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Maclaurin's inequalities: reflections on a STEP question

Published online by Cambridge University Press:  23 January 2015

Stephen Siklos
Stephen Siklos*
Affiliation:
Jesus College, Cambridge CB5 8BL, e-mail: [email protected]
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Extract

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.

Type
Articles
Information
The Mathematical Gazette , Volume 96 , Issue 537 , November 2012 , pp. 499 - 507
Copyright
Copyright © The Mathematical Association 2012

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References

1. http://www.stepmathematics.org.uk Google Scholar
2. Meserve, Bruce E., Fundamental concepts of algebra, Dover (1981).Google Scholar
3. I used http://www.l728.com/cubic.htm, but Googling ‘cubic equation solver’ shows that there are plenty of others out there.Google Scholar
4. Dörrie, Heinrich, 100 Great problems in elementary mathematics, Dover (1965).Google Scholar
5. Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Cambridge University Press (1934).Google Scholar
6. Steele, J. Michael, The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities, Cambridge University Press (2004).Google Scholar