Mathematical Olympiad Treasures by Titu Andreescu & Bogdan Enescu

Author:Titu Andreescu & Bogdan Enescu
Language: eng
Format: epub
Publisher: Birkhäuser Boston, Boston, MA

in which there are m+(n−1)=k−1 terms. For this, we have to check that the sum s′=y 2+y 3+⋯+y n verifies the condition s′<m(n−1). Because y 1≥y 2≥⋯≥y n , it follows that , so that

as needed.

Problem 1.87

The sequence (x n ) n≥1 is defined by x 1=1, x 2n =1+x n and for all n≥1. Prove that for any positive rational number r there exists an unique n such that r=x n .

Solution

Note that all the terms of the sequence are positive numbers and that x 2n >1, x 2n+1<1 for all n≥1. We prove by induction on k≥2 the following statement: for all positive integers a,b such that and a+b≤k, there exists a term of the sequence equal to . If k=2, then a=b=1 and . Suppose that the statement is true for some k>2, and let a,b be coprime positive integers such that a+b=k+1. If a>b, then we apply the induction hypothesis to the numbers a−b and b. Clearly, and (a−b)+b=a≤k; therefore, there exists n such that . But then

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