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

Let $n$ be a positive integer such that $$n=1 \mod 4$$

Then the problem is about finding rational $x$ in function of $n$ such that $ \dfrac{3n+3x+n^{2}}{12}$ and $\dfrac{n(n+3)(3n+3x+n^{2})}{36x}$ are both strictely positive integers. Or at least how one can proves that such a $x$ exists.