5. Suppose $N_{1}, N_{2}, \ldots, N_{2011}$ are positive integers, Let
$$
\begin{array}{l}
X=\left(N_{1}+N_{2}+\cdots+N_{2010}\right)\left(N_{2}+N_{3}+\cdots+N_{2011}\right), \\
Y=\left(N_{1}+N_{2}+\cdots+N_{2011}\right)\left(N_{2}+N_{3}+\cdots+N_{2010}\right) .
\end{array}
$$

Which one of the following relationships always holds?
(A) $X=Y$;
(B) $X>Y$;
(C) $X<Y$;
(D) $X-N_{1}<Y-N_{2011}$;
(E) None of the above.
5. Answer: (B).
Let $K=N_{2}+\cdots+N_{2010}$. Then $X=\left(N_{1}+K\right)\left(K+N_{2011}\right)$ and $Y=\left(N_{1}+K+N_{2011}\right) K$.
$$
X-Y=\left(N_{1} K+K^{2}+N_{1} N_{2011}+K N_{2011}\right)-\left(N_{1} K+K^{2}+N_{2011} K\right)=N_{1} N_{2011}>0 .
$$