IMO Shortlist 1992 problem 18
Dodao/la:
arhiva2. travnja 2012. Let
![\lfloor x \rfloor](/media/m/c/c/2/cc22bc897f71e3436c8e79a0a632e862.png)
denote the greatest integer less than or equal to
![x.](/media/m/9/d/1/9d104e6d78783bc388eb8eec4a4db914.png)
Pick any
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
in
![[0, 1)](/media/m/1/7/2/172b07abde6a2af9c062129ef0d68874.png)
and define the sequence
![x_1, x_2, x_3, \ldots](/media/m/e/7/e/e7eab3322da37c596b1e9d6c58193bdd.png)
by
![x_{n+1} = 0](/media/m/a/4/b/a4bb46ed3d19bc29d0e36791df1f90a8.png)
if
![x_n = 0](/media/m/4/8/c/48cfc769adb076ddc81bed76c538fe5d.png)
and
![x_{n+1} = \frac{1}{x_n} - \left \lfloor \frac{1}{x_n} \right \rfloor](/media/m/2/b/3/2b3d7a92eaea0ff6dd2a0a1bbf796201.png)
otherwise. Prove that
where
![F_1 = F_2 = 1](/media/m/6/f/d/6fd325a1f132d920abb9ad203aeaf285.png)
and
![F_{n+2} = F_{n+1} + F_n](/media/m/a/8/5/a85373a4d9205d55f5e2b968ade1c298.png)
for
%V0
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Pick any $x_1$ in $[0, 1)$ and define the sequence $x_1, x_2, x_3, \ldots$ by $x_{n+1} = 0$ if $x_n = 0$ and $x_{n+1} = \frac{1}{x_n} - \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
$$x_1 + x_2 + \ldots + x_n < \frac{F_1}{F_2} + \frac{F_2}{F_3} + \ldots + \frac{F_n}{F_{n+1}},$$
where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \geq 1.$
Izvor: Međunarodna matematička olimpijada, shortlist 1992