MathDB

Let

A

be a non-empty set of positive integers. Suppose that there are positive integers

b_1,\ldots b_n

and

c_1,\ldots,c_n

such that - for each

i

the set

b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}

is a subset of

A

, and - the sets

b_iA+c_i

and

b_jA+c_j

are disjoint whenever

i\ne j

Prove that

{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.

Problem Statement