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x must be an integer

Source: Pan African Olympiad 2009

October 1, 2011
number theory proposednumber theory

Problem Statement

Let xx be a real number with the following property: for each positive integer qq, there exists an integer pp, such that ∣x−pq∣<13q.\left|x-\frac{p}{q} \right|<\frac{1}{3q}. Prove that xx is an integer.
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