course Description and Grading Breakdown

The William Lowell Putnam Mathematics Competition is the most prestigious annual contest for college students. While the problems only employ machinery from the standard undergraduate curriculum, the ability to solve them requires a great deal of ingenuity. This can be developed through systematic and specific training, and it is this class' aim to assist the interested students in the preparation for the Putnam exam. Beyond the competition itself, problem solving is an essential skill in every discipline, so at the same time we will use the contest as an excuse to develop this ability through challenging (but fun) problems. These problems will generally come from mathematical competitions, but the syllabus will be constructed around carefully-selected topics which simultaneously develop problem solving techniques and inspire discussions about more advanced mathematics. Our aim is to use the competition problems to provide a tour through many interesting topics, and to expose a bridge to higher mathematics.   

Grading will be based on attendance (class and one hour weekly Putnam simulations) and class participation. Caltech students who are not officially enrolled are welcome to attend.

Course Meeting Time and Location

First meeting on Thursday, October 4th. 
7:00-8:55pm
187 Linde

Tuesday starting with October 9th.
7:00 - 8:55 pm
387 Linde


Course Instructor Contact Information

302 Linde

164 Linde

Weekly Putnam Simulations (Baby Putnams)

Mondays 7:00-8pm starting with 10/8.
Location 387 Linde

/Might involve snacks and refreshments!


Contest and Course Schedule

The Seventy Ninth Putnam Examination will be held on Saturday, December 1st, 2018. It will consist of two sessions of three hours each. Also, do not forget to register. You can do that by sending an e-mail to Sofia LeonThe deadline is Monday October 8th at noon

The rough schedule for the quarter below.

 DateTopic 
      10/04
   (Thursday)
 combinatorial ideas: extremal argument, induction (going back and making a   better removal choice), double counting isosceles triangles, linear   independence trick (odd town and an application to bounding the number of   matrices with a certain commutativity constraint) 
      10/9 algebraic inequalities: integration tricks, AM-GM, Cauchy-Schwarz,   Muirhead, several side notes on the baby Putnam exam
     10/16 convergence questions involving soft number theoretic ideas: sum of   reciprocals of numbers without 7 in base 10 converges, existence of   multiples which have all digits no matter the base choice, polynomials   that send numbers with 1's to numbers with 1's, a number theoretic   application which also used a convergent sequence of integers; there are   infinitely many prime numbers n such that \pi(n) divides n, some related analytic number theory
     10/23 functional equations (Cauchy, Jensen equations and how to reduce functional equations to them via substitutions; how to use injectivity, surjectivity), polynomial equations (two polynomials f,g are equal if they agree in > max{deg f, deg g} places and applications; tricks involving roots with multiplicities uing the derivative of a polynomial); f(x)^2+1 has an irreducibile factor of large degree
     10/30 further properties of polynomials; Lagrange interpolation, Chevalley-Warning, exploiting multiplicativity of certain structures, two proofs that the multiset {a_i+a_j: 1 \leq i < j \leq n} determines {a_i: 1 \leq i \leq n} whenever n is not a power of two by using 1) Newton sums and 2) a reduction to integer case via Dirichlet approximation + generating functions to solve integer case
     11/6
 further discussion on Newton sums and Viete relations; proof that tr(A^k) = 0 for every 1 \leq k \leq n implies A^n = 0; linear algebra tricks mostly related to determinants, Hamilton-Cayley and characteristic polynomial of a matrix, properties of rank
     11/13
 TBD
     11/20 TBD
     11/27 TBD
     12/01       (Saturday) Putnam exam



Course Policies

However, you are required to attend at least 8 of the 9 two-hour classes and at least 7 of the 8 Baby Putnams in order to qualify for a grade of at least A-. 

>= 6 classes, >= 5 Baby Putnams in order to qualify for at least B-.

>= 4 classes, >= 3 Baby Putnams in order to qualify for at least C-.


Assignments

The course this year will have no official homework. There will be optional problem sheets for which you can submit solutions (and which I'll happily read and give comments for).


Resources

I will send frequent e-mails with various reading recommendations based on the topics we will be discussing. In addition to these, you are also encouraged to consult:

Putnam archive by Kiran Kedlaya. This page contains all the problems and solutions from the recent Putnam exams.
Putnam and Beyond by Titu Andreescu and Razvan Gelca. This book is available electronically via the Caltech library.