EGMO 2018 Problem 6

(a) Prove that for every real number $t$ such that $0<t<\frac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \geq 0$ ), such that

$$|x-m y| \leq t y$$

(b) Determine whether for every real number $t$ such that $0<t<\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that

$$|x-m y|>t y$$

for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ). (Merlijn Staps, The Netherlands)