nAalniaOMliO wrote: Prove that for any positive integer $n$, there exist pairwise distinct positive integers $a,b,c$, not equal to $n$, such that $ab+n, ac+n, bc+n$ are all perfect squares. For $n>2$, $(1, n^2-n, n^2+n+1)$ are suitable. At $n=2$, $(1,7,14)$ are suitable. At $n=1$, $(2,4,12)$ are suitable.

Any triple of the form $(1,a^2-n,(a+1)^2-n)$ works for large enough $a$. This works since $$[a^2-n][(a+1)^2-n]=(a^2+a-n)^2-n.$$We can also choose $a=n+1$ explicitly to obtain the triple $(1,n^2+n+1,n^2+3n+1)$.