IMO Shortlist 1971 problem 1
Dodao/la:
arhiva2. travnja 2012. Consider a sequence of polynomials
![P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots](/media/m/0/d/a/0dac5afc03d7e0973844d43e508a7e5b.png)
, where
![P_0(x) = 2, P_1(x) = x](/media/m/9/2/7/9273223d7107a6efe859a6453a3ce744.png)
and for every
![n \geq 1](/media/m/a/9/8/a982fcac3e2c9e0d94e965d6efb5a582.png)
the following equality holds:
![P_{n+1}(x) + P_{n-1}(x) = xP_n(x).](/media/m/f/b/b/fbbfc5d97bb1c985dd1a799643f22d22.png)
Prove that there exist three real numbers
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
such that for all
%V0
Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds:
$$P_{n+1}(x) + P_{n-1}(x) = xP_n(x).$$
Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$
$$(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1971