50. (USS 4) Determine all positive integers $n$ for which there exists a polynomial $P_{n}(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.
50. Suppose that $P_{n}(x)=n$ for $x \in\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}$. Then
$$
P_{n}(x)=\left(x-x_{1}\right)\left(x-x_{2}\right) \cdots\left(x-x_{n}\right)+n .
$$

From $P_{n}(0)=0$ we obtain $n=\left|x_{1} x_{2} \cdots x_{n}\right| \geq 2^{n-2}$ (because at least $n-2$ factors are different from $\pm 1$ ) and therefore $n \geq 2^{n-2}$. It follows that $n \leq 4$.
For each positive integer $n \leq 4$ there exists a polynomial $P_{n}$. Here is the list of such polynomials:
$$
\begin{array}{ll}
n=1: \pm x, & n=2: 2 x^{2}, x^{2} \pm x,-x^{2} \pm 3 x, \\
n=3: \pm\left(x^{3}-x\right)+3 x^{2}, & n=4:-x^{4}+5 x^{2} .
\end{array}
$$