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Izho 2016 p6

Source: IZhO 2016 Day 2 P6

January 16, 2016
number theoryirrational number

Problem Statement

We call a positive integer qq a convenient   denominator for a real number α\alpha if ∣α−pq∣<110q\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q} for some integer pp. Prove that if two irrational numbers α\alpha and β\beta have the same set of convenient denominators then either α+β\alpha+\beta or α−β\alpha- \beta is an integer.
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