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(a^2+b^2+c^2)/(a+b+ c) is an integer if (a\sqrt{3}+b)/(b\sqrt3+c) is rational

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 10, 2022

number theoryalgebrarationalInteger

Problem Statement

Let

a,b

and

c

be positive integers that

\frac{a\sqrt{3}+b}{b\sqrt3+c}

is a rational number, show that

\frac{a^2+b^2+c^2}{a+b+ c}

is an integer.
(Anonymous314)