What relationship exists between the variables $x$ and $y$ if they depend on the parameter $z$ as follows:

$$
x=\frac{z^{4}+z^{3}+z^{2}+z+1}{z^{2}+1}, \quad y=\frac{z^{3}+z^{2}+z+1}{z^{2}+1}
$$
Using the second relation, we can easily eliminate $z$. Indeed, by factoring out,

$$
y=\frac{\left(z^{2}+1\right)(z+1)}{z^{2}+1}=z+1, \quad \text { and thus } \quad z=y-1
$$

and this transformation is valid for all real values of $z$, because $z^{2}+1 \geq 1$, so the division is always possible. Thus, according to the first relation,

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

in another form,

$$
\left(y^{2}-2 y+2\right)\left(x+y-y^{2}\right)-1=0
$$

Scsaurszky Péter (Pannonhalma, Benedek-rendi g. IV. o. t.)