Question 79
Let $P$ be a polynomial with real coefficients and odd degree. Suppose that the number of real solutions to
$$
P(P(x)) = P(x), \quad P(x) \neq x
$$
is finite and odd. Show that there exists a real $c$ such that $P(c) = c$ and the polynomial $P(x) - c$ has a real root of multiplicity at least two. (That is, it is divisible by $(x - r)^2$ for some real $r$.)
$$
P(P(x)) = P(x), \quad P(x) \neq x
$$
is finite and odd. Show that there exists a real $c$ such that $P(c) = c$ and the polynomial $P(x) - c$ has a real root of multiplicity at least two. (That is, it is divisible by $(x - r)^2$ for some real $r$.)