1. For which $a$
1) the equation $f(f(x))=x$ has an infinite number of solutions;
2) the equation $f(f(f(f(x))))=x$ has an infinite number of solutions for functions of the form

$$
f(x)=\frac{(2 a+8) x-5}{2 x-a} ?
$$
Answer: 1) $a=-8, f(x)=\frac{-8 x-5}{2 x+8}$

$$
\text { 2) } \begin{aligned}
a & =-8, f(x)=\frac{-8 x-5}{2 x+8} ; a=-2, f(x)=\frac{4 x-5}{2 x+2} \\
a & =-\frac{22}{5}, f(x)=\frac{-4 x-25}{10 x+22}
\end{aligned}
$$