IMO Shortlist 1976 problem 4
Dodao/la:
arhiva2. travnja 2012. A sequence
![(u_{n})](/media/m/a/e/0/ae08f78a4dbf96c714293a6b7f81d5c7.png)
is defined by
![u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots](/media/m/4/e/f/4ef11d6587151600d74341ad0d7f1fd7.png)
Prove that for any positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
we have
![[u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}}](/media/m/0/a/f/0afc47b8329bee70b003a95254582787.png)
(where {{ Nevaljan tag "x" }} denotes the smallest integer
![\leq](/media/m/d/4/9/d4985bb43ccc5abccba4d93ed5de807d.png)
x)
%V0
A sequence $(u_{n})$ is defined by $$u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots$$ Prove that for any positive integer $n$ we have $$[u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}}$$(where [x] denotes the smallest integer $\leq$ x)$.$
Izvor: Međunarodna matematička olimpijada, shortlist 1976