University of Massachusetts, Amherst                                                                                                                                                                                     Fall 2019

MATH 491: Putnam competition preparation seminar

• Homeworks This is material to think about, there is no need to submit anything. H1. due Septemeber 11. Jacob-Cohen-Kilam 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) Re-tell 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 (1--3) above. Repeat this process as many times as necessary.