Lecture 5 (More Useful Tools) Invariants

The Use of Invariants  Let f be a function from X to X with a fixed point a  X, 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 1 Let k and n be positive integers. Show that the sum of the k th 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 2 Let a 1, a 2, …, a 2009, be an arrangement (permutation) of the numbers 1, 2, …, Show that (a 1  1)(a 2  2)…(a 2009  2009) is even.

Hint: When is a difference of two integers even? When is a product of n integers even?

Problem 3 Consider a set of 3074 integers with no prime factors larger than 30. Prove that there are 4 of these integers whose product is the 4 th power of an integer.

Hint: Any of these numbers must have the form: 2 a 1 3 a 2 5 a 3 7 a 4 11 a 5 13 a 6 17 a 7 19 a 8 23 a 9 29 a 10 What are the possible parities (even/odd) of the sequence a 1,…,a 10 ?

Problem 4 Is it possible to tile a 66  62 rectangles with 12  1 rectangles?

Hint: Color each of the small squares with one of six different colors.

Thank You for Coming Wafik Lotfallah