Question 149
Let $S$ be a finite set of nonzero real numbers, and let $f: S \rightarrow S$ be a function with the following property: for each $x \in S$, either
$$
f(f(x))=x+f(x) \quad \text{or} \quad f(f(x))=\frac{x+f(x)}{2}
$$
Prove that $f(x)=x$ for all $x \in S$.
$$
f(f(x))=x+f(x) \quad \text{or} \quad f(f(x))=\frac{x+f(x)}{2}
$$
Prove that $f(x)=x$ for all $x \in S$.