Let $m, n, p$ be positive integers such that $$ m+n+p-2\sqrt{mnp}=1$$
Prove that at least one of $m, n, p$ is a perfect square.
My attempt :
$ (m+n+p-1)^2=4mnp$
so $4mnp$ is a perfect square, i.e., $mnp$ is a perfect square.
Let $q$ be a prime such that $q\mid mnp$, so $q^2\mid mnp$.
Please suggest on how to proceed.
