Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways. 
Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.($a^2+b^2$, is same $b^2+a^2$), $n \in \mathbb{N}.$

I have proved the above question which appeared in one of the Math-Olympiad. And I do know the solution. Sharing the question only because the question has a cute solution. 

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*Let $n = 4x^4$, we have:
$$n(n+1) = (4x^4)^2 + (2x^2)^2 = (4x^4-2x^2)^2 + (4x^3)^2$$

*Let $n = (u^2 + v^2)^2$, we have 
$$\begin{align}
n(n+1) = & ((u^2 + v^2)^2)^2 + (u^2+v^2)^2\\
       = & (u^4 - 2uv - v^4)^2 + (2uv^3-v^2+2u^3v+u^2)^2\\
       = & (u^4 + 2uv - v^4)^2 + (2uv^3+v^2+2u^3v-u^2)^2
\end{align}$$

*Let $n = (x+y)^2 + (2xy)^2$, we finally have an example that $n$ is not a square:
$$\begin{align}
n(n+1) = & (4x^2y^2+2xy+y^2-x^2)^2 + (4x^2y+y+x)^2\\
       = & (4x^2y^2+2xy-y^2+x^2)^2 + (4y^2x+y+x)^2
\end{align}$$

