ILL 1987 Problem 15

Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ 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 $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements.

Prove that $$S_1 + S_2 + S_3 = T_1 + T_2 + T_3.$$