IMO Shortlist 1992 problem 18


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2. travnja 2012.
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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