3. Let $x$ be a real number. If $a=2011 x+9997, b=2011 x+9998$ and $c=2011 x+9999$, find the value of $a^{2}+b^{2}+c^{2}-a b-b c-c a$.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
3. Answer: (D)
We have
$$
\begin{aligned}
a^{2}+b^{2}+c^{2}-a b-b c-c a & =\frac{1}{2} \cdot\left[(a-b)^{2}+(b-e)^{2}+(c-a)^{2}\right] \\
& =\frac{1}{2} \cdot\left[(-1)^{2}+(-1)^{2}+2^{2}\right]=3
\end{aligned}
$$