MathDB
Continued sequence fraction

Source: IMO Shortlist 1992, Problem 18

August 13, 2008
floor functionalgebraSequenceFibonacciFibonacci sequenceInequalityIMO Shortlist

Problem Statement

Let ⌊x⌋ \lfloor x \rfloor denote the greatest integer less than or equal to x. x. Pick any x1 x_1 in [0,1) [0, 1) and define the sequence x1,x2,x3,… x_1, x_2, x_3, \ldots by x_{n\plus{}1} \equal{} 0 if x_n \equal{} 0 and x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor otherwise. Prove that x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}}, where F_1 \equal{} F_2 \equal{} 1 and F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n for n≥1. n \geq 1.
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