National Math Olympiad

The given equation is equivalent to $8b^2-4ab-4bc+a^2+c^2=4b^2-2ac$ or $$4b^2-4ab-4bc+a^2+2ac+c^2=0.$$ This can be further rewritten as $(a+c{)}^2-4b(a+c)+4b^2=0$ and finally as $$(a+c-2b{)}^2=0.$$ From here we conclude that $a+c=2b$ or, equivalently, $b-a=c-b$. Hence, $a$, $b$ and $c$ are consecutive terms of an arithmetic sequence.