MathDB

Let

A

be a non-empty set of integers with the following property: For each

a \in A

, there exist not necessarily distinct integers

b,c \in A

so that

a=b+c

.
(a) Proof that there are examples of sets

A

fulfilling above property that do not contain

0

as element.
(b) Proof that there exist

a_1,\ldots,a_r \in A

with

r \ge 1

and

a_1+\cdots+a_r=0

.
(c) Proof that there exist pairwise distinct

a_1,\ldots,a_r

with

r \ge 1

and

a_1+\cdots+a_r=0

Problem Statement