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Interesting problem from ILL 1987 FRA2

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September 5, 2010
algebra proposedalgebra

Problem Statement

Let a1,a2,a3,b1,b2,b3,c1,c2,c3a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3 be nine strictly positive real numbers. We set S_1 = a_1b_2c_3,   S_2 = a_2b_3c_1,   S_3 = a_3b_1c_2;T_1 = a_1b_3c_2,   T_2 = a_2b_1c_3,   T_3 = a_3b_2c_1. Suppose that the set {S1,S2,S3,T1,T2,T3}\{S1, S2, S3, T1, T2, T3\} has at most two elements.
Prove that S1+S2+S3=T1+T2+T3.S_1 + S_2 + S_3 = T_1 + T_2 + T_3.
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