Subject 1. Determine the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ of the form $f(x)=a x+b$, with $a, b \in \mathbb{R}$, such that

$$
(f \circ f \circ \ldots \circ f)(-x)+(f \circ f \circ \ldots \circ f)(x)=0
$$

for any $x \in \mathbb{R}$ (the function $f$ is composed with itself $n$ times in both terms).

Mathematical Gazette 1981
Subject 1. $f(f \ldots f(x))+f(f \ldots f(-x))=2 b\left(a^{n-1}+a^{n-2}+\ldots+1\right)(2 \mathbf{p})$; $b=0$ is a solution ( $\mathbf{1} \mathbf{p}) ; a^{n-1}+a^{n-2}+\ldots+1=0 \Rightarrow a^{n}=1$ (2p); if $n$ is odd, there are no solutions for $a(\mathbf{1} \mathbf{p})$; if $n$ is even, $a=-1(\mathbf{1} \mathbf{p})$.