Let $a,b,c$ be the sides of a triangle and $a+b+c=3$  Prove that:  $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$  My Attempt   I think this inequality could be proved using the A.M. - G.M. however, I am not sure if I am correct or not. I request someone to please post the solution for the problem as soon as possible.
Give $a, b. c$ be the lengths of a triangle and $a+ b+ c= 3$. Prove that: $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq  \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$$ My try  We have England MO inequality: $$\left ( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \right )^{2}\geq  \left ( a+ b+ c \right )\left ( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c} \right )$$  and $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \geq  3=  a+ b+ c$$  But it cannot be used to prove the above inequality and how to use triangle inequality in this inequality. I need the help. Thanks!
I think everybody with a history of participation can't help but notice the explosive growth of repeated duplicate questions. The work of looking for duplicates and then closing questions has overtaken the work of actually answering questions.  It's become virtually impossible to ask something which hasn't been yet asked. Lately I feel boredom each time I see a new question and start answering it only to realize after the first written sentence that I have answered similar questions five times over or seen them answered some ten times or more. I know where it's going to be going and what answers with what votes it is going to receive before any answer is even written.  It's obviously not the administration's fault, it is simply so. The questions is, what to do about it? Engaging more moderators to close and close duplicates won't work because it's already starting to repel users who get mad at their questions closed and become afraid of asking more or perhaps just go away.  The original idea of Jeff stated the the site is for questions that can be answered. Meaning no open-ended discussions. When I first read this I had a casual thought - well, and what's gonna happen when they all are asked and answered? Will it become a museum, a monument of wisdom and knowledge or what?  I'm not sure what to do about it. Perhaps deprecation/versioning of older questions so that their new versions can sparkle the tiresome evening...  POSTSCRIPTUM.  I recall Star Trek Voyager episode  when another Q asked Captain Janeway for asylum. He was tired of immortality and wanted to die. They opened a hearing and went on to visit the continuum to evaluate the living conditions there which caused the suffering and the wish to escape it in death. A lifeless picture presented itself to the visitors... a house lone in an open space, two people going about their business, nobody talking. When asked why they were silent, the following was said. It's been so long, all has already been said, all stories told, all jokes are made... there is nothing left to do...  Are we there as well?  UPDATE: Just saw this answer in another question:       Also, some of the core questions got asked early on in the life of the site. Great questions still come around from time to time, but many have already been asked.   Doesn't that confirm my point? Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:  $$ |d(x,a) - d(y,a)| \leq d(x,y) $$ for any $a\in A$ - but we cannot just replace it by $|d(x,A) - d(y,A)|\leq d(x,y)$ since the minimum (or infimum in general) can be attained in different points $a\in A$ for $x$ and $y$, so we only have that $$ |d(x,A)-d(y,A)|\leq d(x,y)+\sup\limits_{a,b\in A}d(a,b) $$ which does not mean continuity. Book about classical mechanics I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).  I have , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.  Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing. How can I run a program on startup, minimized? I just want Telegram to be run and I have added it to startup apps. The point is that I need it to be minimized. Any commands? Unless I missed something key during Descent 1/2, how is it that the Borg are once again a collective? How to plot a surface from a set of data? Disclaimer: I already read questions like  and  but these did not solve my problem.  I have this set of data:  x       y       z 2.17    0.001   0.82044815 2.17    0.002   0.82345825 2.17    0.004   0.82679255 2.17    0.008   0.83334715 2.17    0.016   0.84395915 2.17    0.032   0.8584953 2.21    0.001   0.77582165 2.21    0.003   0.78520505 2.21    0.009   0.80205985 2.21    0.027   0.83085105 2.24    0.001   0.7227885 2.24    0.002   0.73391615 2.24    0.005   0.7543979 2.24    0.015   0.78798745 2.24    0.003   0.74176635 2.24    0.009   0.77064805 2.24    0.027   0.81042375 2.26    0.001   0.66545585 2.26    0.003   0.7012046 2.26    0.005   0.721067 2.26    0.009   0.7447984 2.26    0.015   0.76715245 2.26    0.027   0.794177 2.27    0.001   0.62916195 2.27    0.003   0.6774642 2.27    0.009   0.72961785 2.27    0.027   0.7861086 2.28    0.001   0.5750828 2.28    0.003   0.65059675 2.28    0.005   0.6802631 2.28    0.009   0.7145367 2.28    0.015   0.74447695 2.28    0.027   0.7774403 2.29    0.001   0.51357255 2.29    0.002   0.581053 2.29    0.003   0.6173075 2.29    0.009   0.6972096 2.29    0.027   0.76793225 2.31    0.001   0.36997965 2.31    0.002   0.474415 2.31    0.003   0.53649295 2.31    0.009   0.6587164 2.31    0.016   0.70870255 2.31    0.027   0.7482423 2.31    0.05    0.7912395 2.34    0.001   0.2204104 2.34    0.002   0.316308 2.34    0.003   0.39256745 2.34    0.004   0.45240835 2.34    0.009   0.5883453 2.34    0.016   0.6590771 2.34    0.027   0.71444205 2.34    0.05    0.7690014 2.38    0.001   0.13286995 2.38    0.002   0.1828288 2.38    0.004   0.2980268 2.38    0.008   0.4507145 2.38    0.016   0.58417075 2.38    0.032   0.6833616   And if I plot it in Mathematica (with ListPlot3D[data]) it looks, without further tweaking and fiddling around, like:    But if I try this pgfplots code  \begin{tikzpicture}   \begin{axis}     \addplot3 [surf] table {data.txt};   \end{axis} \end{tikzpicture}   I only get this:    What do I have to do to get a surface like in Mathematica (color, smoothness, style, etc. does not matter, just a closed surface)? What is "summoning sickness" and how does it affect my creatures? Note: this is intended to be a canonical question about summoning sickness.  "Summoning sickness", broadly, is the rule that stops you from attacking or paying {T} costs with a creature that you just played. But how exactly does it work? How does it interact with other rules and effects?  For example, if I'm trying to figure out how a creature is affected:   Does the status matter (tapped/untapped, face down/face up, etc.)? What if it came into play, but I didn't cast it? What if it's a token? What if it was on the battlefield when my turn started, but it wasn't a creature? What if it stops being a creature? What if it's a creature and also another card type? What if it's a double-faced card that just transformed? What if it changes control? What if it's a copy of another creature? Which of the creature's activated abilities can I use? Are any of its other abilities affected? Can I tap the creature to pay a cost that is not a {T} cost?   (These examples are primarily taken from questions about summoning sickness that have already been asked on the site) Configure tab-switch hotkey in gnome-terminal I used to use Alt+number to switch between gnome-terminal tabs. In my recent 19.10 installation, this does not work. When I press Alt+1, instead of switching tab, the command line shows this: (arg: 1)   and if I press Alt+2, it becomes: (arg: 12)   I checked the options in gnome-terminal, it is set to use Alt+number to switch between tabs. I would like to know which application is hijacking the hotkey, and how to fix it. PS. I do NOT want to use Ctrl+PgUp/PgDn for this. Thanks. I thought the new Apple font was called San Francisco, but I cannot find it in the font list of the Keynote.app for my Mac. Is it called another way or it is not available for Keynote?  I am currently using OS X El Capitan. Insert data only if record does not exist What books would you recommend to start learning about software testing I'm trying to learn more about software testing, so I want to know which books you would recommend on this topic Should black holes be considered two-dimensional or three-dimensional? (Google) Title of page not updated, everything else did My website changed significantly this month.  Amongst the changes was the website's title.  Two weeks ago, I finally used Webmaster Tools and submitted the page.  So far, Yahoo and Bing have updated the title in the search results, but not Google.  I've resubmitted via Webmaster Tools the website, and the bots have crawled the page, and have added "content keywords".  But the title still doesn't match.  The "content keywords" lists many of the words in the new title (but contains keywords of the existing title also).  Must say that 4 of the top 5 key words in the "significance" list are the new title's.  Still, no change.  Any remedies for this problem?  Is adding the "alt" tag for the logo significant enough to trigger the change?  There isn't an appropriate place to put the company name between  tags on the front page.  Any suggestions?  Thanks. I constantly stay up to date with each new release of Blender, but this means my preferences are wiped clean each time. How do I export/import these preferences to avoid loosing my carefully crafted set up each time? How do I pass variables and data from PHP to JavaScript? For example, if the ex-dividend date is next Monday, and I buy the stock on next Monday, would I get the dividend? Or would the person who sold the stock to me get the dividend? Determine which app is drawing on top of other apps? Is there a way to determine which application is drawing on top of other applications?  For example, Facebook Messenger app can draw the chat bubbles on top of everything, but how do I know which app is doing this?  My problem:  I have an app which is drawing ads on top of everything and I wanna know which app is doing this. Not just to uninstall it, but to warn everyone that uses this app. (I didn't install any application recently and I already verified the recent updates.)  I tried to use the Dump View Hierarchy for UI Automator from the Android Studio, but it seems that the applications that draw on top can't be selected this way.  (Click image to enlarge) I have often heard it recommended that a user account should be disabled by setting its shell to /bin/false. But, on my existing Linux systems, I see that a great number of existing accounts (all of them service accounts) have a shell of /sbin/nologin instead.  I see from the man page that /sbin/nologin prints a message to the user saying the account is disabled, and then exits. Presumably /bin/false would not print anything.  I also see that /sbin/nologin is listed in /etc/shells, while /bin/false is not.  The man page says that FTP will disable access for users with a shell not listed in /etc/shells and implies that other programs may do the same. Does that mean that somebody could FTP in with an account that has /sbin/nologin as its shell?  What is the difference here? Which one of these should I use to disable a user account, and in what circumstances? What other effects does a listing in /etc/shells have? Currently the migration path options from SO are:   Meta.SO Serverfault SuperUser Webmasters Programmers   If we are going to be limited to 5 options, shouldn't we periodically re-evaluate which sites are most appropriate?   (10k only) indicate that the top sites for the last 90 days are:   Superuser - 564 - 5% rejected ServerFault - 278 - 10% rejected Tex - 157 - 13% rejected DBA - 144 - 2% rejected Sharepoint - 127 - 1% rejected Programmers - 126 - 26% rejected CodeReview - 100 - 3% rejected Meta.SO - 81 - 18% rejected Wordpress - 75 - 14% rejected Webmasters - 58 - 8% rejected   All these migrations for sites out of the list required a flag and a moderator to process it.  This is just extra work for the mod team!  If we had the top 5 destinations by volume as the 5 options, we would have had a net savings of 160 flags to be processed.  I suspect we will actually see these numbers change quite a bit if it's easier for users to put questions in the appropriate spot without moderator intervention.  Can we have a review of the migration path list based on migration volume?  Think of the flags! Calculate sum of squares of first $n$ odd numbers Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2,$$ and how do you derive it?