KöMaL problem A, 2015

Algebra Difficulty 6.0 National Olympiad Prove it Hungary

Let $f(x,y)$ be a polynomial with two variables and integer coefficients such that $f$ is constant neither in $x$ - nor in $y$ -direction. Prove that $\max_{a,b\in[-2,2]}\big|f(a,b)\big|\ge4$ .
Based on the idea of Tamás Erdélyi, College Station, Texas
(5 pont)

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: KöMaL, licensed Rights held by KöMaL and the MATFUND Foundation. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.