$\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}$ is an integer Let $x$ be a positive integer. Does there always exist integers $y,z$ such that the sum
$$\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}$$
is an integer, but none of the three terms of the sum is an integer?
For instance: Take $x=1$. Then we need the sum $$\frac{y}{z+1}+\frac{yz}{2}+\frac{z}{y+1}$$ to be an integer, with $y,z$ odd and $\frac{y}{z+1},\frac{z}{y+1}$ non-integers. We can take $y=z=3$.
For $x=2$ this becomes $\frac{2y}{z+1}+\frac{yz}{3}+\frac{2z}{y+1}$, and we can take $y=z=2$.
In general, if we take $y=z$ we have the sum $$\frac{2xy}{y+1}+\frac{y^2}{x+1}.$$

Let $y=z=2x(x+1)-1$. Then
$$
  \frac{xy}{z+1}=\frac{zx}{y+1}=x-\frac{1}{2(x+1)}
$$
and
$$
  \frac{yz}{x+1}=4x^2(x+1)-4x+\frac{1}{x+1}.
$$
Since $x+1\geq2$, neither of these are integers, but
$$
  \frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}=4x^2(x+1)-2x
$$
is an integer.
