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.
Let \(a,b,c >0\) be positive real numbers. Prove that \[(1+a)(1+b)(1+c)\geq 8\sqrt{abc}.\]
For a natural number \(n\) prove that \(n! \leq (\frac{n+1}{2})^n\), where \(n!\) is the factorial \(1\times 2\times 3\times ... \times n\).
Prove the \(AM-GM\) inequality for \(n=2\). Namely for two non-negative real numbers \(a\) and \(b\) we have \(2\sqrt{ab} \leq a+b\).
Prove the Cauchy-Schwartz inequality: for a natural number \(n\) and real numbers \(a_1\), \(a_2\), ..., \(a_n\) and \(b_1\), \(b_2\), ..., \(b_n\) we have \[(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2).\]
Prove the \(GM-HM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + ... \frac{1}{a_n}}.\]
From IMO 1999. Let \(n\geq 2\) be an integer. Determine the least possible constant \(C\) such that the inequality \[\sum_{1\leq i<j\leq n} x_ix_j(x_i^2 + x_j^2) \leq C(\sum_{1\leq i\leq n}x_i)^4\] holds for all non-negative real numbers \(x_i\). For this constant \(C\) find out when the equality holds.
Find all pairs of whole numbers \((x,y)\) so that this equation is true: \(xy+1 = y+x\).
Albert was calculating consecutive squares of natural numbers and looking at differences between them. He noticed the difference between \(1\) and \(4=2^2\) is \(3\), the difference between \(4\) and \(9=3^2\) is \(5\), the difference between \(9\) and \(16=4^2\) is \(7\), between \(16\) and \(5^2=25\) is \(9\), between \(25\) and \(6^2=36\) is \(11\). Find out what the rule is and prove it.