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Divisibility problem with fractions

Source: IMOTC 2015 Practice Test 1 Problem 1

July 11, 2015
number theoryAPMO

Problem Statement

Find all positive integers a,ba,b such that a2+bb2a\frac{a^2+b}{b^2-a} and b2+aa2b\frac{b^2+a}{a^2-b} are also integers.