MATH 642:491Fall, 2012
Problems on power series
1. Use the technique of generating functions to solve the recurrence relation a0 = 1, a1 =
0, a2 = 5 and for n > 3,
an = 4an1 5an2 + 2an3 .
2. Prove that the value of the nth derivative of x3 /(x2 1) at x =
MATH 642:491Fall, 2012
Problems on polynomials
1. Suppose P (X ) C[X ] is such that for every x R, P (x) R. Show that all the
coecients of P (X ) are real numbers.
2. Let P (X ) = X r + a1 X r1 + . . . + ar1 X + ar be a polynomial with complex coecients
s
MATH 642:491Fall, 2012
The Pigeon-Hole Principle
(Note: The book has solutions to some of these problemsdont read them!)
1. Let A be a subset of integers of size n. Prove that there is a nonempty subset of A
whose sum is divisible by n.
2. Given 19 distin
MATH 642:491Fall, 2012
Problems in elementary number theory
(1) Prove that any two successive Fibonacci numbers are relatively prime.
(Two integers are relatively prime if they have no common factor other
than 1).
(2) Show that the tens digit of any power
Linear Diophantine Equations
A Diophantine equation is an equation which is to be solved over the integers.
A linear Diophantine equation of the form
has solutions if and only
if
. There is a similar result for linear Diophantine equations in more
than 2
MATH 642:491Fall, 2012
Problems
1. How many 0s does 400! end with?
2. Find the smallest positive integer having exactly 100 positive divisors.
3. You have a 7 gallon jug and a 10 gallon hat (both the jug and the hat have no markings).
You have access to a
MATH 642:491Fall, 2012
Problems on Analysis and Geometry
1. Suppose f : R R is a continuous function such that for every a, b, if
f a+b . Show that f (x) = x + for some , .
2
f (a)+f (b)
2
=
If we drop the condition that f is continous, does the same conc
MATH 642:491Fall, 2012
Problems
1. Let a, b, c be real numbers in (0, 1). Suppose a + b + c = 2. Show that
abc
8.
(1 a)(1 b)(1 c)
2. Let A1 , . . . , An be a regular n-gon which is inscribed in a circle of radius 1.
Show that
n
|A1 Aj | = n.
j =2
(Hint:
MATH 642:491Fall, 2012
Problems on 2-way counting and inclusion-exclusion
1. A frog hops on the number line from 0 to n. If he is on position i, then he can either
hop to i + 1 or to i + 2. Let Fn be the number of ways he can get from 0 to n. Show
that Fn