Problemset 03

Prepared by Chams Eddine Abdelali Derreche

Problem 1

For a positive integer denote by and the sum and product, respectively, of the digits of . Show that for each positive integer , there exist positive integers satisfying the following conditions:

(We let .)

Problem 2

We call an even positive integer nice if the set can be partitioned into two-element subsets, such that the sum of the elements in each subset is a power of . For example, is nice, because the set can be partitioned into subsets , , . Find the number of nice positive integers which are smaller than .

Problem 3

A permutation of the integers is called fresh if there exists no positive integer such that the first numbers in the permutation are in some order. Let be the number of fresh permutations of the integers .

Prove that for all .

For example, if , then the permutation is fresh, whereas the permutation is not.

Problem 4

Let be integers. Show that for every integer sequence one can choose non-negative integers , satisfying the following conditions:

• for each ,
• all the positive are distinct,
• the sums , , form a permutation of the first terms of a non-constant arithmetic progression.

Problem 5

Find the smallest positive integer for which there exists a colouring of the positive integers with colours and a function with the following two properties:

For all positive integers of the same colour,

There are positive integers such that

In a colouring of with colours, every integer is coloured in exactly one of the colours. In both and the positive integers are not necessarily distinct.