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Six variables inequality - ILL 1987

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September 5, 2010
inequalitiesgeometry3D geometryinequalities unsolved

Problem Statement

Let a1,a2,a3,b1,b2,b3a_1, a_2, a_3, b_1, b_2, b_3 be positive real numbers. Prove that (a1b2+a2b1+a1b3+a3b1+a2b3+a3b2)2≥4(a1a2+a2a3+a3a1)(b1b2+b2b3+b3b1)(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1) and show that the two sides of the inequality are equal if and only if a1b1=a2b2=a3b3.\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.
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