IMO 1986 — Shortlisted problems Question 5

(FRG 1) IMO1. The set $S=\{2,5,13\}$ has the property that for every $a,b\in S$, $a\ne b$, the number $ab-1$ is a perfect square. Show that for every positive integer $d$ not in $S$, the set $S\cup\{d\}$ does not have the above property.