Problem Solving in Math (Math 43900) Fall 2013
Week two (September 3) solutions
Instructor: David Galvin
1. Let f (n) be the number of regions which are formed by n lines in the plane, where no two
lines are parallel and no three meet in a point (e.g. f (
Problem Solving in Math (Math 43900) Fall 2013
Week ve (September 24) problems another grab-bag
Instructor: David Galvin
I sent out a call for peoples favourite problems. This weeks set is a grab-bag from among those
that I received (if yours wasnt includ
Problem Solving in Math (Math 43900) Fall 2013
Week eight (October 15) problems a mock Putnam
Instructor: David Galvin
Think of this as a mock Putnam. The questions are intended to go from easier to harder, as
in the real Putnam papers.
Homework: One impo
Problem Solving in Math (Math 43900) Fall 2013
Week nine (October 29) problems Binomial coecients
Instructor: David Galvin
Binomial coecients crop up quite a lot in Putnam problems. This handout presents some ways
of thinking about them.
Introduction
The
Problem Solving in Math (Math 43900) Fall 2013
Week eleven (November 12) problems Polynomials
Instructor: David Galvin
These problem are all about polynomials, which come up in virtually every Putnam competition.
Things to know about polynomials
Fundamen
Problem Solving in Math (Math 43900) Fall 2013
Week seven (October 8) problems inequalities
Instructor: David Galvin
A list of some of the most important general inequalities to know
Many Putnam problem involve showing that a particular inequality between
Problem Solving in Math (Math 43900) Fall 2013
Week six (October 1) problems recurrences
Instructor: David Galvin
Denition of a recurrence relation
We met recurrences in the induction hand-out.
Sometimes we are either given a sequence of numbers via a rec
Problem Solving in Math (Math 43900) Fall 2013
Week two (September 3) problems induction
Instructor: David Galvin
Induction
Suppose that P (n) is an assertion about the natural number n. Induction is essentially the following:
if there is some a for which
Problem Solving in Math (Math 43900) Fall 2013
Week four (September 17) problems number theory
Instructor: David Galvin
Some useful principles/denitions from number theory
1. Divisibility: For integers a, b, a|b (a divides b) if there is an integer k with
Problem Solving in Math (Math 43900) Fall 2013
Week one (August 27) problems a grab-bag
Instructor: David Galvin
Most weeks handouts will be a themed collection of problems all involving inequalities of
one form or another, for example, or all involving l
Problem Solving in Math (Math 43900) Fall 2013
Week eleven (November 12) solutions
Instructor: David Galvin
1. For which real values of p and q are the roots of the polynomial x3 px2 + 11x q three
successive (consecutive) integers? Give the roots in these
Problem Solving in Math (Math 43900) Fall 2013
Week six (October 1) solutions
Instructor: David Galvin
A non-Putnam warm-up exercise
Using the trick of repeatedly dierentiating the identity
1
= 1 + x + x2 + . . . ,
1x
nd a nice expression for the coecient
Problem Solving in Math (Math 43900) Fall 2013
Week three (September 10) solutions
Instructor: David Galvin
1. (a) Show that among any n + 1 numbers selected from cfw_1, . . . , 2n, there must be two
(distinct) such that one divides the other.
Solution: S
Problem Solving in Math (Math 43900) Fall 2013
Week one (August 27) solutions
Instructor: David Galvin
1. A locker room has 100 lockers, numbered 1 to 100, all closed. I run through the locker room,
and open every locker. Then I run through the room, and
2013 UI MOCK PUTNAM EXAM
September 25, 2013, 5 pm 7 pm
Solutions
1. Let f (n) denote the n-th term in the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . . , obtained by writing one 1, two 2s, three 3s, four 4s, etc.
(a) Find, with proof, f (201
2010 U OF I MOCK PUTNAM EXAM
Solutions
1. (a) Given a set with n elements (where n is a positive integer), prove that exactly 2n1 of its subsets have an odd
number of elements.
(b) Determine, with proof, the number of 8 by 8 matrices in which each entry i
Problem Solving in Math (Math 43900) Fall 2013
Week four (September 17) solutions
Instructor: David Galvin
1. Let a and b be two integer for which a b is divisible by 3. Prove that a3 b3 is divisible by
9.
Solution: a3 b3 = (a b)(a2 + ab + b2 ). Since a b
Problem Solving in Math (Math 43900) Fall 2013
Week ve (September 24) solutions
Instructor: David Galvin
1. Let x be a real number such that x + 1/x is an integer. Prove that
xn +
1
xn
is also an integer for any positive number n.
Solution: Note that x =
Problem Solving in Math (Math 43900) Fall 2013
Week ten solutions
Instructor: David Galvin
1. A chocolate bar is made up of a rectangular m by n grid of small squares. Two players take
turns breaking up the bar. On a given turn, a player picks a rectangul
Problem Solving in Math (Math 43900) Fall 2013
Week nine (October 29) solutions
Instructor: David Galvin
Easy warm-up problems
1. Give a combinatorial proof of the upper summation identity (
n
m
m=k k
=
n+1
k+1
).
Solution: RHS is number of subsets of cfw
Problem Solving in Math (Math 43900) Fall 2013
Week seven (October 8) solutions
Instructor: David Galvin
Some warm-up problems
1. n! <
n+1 n
2
for n = 2, 3, 4, . . .
Solution: Use the geometric mean - arithmetic mean inequality, with (a1 , . . . , an ) =
Problem Solving in Math (Math 43900) Fall 2013
Week three (September 10) problems pigeonhole principle
Instructor: David Galvin
The pigeonhole principle
If n + 1 pigeons settle themselves into a roost that has only n pigeonholes, then there must be at
lea