Baltic Way 1991 · Problem 4

Solution: As $a^{n} - b^{n} = (a - b)\left(a^{n-1} + a^{n-2} b + \cdots + b^{n-1}\right)$, then for any distinct integers $a, b$ and for any polynomial $p(x)$ with integer coefficients, $p(a) - p(b)$ is divisible by $a - b$. Thus, $p(n) - p(-n) \neq 0$ is divisible by $2n$ and consequently $p(-n) \leq p(n) - 2n < n - 2n = -n$.