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\begin{document}
\title{Inequalities (19 May 2018)}
\author{Peter McNamara}
% \maketitle
\begin{center}
{\bf Inequalities (19 May 2018)
Peter McNamara}
\end{center}
\section*{Some Theory}
This is not an exhaustive list by any stretch of the imagination.
The most basic inequality is
$$x^2\geq 0$$
Always keep in mind what you know and what you have to prove. It is very frustrating to spend lots of time proving $a\geq c$ and $b\geq c$ when the question wants you to prove $a\geq b$, since you have made zero progress.
\
The first non-trivial inequality usually encountered is the AM-GM inequality:
\[
\frac{x_1+x_2+\cdots+x_n}{n}\geq \sqrt[n]{x_1x_2\cdots x_n}
\]
for all non-negative reals $x_1,x_2,\ldots,x_n$, with equality if and only if $x_1=x_2=\cdots =x_n$.
Various generalisations to look up: Quadratic mean $\geq$ Arithmetic mean $\geq$ Geometric mean $\geq$ Harmonic mean. This sequence of inequalities generalises to the Power mean inequality. Muirhead's inequality is also a generalisation, though is often simply provable by repeated applications of AM-GM.
\
Jensen's Inequality:
If $f$ is a convex function on an interval $I$, then
\[
f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)\leq \frac{f(x_1)+f(x_2)+\cdots+f(x_n)}{n}
\]
for all $x_1,x_2,\ldots,x_n\in I$. To apply this, you need to know what a convex function is. The following are equivalent (for reasonable functions)
\begin{itemize}
\item $f(x)$ is convex
\item $f(\frac{x+y}{2})\leq \frac{f(x)+f(y)}{2}$ for all $x,y\in I$.
\item $f''(x)\geq 0$ for all $x\in I$.
\end{itemize}
There is a corresponding version for concave functions with all the inequality signs reversed.
\
Geometric Inequalities:
If $a$, $b$ and $c$ are the sides of a triangle, always make the substitution $a=y+z$, $b=z+x$, $c=x+y$. Then by the triangle inequality, $x$, $y$, and $z$ are all positve real numbers.
\section*{Problems}
\begin{enumerate}
\item If $x+y=1$ and $x>0$, $y>0$, prove that
\[
\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\geq 9.
\]
\item Prove the following inequality for all positive real numbers $a$, $b$ and $c$:
\[
\left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{a}\right)\geq 8
\]
\item If $a$, $b$ and $c$ are all positive reals, prove that
\[
a^7+b^7+c^7\geq a^4b^3+b^4c^3+c^4a^3.
\]
\item Find all real solutions to the following system of equations
\begin{align*}
x+y&=2 \\
xy-z^2&=1.
\end{align*}
\item Use Jensen's inequality for $f(x)=\log(x)$ to prove the AM-GM inequality.
\item If $a$, $b$ and $c$ are the lengths of the sides of a scalene triangle, prove that
\[
\frac{1}{b+c-a}+\frac{1}{c+a-b}+\frac{1}{a+b-c} > \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.
\]
\item (Nesbitt) Let $a$, $b$ and $c$ be positive real numbers. Prove that
\[
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}.
\]
\item Prove that $n!\geq n^{n/2}$.
\item Prove that $1\cdot 3\cdot 5\cdots (2n-1)\leq n^n$.
\item Amongst all triangles with the same perimeter, show that the equilateral triangle is the one with the greatest area.
\item Find the least value of $3x+4y$, if $x$ and $y$ are positive numbers satisfying $x^2y^3=6$.
\item (2000 AMO Q3)
Let $x_1,x_2,\ldots x_n$ and $y_1,y_2\ldots,y_n$ be real numbers such that
\begin{enumerate}
\item $0