IrMO 1991 Problem 10

Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties:

(a) $0$ is not in $S$ ;

(b) for each $s_1,s_2$ in $S$ , the rational number $s_1/s_2$ is in $S$ ;

(c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$ .

Prove that if $x$ belongs to $S$ , then there exists elements $y,z$ in $S$ such that $x=y+z$ .