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|x- z|/ |y - z |= 2 or |y - z|/ |x- z |= 2

Source: 2014 Saudi Arabia GMO TST III p4

July 26, 2020
rationalsetalgebra

Problem Statement

Let XX be a set of rational numbers satisfying the following two conditions: (a) The set XX contains at least two elements, (b) For any x,yx, y in XX, if x≠yx \ne y then there exists zz in XX such that either ∣x−zy−z∣=2\left| \frac{x- z}{y - z} \right|= 2 or ∣y−zx−z∣=2\left| \frac{y -z}{x - z} \right|= 2 . Prove that XX contains infinitely many elements.
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