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Interval without integer numbers

Source: Romanian IMO Team Selection Test TST 1988, problem 15

October 1, 2005
trigonometryalgebra proposedalgebra

Problem Statement

Let [a,b][a,b] be a given interval of real numbers not containing integers. Prove that there exists N>0N>0 such that [Na,Nb][Na,Nb] does not contain integer numbers and the length of the interval [Na,Nb][Na,Nb] exceedes 16\dfrac 16.
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