Let $a,b,c$ be the length of sides of a triangle then prove that: $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$  Please help me!!!
For any triangle with sides $a$, $b$, $c$, prove the inequality $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0 .$$   This is IMO 1983 problem 6.  I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to nothing (please don't mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn't lead to anything for me. Could you give me a hint for finding the proper substitution?
iPhone App › Add voice recognition? Do airlines have to check the visa status before selling a ticket? Google's CDN is nice, but it's not 100% reliable for all users. It occasionally fails to load for me. Other users , and some .  It's one more possible point of failure. It would be preferable if Stack Exchange didn't depend on it.  If you don't want to stop using it, consider adding a fallback if it fails to load. Something like this:  &lt;script src="https://ajax.googleapis.com/ajax/libs/jquery/1.7.1/jquery.min.js"&gt;&lt;/script&gt; &lt;script&gt;   if (!window.jQuery) {     document.write('&lt;script src="https://cdn.sstatic.net/stackoverflow/content/js/jquery-1.7.1.min.js"&gt;&lt;\/script&gt;');   } &lt;/script&gt; external variable in awk I have a new laptop which I am setting up as dual boot Windows/Linux.    I would like to access files from both Ubuntu and Windows 7.  Should I create a separate NTFS partition? How do I distinguish between a command, utility, or builtin for getting documentation? "You just can't" vs. "you can't just " I'm a bit confused about this. Which expression is correct?     You can't just do that.    or      You just can't do that.   I'm trying to say:     You can't just bash an ideology because of what someone has said/done. Self-deletion and un-deletion without any changes should reinstate the original close/delete votes How to disable X server autostart in Debian Jessie? Berlin Public Transport Ticket Validity I have purchased an AB zone Berlin 7 day transport ticket and validated it on Sunday 17th April so hopefully it will be valid until Sunday 24th.  I validated the ticket at about 10:00am so I guess 1000 is the time I validated? But what are the numbers 15 and 6 and letters 'So' correspond to? How (if checked by a guard) can they tell if the ticket is valid? And how can I as a customer tell when it is validated to? I accidentally unset all the directories of $PATH while trying to add a new one in ~/.bashrc. I opened a new terminal window as I was editing and now $PATH is empty. I'm worried if I boot from another drive to find the $PATH I won't be able to boot into this drive again.  Basically, what is the default result of echo $PATH? How to resolve merge conflicts in Git repository? How do I generate a smooth random horizontal 2D tunnel? I'd like to create a smoother version of the navigable and quite natural-looking random tunnel found in .  It should ideally be...   infinite, so more can be generated as the player progresses; parametrised, allowing control of its thickness over time; made of smooth curves, not rectangles as in the above game; precomputable, as knowing its bounds in advance allows collision detection for e.g. positioning powerups inside the tunnel.   I'm looking for a generic method I could implement myself. Further parameters and optimizations are welcome.  Asking here was suggested . I think this fits in both places, as it's as much about the algorithm as about gamedev. How to invoke a system call via syscall or sysenter in inline assembly? I'm having trouble with \includegraphics I'm using MacTeX Firstly and fore-mostly I am a complete new comer to the world of LaTeX and so please do bear with me. I will try to be as accurate as I can but if you need any more info please let me know.  All I want is to figure out how to import a graphic into LaTeX (MacTeX).  I am using the following in my code:  \documentclass[10pt,a4paper,draft]{report} \usepackage{graphicx}   Then in my document section I'm using   \begin{figure} \includegraphics[width = 50mm]{Name.jpg} \caption{Label} \end{figure}   Once I compile the document, the area for the picture has been allocated but there is no image, I have tried .png and .pdf files also.  I know that this is a very simple question and I hope there is a simple solution but I cannot find anything on the net about this (which may mean I'm being dim).  Also I'm using Latexian but I have tried 4 different editors and it doesn't work in any. When is the formula $$S_{\infty} = \dfrac{a}{1-r}$$ valid?   Does |$r| &lt;1$? Rootspaces are $\mathop{ad}$ nilpotent I have a PDF with 100 pages. How can I make a new PDF that displays 4 pages on one?  I know that I can do it with Chrome by printing the PDF to new PDF and change this in the options. But I want to make it for a lot of files and put it as a script. What are the most important things I need to do to encourage Google Sitelinks? My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.  Here is how sitelinks might look for a site:    What are some things that I can do to improve my chances of getting sitelinks? How can I make a dispenser that has unlimited arrows?