Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.
Thanks, I did not notice that! (Hard to search for the problem because it's $ab+1$ instead of $uv+1$...)
I now found that this already appears in PEN as Problem A2. See here for a discussion of the problem from 2007.
Notice though that the problem from Germany 2021 is much harder because it did not tell the students that there are infinitely many solutions. Many students spend a lot of time trying to prove that there are only finitely many (or even no) solutions.