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 tha
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 + . .
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 subs
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 o
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
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
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 dro
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 cir
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