nAalniaOMliO wrote: Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$. Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square. $S=ab+c(a+b)=ab-c^2$ and so $S+C=S+c^2+c+1=ab+c+1=ab-a-b+1=(a-1)(b-1)$ And, same :$S+A=(b-1)(c-1)$ and $S+B=(a-1)(c-1)$ So $\boxed{(S+A)(S+B)(S+C)=(a-1)^2(b-1)^2(c-1)^2}$