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turkey 1993 q4

Source:

October 27, 2006

algebra unsolvedalgebra

Problem Statement

a_{n}

is a sequence of positive integers such that, for every

n\geq 1

0<a_{n+1}-a_{n}<\sqrt{a_{n}}

. Prove that for every

x,y\in{R}

such that

0<x<y<1

x< \frac{a_{k}}{a_{m}}<y

we can find such

k,m\in{Z^{+}}