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
li
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
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 h
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 prese
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
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 probl
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.
Someti
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
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 integer
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
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
suc
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
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
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,
a
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 2
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 pro
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
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 inte
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
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
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 -
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