Maclaurin's inequalities: reflections on a STEP question

Step is a public examination administered by the admissions tests division of Cambridge Assessment (which is the parent body of the OCR examination board). It used as a basis for conditional offers by Cambridge and Warwick.A grade one on a STEP paper is awarded to candidates who produce nearly complete solutions to four questions. The duration of the paper is three hours, so each question is designed to take a good candidate about 45 minutes to complete. Not surprisingly, then, the questions can be quite profound, often having depths far beyond what is required for the examination. One such question haunted me, on and off, for many years. It turns out that the underlying idea is elementary and far from new; but it is nevertheless very striking and not, I suspect, well known. A pleasing by-product is a relatively easy proof of the AM–GM (algebraic mean–geometric mean) inequality.