MHB Proving $f(-k)<-k$ when $f(k)<k$

[anemone]

Let $f$ be a polynomial with integer coefficients such that $f(-k)<f(k)<k$ for some integer $k$. Prove that $f(-k)<-k$.

 

[anemone]

Solution of other:

Spoiler

As $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+b^{n-1})$, then for any distinct integers $a$ and $b$ and for any polynomial $f(x)$ with integer coefficients $f(a)-f(b)$ is divisible by $a-b$.

Thus, $f(k)-f(-k)\ne 0$ is divisible by $2k$ and consequently $f(-k)\le f(k)-2k<k-2k=-k$.