ALG-B2-M04-P013 Reverse reciprocal order

Hint. Compare two permutations of the sequences \(a,b,c\) and \(\frac1c,\frac1b,\frac1a\).

The sequences \(a,b,c\) and \(\frac1c,\frac1b,\frac1a\) are both increasing. Therefore \(\frac ac+\frac bb+\frac ca\) is the maximum among all permutations. The sum \(\frac ab+\frac ba+\frac cc\) is one such permutation, so it is not larger.

It is useful to emphasize that reciprocals reverse order, but we rewrote them as an increasing sequence.