A, B and C are three positive real numbers such that: A*B*C = 1, and: A + B + C > 1/A + 1/B + 1/C

Prove that precisely one (and, no more) of A, B and C is greater than 1.

(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.

Posted by Harry on 2010-11-26 23:21:01