We have primes $p\leq q\leq r$ such that $pqr(p+q+r)$ is a perfect square. Find $\max(p+q+r)$.
The only thing I've noticed is that all three can be the same. Let's say $pqr(p+q+r)=a^2$. Then if $p=q=r$, $a^2=3a^4$ but $3$ is not a square.
I think we might be able to show that two of them have to be the same (start from $p<q<r$, show a contradiction) but I'm not sure how to account for this. There's other cases that I think need to be considered, like $p=q<r$, but I'm not sure how to approach them.
I know this came from some sort of Australian contest, but I'm not sure which one. Possibly the AMC?