(A - 1)(B - 1)(C - 1) = ABC + A + B + C - (BC + CA + AB) - 1 = A + B + C - (BC + CA + AB)since ABC = 1 = A + B + C - (1/A + 1/B + 1/C) > 0

A, B and C cannot all be greater than 1 (since ABC = 1), so it follows that exactly two of the factors in the LHS are negative and one is positive. Thus, exactly one of A, B and C is greater than 1.