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(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q => (a^2+b^2+c^2)/(a+b+c) in Z

Source: Greece JBMO TST 2009 p3

April 29, 2019
algebrarationalintegrationradical

Problem Statement

Given are the non zero natural numbers a,b,ca,b,c such that the number a2+b3b2+c3\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3} is rational. Prove that the number a2+b2+c2a+b+c\frac{a^2+b^2+c^2}{a+b+c} is an integer .
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