, 2017

Solution:
Assume WLOG that the side lengths of the triangle are pairwise coprime. Then they can be written as $m^{2}-n^{2}$, $2 m n$, $m^{2}+n^{2}$ for some coprime integers $m$ and $n$ where $m>n$ and $m n$ is even. Then we obtain
$$\frac{P^{2}}{A}=\frac{4 m(m+n)}{n(m-n)}$$
But $n$, $m-n$, $m$, $m+n$ are all pairwise coprime so for this to be an integer we need $n(m-n) \mid 4$ and by checking each case we find that $(m, n)=(5,4)$ yields the maximum ratio of $45$.