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A fraction is an integer

Source: Finnish Mathematics Competition 2004, Final Round, Problem 2

November 14, 2011

algebrarational numberintegrality

Problem Statement

a, b

and

c

are positive integers and

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

is a rational number. Show that

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

is an integer.