Baltic Way 1994 · Problem 5

Solution:

Observe first that if $a$ and $b$ are two different integers then $p(a)-p(b)$ is divisible by $a-b$. Suppose now that $p(a)=1$ and $p(b)=3$ for some integers $a$ and $b$. If we have $p(c)=2$ for some integer $c$, then $c-b= \pm 1$ and $c-a= \pm 1$, hence there can be at most one such integer $c$.