CMO 2013 Problem 3

Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality $$\max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td$$ holds for all (not necessarily distinct) $x,y,z\in X$ , all real numbers $a$ and all positive real numbers $d$ .