IMO Shortlist 1994 problem A1
Dodao/la:
arhiva2. travnja 2012. Let
![a_{0} = 1994](/media/m/6/b/d/6bd6fe2ac9007abb7708ce3bcb9e2f50.png)
and
![a_{n + 1} = \frac {a_{n}^{2}}{a_{n} + 1}](/media/m/c/8/b/c8bfe5d84c40e15fa938839fa06277d5.png)
for each nonnegative integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
. Prove that
![1994 - n](/media/m/3/4/0/34044240ef9de3b114599b5dae300f53.png)
is the greatest integer less than or equal to
![a_{n}](/media/m/e/1/b/e1bf963ddae5d084fba54d8a7aa04acc.png)
,
%V0
Let $a_{0} = 1994$ and $a_{n + 1} = \frac {a_{n}^{2}}{a_{n} + 1}$ for each nonnegative integer $n$. Prove that $1994 - n$ is the greatest integer less than or equal to $a_{n}$, $0 \leq n \leq 998$
Izvor: Međunarodna matematička olimpijada, shortlist 1994