MathDB

Let

A

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

b_{1}

\cdots

b_{n}

and

c_{1}

\cdots

c_{n}

such that [*] for each

i

the set

b_{i}A+c_{i}=\{b_{i}a+c_{i}\vert a \in A \}

is a subset of

A

, [*] the sets

b_{i}A+c_{i}

and

b_{j}A+c_{j}

are disjoint whenever

i \neq j

. Prove that

\frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\le 1.

Problem Statement