University of Massachusetts, Amherst
Fall 2019
MATH 491: Putnam competition preparation seminar

Description:
There are two mathematics competitions for our students:
Putnam (national)
and JacobCohenKilam (UMass).
This seminar is an introduction to problems at these competitions and to
some of the mathematical methods that may be useful.

Method:
Each week there will be a list of problems on this web page, that you are supposed to think about and then the
solutions will be presented in class at the next meeting
and the remaining problems will be discussed.
To make this useful it is essential that you think
of problems in advance. What is the problem about? How would you approach it, i.e., think about it? What are other possible points of view?

Grade formation:
It is based on attendance, participation and taking the Putnam exam.
The competition traditionally takes place the first Saturday in December.

The basic information:

Office :
1235I Lederle Graduate Tower.

Email :
mirkovic@math.umass.edu

Phone :
(413) 5456023.

Meet :
W
4:005:00 in LGRT
room ....

Resource :
A collection of problems from past competitions:
Putnam archive.

Homeworks
This is material to think about,
there is no need to submit anything.

H1. due Septemeber 11.

JacobCohenKilam Exam, 2018.
These are problems from the JCK competition in Spring 2018.
One of our goals is to solve all these problems in class over the semester.
So, at any time you should let me know if you have thoughts on any
of the remaining problems.

H2. due September 18th.
This is Putnam competition from 2017.
Choose a problem to think about and report on it in class.

H3. due September 25th.
This is Putnam competition from 2016.
Choose a problem to think about and to report on it in class.
Jenia Tevelev's
``general but surprisingly effective advice'':
Work in groups. Try small cases. Do examples.
Look for patterns. Use lots of paper. Talk it over.
Choose effective notation. Try the problem with different numbers.
Work backwards. Argue by contradiction. Modify the problem.
Generalize.
Don't give up after five minutes. Don't be afraid of a little algebra.
Sleep on it if need be. Ask.
HOW TO LEARN abstract MATHEMATICS.
The following is what I see as the {\em basic} approach
towards learning mathematics at the conceptual level.
The procedure is

(0)
You start by hearing (or reading) of a new idea, new procedure, new trick.

(1)
To make sense of it you check what it means in sufficiently many
examples. You discuss it with teachers and friends.

(2)
After you see enough examples you get to the point where you
think that you more or less get it. Now you attempt
the last (and critical)
step:

(3)
Retell this idea or procedure, theorem, proof or
whatever it is, to yourself in YOUR OWN words.

More on step (3).

Trying to memorize someone else's formulation,
is a beginning but it is far from what you really need.

You should get to the stage where you can
tell it as a story,
as if you are teaching someone else.

When you can do this, and your story makes sense
to you,
you are done. You own it now.

However, if at some point you find
a piece that does not make sense, then you have to
return to one of the earlier steps (13) above.
Repeat this process as many times as necessary.