IZhO 2016 Problem 6

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