Question 66
Prove that for any collection $a_1, a_2, \dots, a_{2011}$ of real numbers with $a_{2011} \neq 0$ there exists a function $f: \mathbb{R} \to \mathbb{R}$, such that for any real $x$ we have:
$$
a_1 f(x) + a_2 f(f(x)) + \dots + a_{2011} \underbrace{f(f(f\dots f(x)\dots))}_{2011} = x.
$$
$$
a_1 f(x) + a_2 f(f(x)) + \dots + a_{2011} \underbrace{f(f(f\dots f(x)\dots))}_{2011} = x.
$$