The 77th
Putnam Mathematical Competition

will be held
on Saturday, December 3

8-11 am and 1-4 pm in 10 Evans Hall.

Come at least 20 min early, that is no later than 7:40 am:

We'll have some paperwork to do.

Bring your student ID, pencils, and erasers.

## Fall-16. Math 191 (ccn 19054):
Putnam Workshop

** Instructor: ** Alexander Givental ** **

Meetings:
MWF 4-5 p.m. in 9 Evans

** Office hours:** MWF 2-4 p.m. in 701 Evans

This class will function as a problem-solving seminar intended, in the
first place, for those who plan to take the Putnam Exam, the US and Canada
math Olympiad for college students (which will be held on Saturday,
December 3, 2016). Our objectives will be to build participant's confidence
in solving
challenging math problems and, more importantly, to learn interesting
mathematics, using materials of various Mathematical Circles and Olympiads as
motivation.

** Useful links: **

Past Putnam Exams 1980--2010

Introduction to LaTeX --
Harvard Math Dept

LaTeX help at Art of Problem Solving website

** Final: ** In lieu of the final exam, your participation in the Putnam
Exam on Dec. 3, 2016 is required.

** Homework: ** is due weekly on Mondays in class.

** Grading:** Your best effort during the semester plus full participation in
the Putnam competition will be rewarded with an A.

Three rules:

Use of outside materials or collaboration are not prohibited, but each
instance must be explicitly acknowledged.

Failure to acknowledge one's use of
somebody else's ideas is commonly known as * academic plagiarism.*

As a means for developing clear writing style, you are allowed maximum
1/2-page per problem.

As a means for offsetting the burden of the previous rule, you are advised
to typeset your solutions in LaTeX (see the links above).

** HW1, due Monday, Aug. 29 **

** 1.** Find the maximal number of kings that can be placed on an 8x8 chess
board so that no two of them attack each other?

** 2.** 45 points lie on the line AB outside the segment [A,B]. Prove that
the sum of their distances to A differs from the sum of their distances to B.

** 3.** 51 points are scattered in the square of 1 meter x 1 meter.
Prove that one can place a square of size 20 cm x 20 cm in such a way that
it covers at least 3 of the points.

**4.** Prove that for every integer n, divisible by neither 2 nor 5,
there exists an integer m, whose decimal representation consists only of 1s,
which is divisible by n (e.g. for n=13, we have m=111111 divisible by 13).

** 5.** Prove that the connected components of the complement to a closed (possibly self-intersecting) polygonal line on the plane can be colored into 2 colors so that no two components of the same color share a border segment.

** HW2, due Wed., Sep. 7:**

** 1.** Prove that 11...11 (243 ones) is divisible by 243.

** 2.** Several squares are given. Prove that it is possible to cut them into pieces and arrange them to form a single large square.

** 3.** Prove that the sum of the squres of the binomial coefficients "n choose k" is equal to "2n choose n" (e.g. 1^2+3^2+3^2+1^2=6!/3!3!).

** 4.** Prove that (a+b)^p = a^p+b^p mod p, when if p is prime (and
a,b are arbitrary integers). Derive Fermat's Little Theorem: a^p = a mod p
for any integer a and prime p.

** 5. ** Let 1< d_1, d_2, ... , d_{12} < 12 be twelve numbers between 1 and 12.
Prove that there exist three of them which are side lengths of an acute
triangle.

** HW3, due Wed., Sep. 14:**

** 1.**
Let A be the sum of the digits of 4444^{4444} (4444 to the power 4444),
and let B be the sum of the digits of A. Find the sum of the digits of B.

** 2.** Prove that 300^{3000}-1 (300 to the power 3000, minus 1)
is divisible by 1001.

** 3.** Let p be prime other than 3. Prove that 11....1 (p ones) is not
divisible by p.

** 4.** Two players take turns taking one or two stones off a pile of
20 stones. The player taking the last stone wins.
Which player has a winning strategy?

** 5. ** A log is divided by 12 blue marks into 13 equal parts, by 16 red
marks into 17 equal parts, and then cut into 30 equal pieces. Show that each
piece except the first and last ones contains one red or blue mark.

Here is the problem to solve by and discuss on Friday:

Prove that there exist exactly 4 infinite-to-the-left decimal
sequences x such that x^2 =x.

In particular, the square of each integer which ends with k rightmost digits
of one of these sequences ends with the same k digits.

(Examples: 5^2=25, 6^2=36.)

** HW4, due Wed., Sep.21:** It is probably the right moment to start working on problems from past Putnam Competitions.

So, solve A2,A3,A4, B1,B5 from Putnam 1990 .

I don't recommend to look at the solutions probably available somewhere online, but if you do, don't forget to acknowledge the role of this in your hw.

** HW5, due Wed., Sep.28:** Solve A2,A3,B2,B4,B5 from Putnam 1991 .

** HW6, due Wed., Oct. 5:** Finish Mock Putnam problems.

** HW7, due Wed., Oct. 12:** In my linear algebra class, we've just finished the theme "Determinants".
So, let's examine Putnam problems on this subject. You can find the theory on pp. 73-90 at https://math.berkeley.edu/~giventh/la/la_16_ch2.pdf . For HW problems, visit http://www.math.toronto.edu/barbeau/putnamla.pdf and solve problems 2014-A-2, 2009-A-3, 2008-A-2, 1992-B-5, and 1986-B-6.

** HW8, due Wed., Oct. 12:** Solve A1,A2,A4,B2,B4,B6 from Putnam 1994.

** HW9, due Wed., Oct. 19:** Solve A1, A5, A6, B1, B2, B4
from Putnam 2009.

** HW10, due Wed., Nov. 2:** Solve A2, A3, A4, B1, B4, B6 from Putnam 2010.

** HW11, due Wed., Nov. 9:** Solve A3, A5, B2, B4, B5, B6 from Putnam 2012.

** HW12, due Wed., Nov. 16:** Solve A1, A3, A5, A6, B1, B2 from Putnam 2013.

** HW12, due Wed., Nov. 30:** Solve A1, A4, B2, B3 from Putnam 2014 and A1, A2, A3, A4, A6, B1, B4.
from Putnam 2015.