I propose the following problem (Maybe it has a trivial solution):

Let $n$ be a positive integer such that $$n\equiv1 \pmod 4.$$

Then the problem is to find a rational $x$ as a function of $n$ such that 
$$ \dfrac{3n+3x+n^{2}}{12} \quad\text{and}\quad \dfrac{n(n+3)(3n+3x+n^{2})}{36x}$$
are both strictly positive integers. Or at least how one can proves that such a $x$ exists.