Prove the inequality for a triangle with sides $a,b,c$ we have $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} &lt;2$$  Trial: Since $a,b,c$ are sides of a triangle I know  $a+b&gt;c,b+c&gt;a,a+c&gt;b$
How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict.
Can you grapple/shove with the Hunter Ranger's Whirlwind Attack? The  states:     You can use your action to make a melee attack against any number of creatures within 5 feet of you, with a separate attack roll for each target.   The rules on "" state (the rules in shoving are effectively identical):     When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.   I couldn't find any other features spells or abilities that allow a creature to make a flat "melee attack". If they had said "melee weapon attack" instead this would have meant the same thing but specifically have excluded grapple/shove as an option.  I am unsure whether the rules on grappling/shoving mean that the only time you can grapple/shove are when you take the Attack action, which you are not doing when using the Whirlwind Attack feature.   I would like answers to assume that the following ruling made in the  is correct:     Whirlwind Attack is unusual, in that it’s a single attack with multiple attack rolls... Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, how do you calculate the surface area of that intersection?   To clarify, the area is that of the curved surface on the sphere defined by the intersection. At the extreme where both $a_1,a_2 = r$, we would be describing a spherical lune.   Alternatively define the spherical caps by the angles  $\Phi_1 = \arcsin(a_1/r)$ and $\Phi_2 = \arcsin(a_2/r)$. How should I refer to a friend who is a girl but not a girlfriend? When I'm talking about my friend, who is a girl, but not a girlfriend, what word or phrase should I use? If the gender was unimportant, it would not be a problem. But if I want to note that the friend is female, not male, how should I say that, to avoid ambiguity? Is it possible to write any bounded continuous function as a uniform limit of smooth functions Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?   On any bounded interval this is true since by Stone-Weierstrass polynomials are smooth and dense in the continuous functions. I guess the result is still true for $\mathbb{R}$ but I don't know how to prove it. Since I don't have any integrability conditions on $f$ the technique of convoluting with mollifiers is probably not applicable. How to sort text in sqlite3 with specified locale? Sqlite3 by default sorts only by ascii letters. I tried to look in google, but the only thing I found were informations about collations. Sqlite3 has only NOCASE, RTRIM and BIARY collations. How to add support for a specific locale? (I'm using it in Rails application) Prove inequality using Schwarz's Lemma I'm having trouble proving the following:  Let $f$ be analytic on $D(0,R)$ and suppose there is a real constant A for which $\operatorname{Re}f(z) &lt; A$ for all $z \in D(0,R)$. If $f(0) = 0$, show that $|f(z)| \leq \frac{2A|z|}{R-|z|}$.   I was told to first start by showing that $|\frac{f(z)}{2A-f(z)}| \leq 1$ and also use the following result, which I was able to prove:  Let $f$ be analytic on $D(0,R)$ and let $f(0)=0$. If $|f(z)| \leq M$ for all $z \in D(0,R)$, then $|f(z)| \leq M|z|/R$ for all $z\in D(0,R)$.  Thanks in advance! $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$ I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks. I want to unwrap a label which has a transparent area Asking distant cousins for DNA? In my family we have a DNA contradiction – two cousins (at about the 4th cousin level) took a Y-DNA test, but did not match and had two completely different haplogroups.   Since then, I have traced living cousins with the same surname – the only ones left living that I am aware of – and I would love to pay to have them tested to see if we can figure out where the non-paternity event (NPE) or other event happened (i.e. this new cousin could help us determine which line the NPE occurred in depending on which cousin he matches).   Does anyone have a sample letter, or know the best way to go about asking a distant cousin for DNA?   I have their address and am planning to write a letter, but I can't imagine receiving such a letter, so I am not quite sure how to ask without scaring them off. The center of the dihedral group How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. Why is Ubuntu 14.04 so slow on my laptop? Changing units of measurement in a simple regression model What is the difference between String and string in C#? Example (note the case):  string s = "Hello world!"; String s = "Hello world!";   What are the guidelines for the use of each? And what are the differences? I am looking for an Ubuntu version that consumes less Memory and CPU. I have read both Lubuntu and Xubuntu (The homepages, wikipedia, phoronix and other sites comparing both). But from experience, which one uses less memory and is less CPU intensive. I need to install them in very old hardware and want to persuade the owner of the hardware of the benefits of Ubuntu.  in this case I want to install 11.10 or 12.04 when it comes out. How are each behaving in those versions?  The 2 PCs I will be installing either Xubuntu or Lubuntu are:  Granpa PC:  CPU - Pentium 2 450Mhz RAM - 64MB DIMM Video - 16MB Used for - Documents and Internet. No listening to music, no looking at videos. Just using it for document writing.  The other old meat:  CPU - Pentium 3 550Mhz RAM - 128MB DIMM Video - 16MB Used for - Documents and Internet also but they want.. or maybe they are wishing for it to use it to see movies and listen to music. This one has internet. The other one does not.  UPDATE  Wanted to update this question with the test results for over 8 months worth of using Lubuntu and Xubuntu. When it comes to very old hardware, am talking about 64 MB, 128 MB or even 256 MB of RAM computers, Lubuntu is better than Xubuntu when comparing Memory usage. When it comes to CPU usage, both are the same.  This will not improve I/O Disk performance which will be the bottleneck for old computers. Also the difference in performance between Xubuntu and Lubuntu is about 10% to 20% percent in memory usage. Again, CPU usage is the same.  Now, when it comes to how responsive the Desktop is, LXDE (Lubuntu) has a much better response time than XFCE (Xubuntu). I would say about 50% or just simply noticeable to the eye. Lubuntu is similar to how Windows XP looks while Xubuntu is more like how Gnome 2 looked.  The computers (The old ones mentioned here) and 2 more are still working in excellent conditions thanks to the Ubuntu family. I also want to add I am using 13.10 for both distros. I understand the energy and mass can change back and forth according to Einstein. It is fluid; it can go from one to the other. So, what keeps mass from just turning into energy? Is there some force holding a subatomic particle together? What keeps mass in it's state? I hope this is not a silly question but I am clueless. Thanks Possible Duplicate:          When I add a piece of code of figure insertion to the context I'm faced with a problem: a number of irrelevant words appear.    Code:  \begin{figure}     \centering         \includegraphics[width=0.9\textwidth]{C:/Thesis/Latex/thesis_1(1)/Pictures/study area.jpg}         \rule{35em}{0.3pt}     \caption{The Grand St. Bernard wireless sensor network deployment (a) the coordinates of nodes according to the Swiss coordinate system (b) the distribution of the nodes in the study site \citep{r33}}     \label{fig:study area} \end{figure}   Problem: Show the existence of a certain subgroup of F2 So our 6-month-old boy still doesn't sleep through the night. He still wakes up at least once and wants to eat.  He takes 3 naps during the day, ranging from 30 minutes to 1.5 (sometimes 2) hours.  We usually put him down for the night at least 2 hours after his latest nap, but we haven't been able to get him down to 2 naps per day because he gets pretty cranky during the evenings. Depending on the duration and times of his naps, his bedtime ranges between 10 and midnight.  We have started feeding him a small amount (2 tablespoons dry) of rice cereal after his morning and evening naps in the hopes that he'll have a fuller stomach through the night, but he still wakes up hungry after sleeping between 3 and 6 hours. I can handle the 6 hours, since that means I get 6 hours of pretty much uninterrupted sleep, but it's usually more like 3-4.  Any suggestions? Complex differential equation How do I solve $$x'=e^{it}\overline{x}?$$  This is a complex differential equation, but I don't see how to solve it.  Edit: the original ODE is given by $$(x', y')=\begin{pmatrix}\cos t&amp; \sin t\\ \sin t&amp;-\cos t\end{pmatrix}(x,y)$$ I want to show that solutions of this ODE do not remain bounded for all $t$, and the idea was to solve the complex equation. How to delay or slowly loop commands?