I recently came across this problem...and it surpassed my understanding of maths. It goes as follows:
Find the n nonzero natural number, such that it satisfies the expression:
$$[(x+y)^n-(x^n+y^n)]^2=n^2x^2y^2(x+y)^2(x^2+xy+y^2)^{n-3}$$
where $x,y \in \mathbb{R}$
I noticed we could do something with the $x^2+xy+y^2$ term, so I tried to turn that term into a cube:
$$(x^2+xy+y^2)^{n-3}=\frac{(x-y)^{n-3}(x^2+xy+y^2)^{n-3}}{(x-y)^{n-3}}=\frac{(x^3-y^3)^{n-3}}{(x-y)^{n-3}}$$
And this is the point where I can't go on with simplifying terms. I might try squaring the LHS, but it wouldn't lead me nowhere in my opinion.