11. (20 points) Given a positive real number $a$, for any positive integer $n$, let
$$
f(n)=\left[\frac{n+\left[\frac{a}{n}\right]}{2}\right],
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$.
(1) If $f(5)=5$, find the range of values for $a$;
(2) Prove that $f(n)>\sqrt{a}-1$.

保留了源文本的换行和格式。
11. (1) From $f(5)=5$, we know
$$
\begin{array}{l}
{\left[\frac{5+\left[\frac{a}{5}\right]}{2}\right]=5 \Leftrightarrow\left[\frac{a}{5}\right]=5 \text { or } 6} \\
\Leftrightarrow 5 \leqslant \frac{a}{5} < 7 \Leftrightarrow 25 \leqslant a < 35.
\end{array}
$$
$$
\begin{array}{l}
\text { Therefore, } f(n)=\left[\frac{n+\left[\frac{a}{n}\right]}{2}\right] \\
\geqslant \frac{n+\left[\frac{a}{n}\right]-1}{2}>\frac{n+\left(\frac{a}{n}-1\right)-1}{2} \\
=\frac{n+\frac{a}{n}}{2}-1 \geqslant \frac{2 \sqrt{n \cdot \frac{a}{n}}}{2}-1=\sqrt{a}-1 .
\end{array}
$$