Question 34
Let $X$ be a set of rational numbers satisfying the following two conditions:
(a) The set $X$ contains at least two elements,
(b) For any $x, y$ in $X$, if $x \neq y$ then there exists $z$ in $X$ such that either $\left|\frac{x-z}{y-z}\right|=2$ or $\left|\frac{y-z}{x-z}\right|=2$.
Prove that $X$ contains infinitely many elements.
(a) The set $X$ contains at least two elements,
(b) For any $x, y$ in $X$, if $x \neq y$ then there exists $z$ in $X$ such that either $\left|\frac{x-z}{y-z}\right|=2$ or $\left|\frac{y-z}{x-z}\right|=2$.
Prove that $X$ contains infinitely many elements.