IMO Shortlist 1976 problem 9
Dodao/la:
arhiva2. travnja 2012. Let
![P_{1}(x)=x^{2}-2](/media/m/b/2/4/b24c471df42f2bd6eb12c2c29397119a.png)
and
![P_{j}(x)=P_{1}(P_{j-1}(x))](/media/m/9/3/1/93152d013eceb9f59db4bee64fe2d847.png)
for j
![=2,\ldots](/media/m/f/b/9/fb9acbd64da10059292736f6bab61c0c.png)
Prove that for any positive integer n the roots of the equation
![P_{n}(x)=x](/media/m/7/3/9/739117f0f55d78db3eb22f197f332850.png)
are all real and distinct.
%V0
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
Izvor: Međunarodna matematička olimpijada, shortlist 1976