Question 17
(POL 1) (IMO 5) Let $\mathcal F$ be a nonempty set of functions $f:\mathbb R\to\mathbb R$ of the form $f(x)=ax+b$, where $a$ and $b$ are real numbers and $a\ne0$. Suppose that $\mathcal F$ satisfies the following conditions:
(1) If $f,g\in\mathcal F$, then $g\circ f\in\mathcal F$, where $(g\circ f)(x)=g[f(x)]$.
(2) If $f\in\mathcal F$ and $f(x)=ax+b$, then the inverse $f^{-1}$ of $f$ belongs to $\mathcal F$ ($f^{-1}(x)=(x-b)/a$).
(3) None of the functions $f(x)=x+c$, for $c\ne0$, belong to $\mathcal F$.
Prove that there exists $x_0\in\mathbb R$ such that $f(x_0)=x_0$ for all $f\in\mathcal F$.
(1) If $f,g\in\mathcal F$, then $g\circ f\in\mathcal F$, where $(g\circ f)(x)=g[f(x)]$.
(2) If $f\in\mathcal F$ and $f(x)=ax+b$, then the inverse $f^{-1}$ of $f$ belongs to $\mathcal F$ ($f^{-1}(x)=(x-b)/a$).
(3) None of the functions $f(x)=x+c$, for $c\ne0$, belong to $\mathcal F$.
Prove that there exists $x_0\in\mathbb R$ such that $f(x_0)=x_0$ for all $f\in\mathcal F$.