# Problem 3.

A polynomial $P(x)$ with integer coefficients has the properties

$$
P(1)=2019, \quad P(2019)=1, \quad P(k)=k,
$$

where the number $k$ is an integer. Find this number $k$.

#
# Solution.

Since the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.

We get that

$$
\begin{gathered}
P(k)-P(1)=(k-2019) \text { is divisible by }(k-1), \\
P(k)-P(2019)=(k-1) \text { is divisible by }(k-2019) .
\end{gathered}
$$

This can only be true if $|k-1|=|k-2019|$.

The solution to the obtained equation is $k=1010$.

Answer: $\quad k=1010$.