If $x+y^3,x^2+y^2,x^3+y$ are all integers, are $x,y$ both integers? 
Let $x,y$ be both real numbers. If $x+y^3,x^2+y^2,x^3+y$ are integers, are $x,y$ both integers?

This question begins with two real numbers while usual number theory tricks are based on the precondition that the variables are integers. Showing $x,y$ are algebraic numbers is easy by observing $x+((x^3+y)−x^3)^3$ is an integer, but how can algebraic numbers help?

Here's a hint (or at least some useful steps that are too long for a comment): Let $t=xy$. Note that
$$\mathbb Z\ni (x^3+y)(x+y^3)-(x^2+y^2)^2=t^3+t-2t^2=t(t-1)^2$$
and
\begin{align*}
\mathbb Z
&\ni \big[(x^3+y)(x^2+y^2)-(x+y^3)\big]\big[(x+y^3)(x^2+y^2)-(x^3+y)\big]\\
&=t(t-1)^2+(x^2+y^2)^2t(t^2+t-1).
\end{align*}
Therefore $t^3+t^2-t$ is rational (since $x^2+y^2\neq 0$ or we are done). Can you finish from here?
