Today we explore inequalities related to mean values of a set of
positive real numbers. Let \(\{a_1,a_2,...,a_n\}\) be a set of \(n\) positive real numbers. Define:
Quadratic mean (QM) as \[\sqrt{\frac{a_1^2 + a_2^2 +
...a_n^2}{n}}\] Arithmetic mean (AM) as \[\frac{a_1 + a_2 + ...+a_n}{n}\]
Geometric mean (GM) as \[\sqrt[n]{a_1a_2...a_n}\] Harmonic
mean (HM) as \[\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ...
\frac{1}{a_n}}.\] Then the following inequality holds: \[\sqrt{\frac{a_1^2 + a_2^2 + ...a_n^2}{n}} \geq
\frac{a_1 + a_2 + ...+a_n}{n} \geq \sqrt[n]{a_1a_2...a_n} \geq
\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... \frac{1}{a_n}}.\] We
will prove \(QM\geq AM\) and infer the
\(GM \geq HM\) part from \(AM \geq GM\) in the examples. However, the
\(AM\geq GM\) part itself is more
technical. The Mean Inequality is a well known theorem and you can use
it in solutions today and refer to it on olympiads.