The William Lowell Putnam Mathematical Competition 2001
The William Lowell Putnam Mathematical Competition will take place on Saturday, December 1, 2001.
It is a challenging opportunity for you to test your mathematical
mettle. In the 2000 competition, 2818 individuals participated,
representing 434 colleges and universities across Canada and the U.S.
Stanford placed seventh, and many individuals did very well.
The competition emphasizes ingenuity rather than knowledge, so freshmen
are not at much of a disadvantage compared to seniors. Interest in or
experience with problem solving is a plus. Not just math majors
have done well; some recent winners have come from nearby disciplines,
including physics, computer science, and engineering.
Completely solving even one of the twelve problems is a
significant achievement, and in most years would place you well
above the median. (Keep in mind that the particpants are selfselected
from among the best in the continent.)
If you might like to try the Putnam, please let me know as soon as possible,
definitely by November 7. The preliminary list of competitors has
already been submitted, but I can send in changes as late as early November.
If you need to take the exam at a different time, please let me know
as soon as possible as well.
Links:
 The introductory poster.
 Meetings: We will have dinnertime problem solving practice
sessions this quarter (Oct. 9  Nov. 27): Tuesdays, 380383N, 67:30.
Practice problems will include some on the
topics discussed, and some others.
 Here
is the handout for the introductory meeting.
 October 9:
 topics: pigeonhole principle and induction
 notes and
practice problems.

Problem of the week: Find positive integers n and a_1, a_2, ..., a_n such that
a_1 + a_2 + ... + a_n = 1979
and the product a_1 a_2 ... a_n is as large as possible.
(Followup: what year did this problem appear on the Putnam?)
 October 16:
 topic: number theory.
 notes and problems.

link
to one of the "milliondollar problems", the BirchSwinnertonDyer
Conjecture.
 Problem of the week:
Write down the first
few powers of 2 in a column, and the first few powers of 5:
2 5
4 25
8 125
16 625
32 3125
Notice that the total number of digits in each row are 2, 3, 4, 5, 6!
Why is the number of digits in the nth row, n+1?
 October 23:
 topics: linear recursions (recurrence relations) and linear
recursive sequences; fancy linear algebra.
 problems.

Instead of a "Problem of the Week", here's a "Website
of the Week":
Rich Schwartz's "Lucy and Lily" game.
This may not work on some browsers. On my machine at home, it works on
internet explorer, but not on netscape. There's actually a huge amount of
cool math buried in this game, including links to Penrose tiles. Try
playing the "single game" to see how hard it is. Then try the double
game, which looks even harder. Finally, read the simple winning strategy
for the double game. (It's amazing that the double game ends up being
easier than the single game.)
 October 30:
 topics: generating functions; analysis on the real line.
 notes and problems.
 Problem of the week (found on the web; this one is an old chestnut). Followup once you've figured it out: Find
Fibonacci numbers in the puzzle. They
are somehow central to what's going on. Can you make a similar problem with "bigger" Fibonacci numbers?
 November 6
 November 13
 topics: invariants (presented by Vin da Silva), and how to
tackle a general Putnam problem.
 problems (including the 1988 and 1993 Putnams)
 Website of the week:
Why Mathematicians Now Care About their Hat Color.
(Problems: (i) Figure out the 3person strategy before
reading the entire article. (ii) Explain why there's
an n person strategy that's at least as successful as the n1
person strategy. (iii) Find a sevenperson strategy (very
hard, unless you've happened to have seen "Hamming Codes")
 November 20
 topics: the "hat problem" (see the website of last week),
and more practice.
 problems (including the 1988 and 1992 Putnams)
 November 27: No seminar. But here's a
handout with sixteen analysis problems and solutions.
 December 1: The Putnam. Part A: 811 am, Part B: 14 pm.
The solutions are available
here.
 The official Putnam website.
 Recommended reading: I really like Loren Larson's "Problem Solving Through
Problems"  definitely worth
owning. It's great preparation for the Putnam.
It's now on reserve at the math library, in Building 380. It is also great help trying many old problems;
a compilation of old Putnam problems and solutions, collected
by Kiran Kedlaya. Older Putnam problems have appeared in two books; I'll
add the precise reference here soon.
 Some more links are available through this good
UCSD site,
by Patrick Fitzsimmons.
Want more? Additional suggestions for others? Please
let me know.
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