Integer solutions to $y=x^2+\frac{(1-x)^2}{(1+x)^2}$

Question

As part of another problem I've been trying to find the greatest integer solutions to $$y=x^2+\frac{(1-x)^2}{(1+x)^2}$$ but am getting very stuck... Would the fact that it asymptotes to $y=x^2$ help at all? Does this mean it won't pass through any integer coordinates after a certain point? How would I go about finding integer solutions and showing that my list is exhaustive/that I have found the greatest solution?

Answer 1

Since we have

$$ y=x^2+\frac{(1-x)^2}{(1+x)^2}=x^2+1-\frac{4x}{(x+1)^2} $$

and since

$$ \frac{4x}{(x+1)^2}\le 1 $$

for all $x$, equaling $1$ only when $x=1$, the largest integer solutions for both $x$ and $y$ are $(x,y)=(1,1)$.

Answer 2

$$y=x^2+\frac{(1-x)^2}{(1+x)^2}=x^2+\left(\frac{2}{1+x}-1\right)^2$$

If $y\in \Bbb Z$ then $1+x|2$ so $x\in\{0,1,-2,-3\}$.