BR1F1SZ wrote: Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is not an integer. If both $x+\frac 1x$ and $x-\frac 1x$ are integers (and so $2x$ and $\frac 2x$ are integers too), we easily get $x=\pm 1$ which do not fit. So $x-\sqrt 2=n$ and $x^2+2\sqrt 2=m$ for some $m,n\in\mathbb Z$ which easily gives $n=-1$ And so $\boxed{x=\sqrt 2-1}$, which indeed fits