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\homework{7: Inequalities (Triangle, AMGM, $x^2 \geq 0$)}{Due Tuesday, Week 4, at the start of class.}

Solve \textbf{three} of the following \textbf{six} problems.  As always, prove your claims!

The theme for these problems: algebraic inequalities!  There are many useful algebraic inequalities that come up all the time when solving problems.  We list three commonly-occuring ones here:
\begin{itemize}
\item For any real number $x$, $x^2 \geq 0$.  This can be generalized slightly: if $x, y$ are two numbers with the same sign, then $xy \geq 0$.
\item (Triangle inequality\footnote{We call this the \textbf{triangle} inequality because geometrically, it is the following statement: the sum of the lengths of two sides of a triangle is never shorter than the length of the triangle's third side.}.)  Take any three points $\vec{x} = (x_1, \ldots x_n), \vec{y} = (y_1, \ldots y_n), \vec{z} = (z_1, \ldots z_n) \in \mathbb{R}^n$.  Then the distance\footnote{If you haven't seen this before: the distance from $(x_1, \ldots x_n)$ to $(y_1, \ldots y_n)$ is the quantity $\sqrt{ (x_1 - y_1)^2 + \ldots + (x_n - y_n)^2}.$  Think about why this makes sense!} $d(\vec{x}, \vec{y})$ from $\vec{x}$ to $\vec{y}$ is bound above by the sum $d(\vec{x}, \vec{z}) + d(\vec{z}, \vec{y})$.  
\item (Arithmetic mean-geometric mean inequality, i.e. AM-GM.)  The arithmetic mean of $n$ nonnegative real numbers is bounded below by their geometric mean.  In other words, for any nonnegative numbers $x_1, \ldots x_n \in \mathbb{R}$, we have the following inequality:
\begin{align*}
\frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1x_2\ldots x_n}
\end{align*}
\end{itemize}

Try using them to solve some of the problems below, and have fun!
~\\
\begin{enumerate}
\item[0.] Solve any un-signed-up-for problems from HW\#6!
\item Take any three positive real numbers $x,y,z$.  Suppose that for any positive integer $n$, there is a triangle with side lengths $x^n, y^n, z^n$.  Prove that at least two of the values in $\{x,y,z\}$ are equal.
\item Prove the AM-GM inequality.  (If you do this problem, do not use Wikipedia/texts, as they will likely contain the solution.)
\item Suppose that $x,y,z$ are the three side lengths of a triangle with perimeter 2.  Show that 
\begin{align*}
1 < xy +yz + zx - xyz \leq \frac{28}{27}.
\end{align*}
\item Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ denote two lists of nonnegative real numbers.  Show that
\begin{align*}
(a_1 \cdot a_2 \cdot \ldots \cdot a_n)^{1/n} + (b_1 \cdot b_2 \cdot \ldots\cdot b_n)^{1/n} \leq \left( (a_1+b_1) \cdot (a_2 + b_2) \cdot \ldots \cdot (a_n + b_n)\right)^{1/n}.
\end{align*}
\item Suppose that $a,b,c$ are three real numbers such that $|a-b| \geq |c|, |b-c| \geq |a|$ and $|c-a| \geq |b|$.  Prove that one of these three numbers is equal to the sum of the other two.
\item Find\footnote{If you haven't seen this before: the notation
\begin{align*}
\min_{x \in A} f(x)
\end{align*}
just means ``the minimum value that $f(x)$ can attain on the set $A$.  For example, the minimum value that $(x-1)^2$ can attain on $\mathbb{N}$ is 0; therefore, we would write $\min_{x \in \mathbb{N}} x^2 = 0$.  The expression $\max$ denotes maximum in the same way.}
\begin{align*}
\min_{a,b \in \mathbb{R}} \left(\max\left( (a^2 + b, b^2 + a) \right)\right).
\end{align*}
Prove that your claimed minimum is correct.
\end{enumerate}





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\item Let $\{a_1, \ldots a_n\}$ and $\{b_1, \ldots b_n\}$ be any two increasing sequences of positive real numbers.  Suppose that $\{x_1, \ldots x_n\}$ is some rearrangement of the $\{b_1, \ldots b_n\}$ list: that is, the elements of the $\{x_1, \ldots x_n\}$ list are the same as those in the $\{b_1, \ldots b_n\}$ list, just in some other order.
Prove that
\begin{align*}
\sum_{i=i}^n a_ib_i \geq \sum_{i=1}^n a_ix_i.
\end{align*}