4. Let
$$
x=1 \underbrace{000 \cdots 000}_{2011 \text { times }} 1 \underbrace{000 \cdots 000}_{2012 \text { times }} 50 .
$$

Which of the following is a perfect square?
(A) $x-75$;
(B) $x-25$;
(C) $x$;
(D) $x+25$;
(E) $x+75$.
4. Answer: (B).

Note that
$$
\begin{aligned}
x & =\left(10^{1012}+1\right) \times 10^{2014}+50=10^{4026}+10^{2014}+50 \\
& =\left(10^{2013}\right)^{2}+2 \times 10^{2013} \times 5+50=\left(10^{2013}+5\right)^{2}+25
\end{aligned}
$$

So $x-25$ is a perfect square.