25. Let
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}
\end{array}
$$

Determine the minimum integer $s$ such that
$$
s A \geq\binom{ 4020}{2010}
$$
(Note: For a given positive integer $n,\binom{n}{r}=\frac{n!}{r!(n-r)!}$ for $r=0,1,2,3, \cdots, n$; and for all other values of $r$, define $\binom{n}{r}=0$.)
25. Answer: 2011
Note that
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}=\frac{1}{2} \sum_{k=0}^{2011}\left(\binom{2010}{k}-\binom{2010}{k-1}\right)^{2}
\end{array}
$$

Observe that
$$
\sum_{k=0}^{2011}\left(\binom{2010}{k}-\binom{2010}{k-1}\right)^{2}
$$
is the coefficient of $x^{2010}$ in the expansion of the following expression:
$$
\left(\sum_{k=0}^{2011}\left(\binom{2010}{k}-\binom{2010}{k-1}\right) x^{k}\right)\left(\sum_{k=-1}^{2010}\left(\binom{2010}{k}-\binom{2010}{k+1}\right) x^{k}\right) .
$$

We also have
$$
\begin{array}{c}
\sum_{k=0}^{2011}\left(\binom{2010}{k}-\binom{2010}{k-1}\right) x^{k}=\sum_{k=0}^{2011}\binom{2010}{k} x^{k}-\sum_{k=0}^{2011}\binom{2010}{k-1} x^{k} \\
=(x+1)^{2010}-x(x+1)^{2010}=(1-x)(x+1)^{2010}
\end{array}
$$
and
$$
\begin{array}{c}
\sum_{k=-1}^{2010}\left(\binom{2010}{k}-\binom{2010}{k+1}\right) x^{k}=\sum_{k=0}^{2010}\binom{2010}{k} x^{k}-\frac{1}{x} \sum_{k=-1}^{2009}\binom{2010}{k+1} x^{k+1} \\
=(x+1)^{2010}-\frac{1}{x}(x+1)^{2010}=(1-1 / x)(x+1)^{2010} .
\end{array}
$$

The coefficient of $x^{2010}$ in the expansion of the following expression:
$$
(1-x)(x+1)^{2010}(1-1 / x)(x+1)^{2010}=(2-1 / x-x)(x+1)^{4020}
$$
is
$$
2\binom{4020}{2010}-\binom{4020}{2011}-\binom{4020}{2009}=2\binom{4020}{2010}-2\binom{4020}{2009} .
$$

Hence
$$
A=\binom{4020}{2010}-\binom{4020}{2009}
$$

Consider the inequality:
$$
\begin{array}{c}
s A \geq\binom{ 4020}{2010}, \\
s(1-2010 / 2011) \geq 1, \\
s \geq 2011 .
\end{array}
$$

Hence the answer is 2011 .