My teacher gave me the following lemma to prove :
Let $P(x)$ be a polynomial with integer coefficients. If some integer $t$ satisfies $$P(P((...P(t)...))) =t$$ for some number of iterations, then prove that $t$ should either satisfy $P(t)=t$ or $P(P(t)) = t$.
$(t \in \mathbb{Z})$
I have tried some usual approaches but failed. My teacher told me that this lemma is related to some IMO problem of the year 2005 or 2006. So I expect some clever and ingenious proof. I have tried my best but it does not yield.
Can anyone help me to prove this conundrum ?
Any help will be gratefully acknowledged.
Thanks in Advance ! :-)
Lemma $1$: if $s$ and $t$ are two distinct integer values then $s-t$ divides $P(s)-P(t)$.
Proof: suppose $P(x)=a_nx^n+\dots+a_0$, then $P(s)-P(t)=a_n(s^n-t^n)+a_{n-1}(s^{n-1}-t^{n-1})+\dots a_1(s-t)$. Clearly every summand is a multiple of $s-t$ because of the classical factorization $s^k-t^k=(s-t)(s^{k-1}+s^{k-2}t+\dots+t^k)$.
Suppose that we have elements $s_1,s_2,\dots,s_n,s_{n+1}=s_1$ such that $P(s_i)=s_{i+1}$ for all $1\leq i \leq n$ and the first $n$ elements are all distinct. We then have that $s_1-s_2|s_2-s_3\dots | s_1-s_n$. This implies that the magnitudes of all the distances are the same, and in order for the first $n$ elements to be different we clearly need $n=2$ or $n=1$.