<s>[INST] Determine all pairs of $(x,y)$ such that $\frac{x^2y+x+y}{xy^2+y+11}$ is an integer Here is a problem that I got:

Determine all pairs of positive integer $(x,y)$ such that $\dfrac{x^2y+x+y}{xy^2+y+11}$ is an integer.

My attempt:
First notice that $(11, 11)$ is a solution. And if $\dfrac{x^2y+x+y}{xy^2+y+11}$ is an integer, then $xy^2+y+11|x^2y+x+y$. Hence $x^2y-xy^2+x-11|xy^2+y+11$.
And then I'm stuck from here. Can anyone help me? Thanks.
 [/INST] From $\dfrac{x^2 y+x+y}{x y^2+y+11}=k$ we get
$x^2 y+x+y=k x y^2+k y+11 k\quad(*)$
from which we deduce that
$x^2 y=k x y^2$ and then $x=ky$
plug this in $(*)$
$k^2 y^3+k y+y=k^2 y^3+k y+11 k$
that is $y=11k$ and then $x=11k^2$
Thus $(11k^2;\;11k)$ for any $k\in\mathbb{N};\;k>0$ satisfy the given equation
There are three more solutions that do not enter in the previous formula
$(7;\;1)$ which gives $3$
$(10;\;2)$ which gives $4$
$(121;\;1)$ which gives $111$
BTW
plugging $(11k^2;\;11k)$  in the given equation gives $k$ as result
hope this helps
</s>