IMO Shortlist 1982 problem 3
Dodao/la:
arhiva2. travnja 2012. Consider infinite sequences
![\{x_n\}](/media/m/2/e/a/2ea4568f644fbb1ce6d81084c9fd775f.png)
of positive reals such that
![x_0=1](/media/m/3/4/a/34aa15bf32e3ddf12071ff33bdfebbe1.png)
and
![x_0\ge x_1\ge x_2\ge\ldots](/media/m/a/d/3/ad32c9e5eb4ea3c0783cff15b086f37a.png)
.
a) Prove that for every such sequence there is an
![n\ge1](/media/m/f/0/7/f07d10e2748ccf8b489f78bfa351880e.png)
such that:
![{x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999.](/media/m/1/2/9/129d84575b6a5fb52b80bb4fab49e91f.png)
b) Find such a sequence such that for all
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
:
%V0
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
a) Prove that for every such sequence there is an $n\ge1$ such that: $${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999.$$
b) Find such a sequence such that for all $n$: $${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1982