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Strange student's addition

Source: XIII Ibero American Olympiad For University Students

July 22, 2011
number theory proposednumber theory

Problem Statement

A student adds up rational fractions incorrectly:
\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}  (\star)
Despite that, he sometimes obtains correct results. For a given fraction ab,a,b∈Z,b>0\frac{a}{b},a,b\in\mathbb{Z},b>0, find all fractions xy,x,y∈Z,y>0\frac{x}{y},x,y\in\mathbb{Z},y>0 such that the result obtained by (⋆)(\star) is correct.
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