Kalinin A.

The function $f(x)$ is defined and satisfies the relation

$$
(x-1) f(x+1)-f(x)=x
$$

for all $x \neq 1$. Find all such functions.
Substituting in the given equation instead of $x$ the fraction $\frac{x+1}{x-1}$, we obtain:

$$
\frac{2}{x-1} f(x)-f(x+1)=\frac{x+1}{x-1} \text {. (1) }
$$

Equation (1) together with the equation given in the condition defines a system of two equations with unknowns $f(x)$ and $f\left(\frac{x+1}{x-1}\right)$. Solving it, we find that $f(x)=1+2 x$. Verification shows that the found function satisfies the equation.