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Sequence

Source: Bulgarian TST2/2006 Problem 2

May 31, 2006
algebra unsolvedalgebra

Problem Statement

a) Let {an}n=1∞\{a_n\}_{n=1}^\infty is sequence of integers bigger than 1. Proove that if x>0x>0 is irrational, then \ds x_n>\frac{1}{a_{n+1}} for infinitely many nn, where xnx_n is fractional part of anan−1…a1xa_na_{n-1}\dots a_1x.
b)Find all sequences {an}n=1∞\{a_n\}_{n=1}^\infty of positive integers, for which exist infinitely many x∈(0,1)x\in(0,1) such that \ds x_n>\frac{1}{a_{n+1}} for all nn.
Nikolai Nikolov, Emil Kolev
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