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Subset of the Rationals

Source: 1991 IrMO Paper 2 Problem 5

October 1, 2017

Problem Statement

Let Q\mathbb{Q} denote the set of rational numbers. A nonempty subset SS of Q\mathbb{Q} has the following properties:
(a) 00 is not in SS;
(b) for each s1,s2s_1,s_2 in SS, the rational number s1/s2s_1/s_2 is in SS;
(c) there exists a nonzero number q∈Q\Sq\in \mathbb{Q} \backslash S that has the property that every nonzero number in Q\S\mathbb{Q} \backslash S is of the form qsqs for some ss in SS.
Prove that if xx belongs to SS, then there exists elements y,zy,z in SS such that x=y+zx=y+z.
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