IMO 1976 — Shortlist Question 9

(FIN 2) IMO2. Let $P_1(x)=x^2-2$, $P_j(x)=P_1(P_{j-1}(x))$, $j=2,3,\ldots$. Show that for arbitrary $n$ the roots of the equation $P_n(x)=x$ are real and different from one another.