IMO 1987 LL FRA15

Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real

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IMO 1987 LL FRA15

Origin: FRA

Problem

Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real numbers. We set S1 = a1b2c3, S2 = a2b3c1, S3 = a3b1c2; T1 = a1b3c2, T2 = a2b1c3, T3 = a3b2c1. Suppose that the set {S1, S2, S3, T1, T2, T3} has at most two elements. Prove that S1 + S2 + S3 = T1 + T2 + T3.