Prove that: $$\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ac}{a+c}\leq \frac{a+b+c}{2} : (a, b, c) \in \mathbb{R}^+$$
If $a,b,c$ are positive real numbers, not all equal, prove that  $$\dfrac{bc}{b+c}+\dfrac{ca}{c+a}+\dfrac{ab}{a+b}&lt;\dfrac{a+b+c}{2}$$  Progress: $\frac{bc}{b+c}-\frac{b+c}{2}=$
Extra "lines" are being added when creating/editing questions/answers in Edge Lately, I've noticed that extra lines are being added when I create questions/answers. For example, if I write:   Your problem is X, Y, Z. Fix it by doing:      Code  Then maybe also do some other thing.  Oh, and some other thing as well.   When I post it, then go back into the editor, it's been changed to:   Your problem is X, Y, Z. Fix it by doing:       Code   Then maybe also do some other thing.   Oh, and some other thing as well.   Which is mildly annoying since that causes less to be displayed in the box at once, but whatever.  The weird thing is though, the added line isn't properly treated as a newline, even though it does appear to be caused by a newline. A lot of lines that look like they should be empty don't even get rendered. If I delete the original lines that I entered leaving only the "new" newlines, I get weird behavior:    I tried selecting the characters involved and inspecting them in Python, and they just appear to be newlines, so I have no idea what the problem is.    The only browser I've been able to reproduce it in is Edge (version 44.18362.329.0). It does not happen in Chrome. This has been happening for ~3 days now.    There is also a problem with link embedding and code formatting. If while editing I highlight some text and either try to embed a link (interlinked circle button) or format it as code ({} button), it often puts the wrong text between the […]s when embedding, or formats code that I didn't highlight (and doesn't format the code that I did highlight). new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work? Are global variables thread-safe in Flask? How do I share data between requests? Permutations on word $MISSISSIPPI$. In how many ways can the letters of the word $MISSISSIPPI$ be rearranged ?   I am confused on whether it is  $\dfrac{11!}{4!4!2!}$ or $\dfrac{11!}{4!4!2!}-1$  since it is given rearranged and not arranged. Newbie Q: Is "Field Level Security" the same as "Field Accessibility"? There are two buttons that seem very similar when viewing the detial page for an object: Set Field Level Security and View Field Accessibility. They both seem to allow you to set the field level security.  Are these ("Field Level Security" and "Field Accessibility") basically the same thing? How do I display a decimal value to 2 decimal places? When displaying the value of a decimal currently with .ToString(), it's accurate to like 15 decimal places, and since I'm using it to represent dollars and cents, I only want the output to be 2 decimal places.   Do I use a variation of .ToString() for this? Basically, I want the header fonts to use a different (smaller) font size than the size I set using \documentclass. What should be the option I'd be using ?  \documentclass[12pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[margin=1in, headsep=10pt]{geometry} \usepackage{fancyhdr} \pagestyle{fancy}  \fancyhead[L]{Personal Statement} % What other option should I use to get a custom font size ? Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational. On the photos where the lens flare is present, my Nikon D7000 is drawing strange labyrinth-looking artifacts. Here's an example.  The first image is a screenshot of a little part of the original photo zoomed 2:1 in Lightroom.  The second image is the same photo with Lightroom's contrast, clarity and sharpening set to the maximum to make the labyrinth pattern more visible.      This affects only some of the photos presenting lens flare. Here, for instance, the lens flare is present and light conditions are very close to the first photo, but there is no labyrinth pattern, only usual noise (screenshot of a part of a photo at 1:1 with contrast, clarity and sharpening set to the maximum):    Notes:   I can see this pattern only on some of the photos when the sun is in the frame or nearly in the frame. It is present with any lens I tested. The actual photo was shot at f/8, ISO 200, 1/640 s. I've seen this pattern in photos shot at ISO 100 as well. The photo is in RAW format, so this is not a JPEG artifact.   What is that? How do I avoid it? Setting default values (conditional assignment) In Ruby you can easily set a default value for a variable   x ||= "default"   The above statement will set the value of x to "default" if x is nil or false  Is there a similar shortcut in PHP or do I have to use the longer form:  $x = (isset($x))? $x : "default";   Are there any easier ways to handle this in PHP? Difference between ref and out parameters in .NET Order of operations using Object Initializer Syntax Does the order in which I set properties using the object initializer syntax get executed in the exact same order?  For instance if I do this:  var s = new Person { FirstName = "Micah",                      LastName = "Martin",                      IsLoaded = true                    }   will each property get set in the same order? A recent quandary came up with some friends in a game of Ticket to Ride.  If, in the last turn, you have the cards to play a route longer than the number of trains that you have left, can you do so?  Or are you limited by your number of trains? Zip using Oracle Stored Procedure Determine group G as cyclic If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. How do I split a string so I can access item x? Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"? Can't run "dpkg --add-architecture i386" on 12.04 64-bit to run eclipse ADT Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is $G=\Bbb Z/p\Bbb Z\oplus \Bbb Z/p\Bbb Z$. This is abelian and has order $p^2$. I think the other is $\Bbb Z/p^2 \Bbb Z$.   Now, it should follow from the fact that there is only one cyclic group of order $n$ up to isomorphism that these two are unique up to isomorphism. All I need to show is these two are in fact not isomorphic. It suffices to show that $G$ as before is not cyclic. But this is easy to see, since we cannot generate any $(x,y)$ with $x\neq y$ by repeated addition of some $(z,z)$.  Now, it suffices to show that any other group of order $p^2$ is isomorphic to either one of these two groups. If the group is cyclic, we're done, so assume it is not cyclic. One can see that $G=\langle (1,0) ,(0,1)\rangle$. How can I move on? I'm fairly new to LaTeX and I want to know how to change the way sections are numbered by default. I use LyX, and there it is relatively simple to change the way an enumeration is numbered, i.e.  1 foo  a bar    i baz   But I want to know if there is a way to do this by default for sections and subsections.