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Prove expression is rational

Source: Dutch IMO TST I Problem 3

July 17, 2014
algebra proposedalgebra

Problem Statement

Let aa, bb and cc be rational numbers for which a+bca+bc, b+acb+ac and a+ba+b are all non-zero and for which we have 1a+bc+1b+ac=1a+b.\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}. Prove that (c−3)(c+1)\sqrt{(c-3)(c+1)} is rational.
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