Lecture 5 (More Useful Tools)

Invariants

The Use of Invariants

Let f be a function from X to X with a fixed point aX, i.e. f(a) = a.

Define the functions f n(x) recursively by:

f 1(x) = f(x), and f n+1(x) = f(f n(x)) Then a is also a fixed point for f n,

i.e. f n(a) = a

Problem 1Let k and n be positive integers. Show that the

sum of the kth power of 4n consecutive integers must be even.

Hint:When is a sum of n integers even?

When is a product of k integers even?

When is a power of an integer even?

Problem 2Let a1, a2, …, a2009, be an arrangement

(permutation) of the numbers 1, 2, …, 2009. Show that (a1 1)(a2 2)…(a2009 2009) is

even.

Hint:When is a difference of two integers even?

When is a product of n integers even?

Problem 3Consider a set of 3074 integers with no prime

factors larger than 30. Prove that there are 4 of these integers whose product is the 4th power of

an integer.

Hint:Any of these numbers must have the form:

2a13a25a37a411a513a617a719a823a929a10

What are the possible parities (even/odd) of the sequence a1,…,a10?

Problem 4Is it possible to tile a 6662 rectangles with 121

rectangles?

Hint:Color each of the small squares with one of six

different colors.

Thank You for ComingWafik Lotfallah