If $a,b,c$ are distinct positive numbers , show that $\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ If $a,b,c$ are distinct positive numbers , show that $$\frac{a^8 + b^8 + c^8}{a^3 b^3 c^3} &gt; \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ I am thinking of Tchebycheff's inequality for this question, but not able to proceed. How do I solve this?
Inequality $\frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$ Let $a,b,c$ be positive reals . Prove that $\displaystyle \frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$  I found this one in a book, no hints mentioned, but marked as very hard. I can't make any progress...
Where do I ask where to ask? How to allow certain html tags? Is the Echo of a Echo Knight actually a creature? I've always been certain that the echo created by an Echo Knight fighter using their 3rd-level feature Manifest Echo is not defined in game terms as a creature, since the description of the echo never explicitly tells it is a creature. But the presence of people believing the echo to be a creature in lieu of the fact that it is immune to all conditions and might make saving throws (game characteristics typically associated with creatures only) brought this doubt to my mind.  from Jeremy Crawford implies it is just an image and not a creature, further confirming that the feature should tell you, but  implies that the echo is an object whilst the feature doesn't explicitly say so. (Included for reference and insight, since tweets are not eligible as official ruling) Is the echo a creature? Where to post questions asking for expert opinion? After my lie down, I posted  on SO. As I suspected would happen, it got moved to programmers. Unfortunately, the kind of people I'm interested in getting answers from (litb,skeet et al - the top SO programmers in their respective disciplines) don't frequent Programmers much. So my question is, where to ask somewhat non-specific questions of high-powered users? Pay real money to set bounties (another method, not replacing rep method) It may seem like a stupid thing, but could we please be able to pay real money to set a bounty.  For example, if I had only 1 rep, I want to be able have a bounty.   Here late the suggested prices in USD. Convert money    $0.49 for each 50 reputation $0.99 for each week $1.99 for a month $9.99 for a year   Every month, the bounty fee would come in. This would mean you would have to enable your account with a bounty fee button on your profile. Can you run Android in a VM on PC? I came across a how-to guide for running Android 1.7 in a virtual machine (VirtualBox), but 1.7 is old. I haven't been able to find a Android 2.2 or 2.3 image anywhere, does anyone have any ideas on how to virtualize newer Android OS's? Preferably a free virtualization solution like VirtualBox.  Here is the link about virtualizing 1.7:    They send you to here to download Android disk images:    But I can't find anything newer than 1.7, anyone have any ideas? I'm kind of curious what kinds of criteria other people go through when selecting the correct answer to their questions.  Here's what I consider when selecting the answer to my question:   Is the answer objective?  It's easy to pick out biased answers.  Usually they involve a statement along the lines of "x sucks, you should ALWAYS use y." Do other members seem to support the answer?  Has it gotten a significant number of +1 posts or has it been voted up a number of times? Is the answer the most thorough?  Does it have an example?   With that said, there are a couple of BAD reasons to choose an answer in my opinion:   The answer is what you want to hear.  My biggest pet peeve is when somebody asks what ostensibly appears to be a valid question, but is really just a thinly veiled attempt to get someone to agree with them.  Thus, the question usually ends up showing a certain bias.  And guess which answer gets selected?  The one that confirms that bias. Nobody posted a better answer.  If you don't see a good enough answer, don't select one.   I'm sure there are a few things I'm leaving out, plus some of what I've said may be arguable.  What is everyone else's take on this? Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. What makes a photo a good photo? I don't know if this is a stupid question, this just came across my mind.  Imagine a photo of a dog yawning.  Someone could say that is a very good photo as it is taken just at the right moment, from a perfect angle (eg. level with the dog) and it has a shallow depth of field focusing on its face.  Someone else might dislike it because unfortunately the dog is ugly.  So from that example I could say people look at the following elements when judging a photo (but I could be wrong):   The timing (if it's taken at the right moment) The angle it's taken from (composition?) DOF (composition?) The subject (eg. whether the dog is pretty)   I am interested to know what experts think the elements you look at when judging whether a photo is a good photo.  Thanks in advance for your comments. share login/logout sessions across two installs? Shimano Deore front derailleur occasionally sticking I have a question about my front derailleur sticking occasionally.  I've had this bike for a 1.7 years and have ridden it through 2 winters.  It has only been after this last winter that my front derailleur has been sticking on occasion where it's like grit has jammed in somewhere (which I'm sure it is the case).  When I'm on the fly I just knock it with my heel and it will downshift and I can push the shifter hard enough to go back up when I need to.  The first time I decided to overhaul both the front and back ones.  Took them apart, cleaned/degreased, regreased the two back cogs, and used WD-40 professional silicone on the rest of the parts as to not attract more dirt.  Then after about a month it got stuck again, and I couldn't see anything that could be jamming it (while it was still stiff as heck).  Didn't take it apart this time just used a tooth brush and silicone again and it's fine at the moment.  I'm assuming it may be that since that the roads have been grittier/dirtier from the winter with some recent rain and slush that it may just be normal.  I haven't been on the trails yet this year so I know it's not that.  Is this normal, has some kind of bur that I can't see come up and is catching grit?  EDIT:  I should also note that this is a hardtail mtb that I use for trails and winter commuting (a lot of salt and grit on the roads). I own a Nikon D40X and just the kit lens and I'm considering buying either a macro lens or some extension tubes.  I don't necessarily want to do true 1:1 macro, just some close ups of bugs, fungi and flowers. Is there any way to get the number of pending chat request on omni channel? I have a requirement to get the number of pending chat request in the omni channel.  I have set the presence configuration capacity to 5 so that an agent can have only 5 chat requests. if a case is solved agent will get one of the pending chat request. But right now i can't see the number of pending request.     Lets say i have 10 pending request in the queue, but i don't have any numbers to see how many are pending in the queue.   Is there any way that can show the pending chat requests in the queue? If I call finalize() on an object from my program code, will the JVM still run the method again when the garbage collector processes this object?  This would be an approximate example:  MyObject m = new MyObject();  m.finalize();  m = null;  System.gc()   Would the explicit call to finalize() make the JVM's garbage collector not to run the finalize() method on object m? We know that you can't transfigure food, so you have to grow it or buy it. Ditto money.  I'm sure that aside from some medicinal plants Herbology Professor had there were no farmlands at Hogwarts. So they must have bought food (as well as potions ingredients for Snape etc...). And they needed to pay the teacher salaries.  Where did all that money come from according to canon?  Ministry funding?  Tuition?  Patent money on Dumbledore's inventions?  Philanthropic donations by Malfoys and co?  Other? Standard example where Jacobson radical not equal nilradical Is there a standard example in commutative algebra of a ring where the jacobson radical does not equal the nilradical? If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$. Where can I find documentation on how to use the \nolinebreak command?  The  at  never makes mention of \nolinebreak  The  states that \nolinebreak is implemented by multiplying its input parameter penalty by (-1) and passing the result into \nopagebreak.       I haven't found documentation on how to use the command. Error on \usepackage {geometry} text property The following produces error! The error is solved if no space exists between text= and {5in,8in}. This is really weird! Why is this happening with LaTeX?  \documentclass[a4paper,10pt,onecolumn,oneside, portrait]{article}  \usepackage[text= {5in,8in},centering]{geometry}  \begin{document}      \section{What Endogenous uncertainty is:}  yes  \subsection{Type as it is : Stochastic and determ! } on  \subsubsection{Well there is more than it} Yes    \end{document} How to create fixed width table columns with text raggedright/centered/raggedleft? I would like to create a table with some columns' width specified, while the text in those columns should be centered both horizontally and vertically. I found out that  \usepackage{array} \begin{tabular}{| c | c | m{5cm} |}   vertically centers the text in the last column, but justifies it horizontally (like text in normal paragraphs).  Update: With Jake's method, it seems both \\ and \hline cause an error. How to fix it?