CZ-SK-MO 2019 Problem 3

Let $a,b,c,n$ be positive integers such that the following conditions hold (i) numbers $a,b,c,a+b+c$ are pairwise coprime, (ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$ -th power. Prove, that the product $abc$ can be expressed as a difference of two perfect $n$ -th powers.