MathDB

(a) Prove that given constants

a,b

with

1<a<2<b

, there is no partition of the set of positive integers into two subsets

A_0

and

A_1

such that: if

j \in \{0,1\}

and

m,n

are in

A_j

, then either

n/m <a

n/m>b

. (b) Find all pairs of real numbers

(a,b)

with

1<a<2<b

for which the following property holds: there exists a partition of the set of positive integers into three subsets

A_0, A_1, A_2

such that if

j \in \{0,1,2\}

and

m,n

are in

A_j

, then either

n/m <a

n/m>b

Problem Statement