\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\begin{document}

\title{General Mathematics and Computational Science I}
\author{Exercise 12}
\date{November 9, 2006}
\maketitle

\begin{enumerate}

\item Solve Problem 14 from Ivanov, p.~50.

\item For positive real numbers $a_1, \dots, a_n$, let
\begin{equation*}
  G(a_1, \dots, a_n) = \Biggl( \prod_{i=1}^n a_i \Biggr)^{\tfrac1n}
\end{equation*}
denote their geometric mean, and
\begin{equation*}
  A(a_1, \dots, a_n) = \frac1n \sum_{i=1}^n a_i
\end{equation*}
their algebraic mean.  Show that, if $a_1 \neq a_2$, then
\begin{equation*}
  G(a_1, \dots, a_n) < G(a^*,a^*,a_3, \dots, a_n)
\end{equation*}
and 
\begin{equation*}
  A(a_1, \dots, a_n) = A (a^*,a^*,a_3, \dots, a_n) \,,
\end{equation*}
where
\begin{equation*}
  a^* = \frac{a_1+a_2}2 \,.
\end{equation*}

\item Solve Problem 17 from Ivanov, p.~53.

\end{enumerate}
\end{document}
