<s>[INST] Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$ Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$.
So far,
\begin{align*}
x^2 + y^2 + 121 + 2xy + 22x + 22y &= x^2 + y^2 + 121\\
2xy + 22x + 22y &= 0\\
(2x+22)y &= -22x\\
(x+11)y &= -11x
\end{align*}
At least 1 of $x,y$ must be a multiple of 11?
Dont know where to progress after this. All help appreciated.
 [/INST] Let $x=a-11,y=b-11$ in $xy+11x+11y=0$ to get
$$ab=121$$
Since the change of variables is completely reversible in the integers, every divisor of $121$ corresponds one-to-one to a solution: $a=\pm1,\pm11,\pm121$, correspinding to
$$(x,y)=(0,0),(-22,-22),(-10,110),(110,-10),(-12,-132),(-132,-12)$$
</s>