Prove that $$a^ab^bc^c\ge (abc)^{(a+b+c)/3}$$  My attempt: $$a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}\implies \bigg(\dfrac{a}{b}\bigg)^{(a-b)/{3}}\bigg(\dfrac{b}{c}\bigg)^{(b-c)/{3}}\bigg(\dfrac{c}{a}\bigg)^{(c-a)/{3}}\ge 1$$    I have a slightest hint that symmetry might be helpful here, but wondering how. Please help.  I know that this question has been asked before, but I could not see any soltuion having my approach, so I am wondering if any solution is possible using my approach or it is a dead end.
Prove that the following inequality holds   $$a^a b^b c^c\ge (abc)^{\frac{a+b+c}{3}}$$   if $a,b,c$ are positive.   I'm not sure how to handle these kinds of powers. Are there any "famous" but not so advanced inequalities that involve this kinds of expressions?  I was trying to do something like ordering them without lost of generality because it's symmetric, but I get nowhere and I just make the thing even more complicated.  I'm asking for solution rather than hints since this is my first time encountering these kinds of problems.
Show all inbox items (notifications) in app The SE app currently only shows some inbox items:   Answers Comments Chats   When I see those I assume I'm all caught up, but it turns out there are more on the web which aren't displayed in the app:   Bounty expiration Elections Edits on your own posts Maybe others?   I was about to tag this  but realized it's probably intentional.  I imagine the reason for not showing them is because they're not something you can act on from within the app. Bounties can't be offered or awarded1 and elections can't be voted on.  But then again chats can't be replied to either. Badge descriptions can't be viewed (on the Achievements side). Selecting those launches the browser. Why then shouldn't bounties or elections launch the browser too? Hiding them just means I'll miss it if I haven't visited in browser recently.  1 Though accepting answers does award the bounty Steam not opening in Ubuntu 16.04 LTS Where can I find 32-bit version of Ubuntu? I need to find the 32-bit version, and all I keep finding is the 64-bit. Can anyone provide me with a link? Is there a way of moving unwanted villagers out faster? Is there any way to speed up this process or indirectly 'force a villager to move out'?  I know they move out if it's time for a new one to move in, and I know you can't actually 'force' them persay.  I have a villager who has caused nearly all my other villagers to wear his.. rather hideous 'kiddie shirt'. ((the character's default shirt)) Aside from that his home blocks the perfect spot, very close to my river, for building a convenient bridge...  I've taken to pushing him, hitting him with a net, and I've been trying to find a Toy Hammer... He wanted to leave once and I told him to go, but then he said he wouldn't leave until I, "made an effort to get to know him" or some BS..  Are there any other ways I can get an unwanted, 'pesty', villager to leave town? Name of a special matrix I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like  $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    e &amp; f &amp; g &amp; c \\    h &amp; i &amp; f &amp; b \\    j &amp; h &amp; e &amp; a  \end{array} \right]$$  Does this matrix have a special name in literature? What are it's properties?  And a matrix that is symmetrical by both diagonals   $$A = \left[ \begin{array}{c c c c}    a &amp; b &amp; c &amp; d \\    b &amp; e &amp; f &amp; c \\    c &amp; f &amp; e &amp; b \\    d &amp; c &amp; b &amp; a  \end{array} \right]$$  What's the name of it? Any interesting properties? How can I backup my photos while travelling? I'm going travelling around the world for a year in September and am already panicking about how best to send my photos home while in various countries.  Requirements:   I only need to backup JPEG files (I don't shoot RAW but that's another thread ;) ) I need to be able to upload in a web-based manner (I will be using Internet cafes and so can't assume I can use FTP etc) I need plenty of space, preferably unlimited (I racked up about 200GB last year) I need to store the full image (no lossy compression is acceptable!)   Options I can think of:   Burning and posting DVDs (can I trust these in the post and not to get corrupt? It may be better than waiting for upload speed though...) Flickr (I have a pro account but not sure how hard it'll be to get the photos back down again. Will I even be able to get the original snaps?) Setting up a server at a friend's house and hosting some kind of file server with a web-app front end   What have other people done? Or does anyone else have a better idea? Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$ Origin/reason for the expression "on the bus" instead of "in the bus" This is sort of a follow up to my question .  I was told a while ago that the reason why we use "on the bus" instead of "in the bus" is because back in the day buses were open, that is, they didn't have a roof.   Is this story correct or is there another reason why the correct expression is "on the bus", in spite of the fact that when one gets on the bus, he is actually inside the bus? Allow deleting a tag wiki or excerpt Sometimes a tag wiki excerpt or body ends up with unacceptable or useless content: plagiarized text that must be removed, or a totally unhelpful dictionary definition. (For example:  — and I've seen this situation many times before.)  The only way to get rid of that content is to submit replacement text that meets the minimum length requirement. This is not always easy as writing a decent wiki, and especially a decent excerpt, requires subject knowledge.  Currently the users or even moderators who are alerted to unacceptable content in a tag wiki cannot remove that content without providing their own, which they are not always in a position to do. Yet these wikis would be better off with no content than with either unacceptable or unhelpful content, or some content written by a non-expert in a hurry.  Please provide a way for users to blank a tag wiki excerpt or body. I'm not sure what the reputation requirement should be: same mechanism as for tag wiki edits (suggested edits with approval), or only done directly by a user with the trusted user (20k) privilege, or moderator-only. Do Stack Exchange sites allow editing of comments and posts from other users? Possible Duplicate:         Is this normal behavior in Stack Exchange sites that after just signing up one can edit not only his/her own comments / posts, but also others' comments / posts?  If so, then the Q&amp;A does not look safe to me. Imagine someone maliciously modifying a post and bringing bad reputation on a Q&amp;A member.  Can you check this and perhaps fix it? Will I get OTA updates after installing Android L Developer preview? What home printer would allow for exact alignment of front and back? Note: I'm not a graphic designer, but this seemed like the most relevant place to post. Feel free to send me off somewhere else if I'm not in the right place.  I would like to make some notebooks with double-sided grid paper that I've already designed, but my printer seems like it doesn't allow for absolutely exact alignment. Even if one side prints nice and straight, the other side will be slightly skewed left to right by one to two millimeters, which I read from an earlier post is a pretty reasonable variance for a home printer. So I suspect it's not possible to get both sides completely aligned on this printer – but if I'm wrong, please let me know how.  What kind of printer would allow for absolutely exact alignment? Is there any difference between inkjet and laser in this regard, or a more expensive printer versus a cheaper one? How many spell slots does a Level 3 Warlock have? The Warlock Table in the PHB (p106) states that a Warlock has the following:   1 First-level spell slot at level 1. 2 First-level spell slots at level 2. 2 Second-level spell slots at level 3.   Does this mean that a third level Warlock has a cumulative 5 Spell Slots? (3 First-Level slots and 2 Second-Level slots)  Or do the new slots just replace old slots between levels such that a 3rd-level Warlock ultimately has only 2 second-level spell slots? Does deleting iCloud photos affect them on devices? If I delete a photo from my iCloud will it also be removed from my devices that are connected to my account? Is C++ CLI a superset of C++? Would a C++  compiler be able to compile some large sets of C++ classes without modifications?   Is C++ CLI a superset of C++? How do I use filenames that start with a dash "-" as command arguments? I have a master's degree in International Studies, and a double major with Computer Science from undergrad.  My transcripts suck.  There's no other way to dress it up.  I have pretty good teaching experience, and my GRE scores are awesome, and I suspect my recommendations are as bland as everyone else's.  Basically, to an admissions committee, I suspect I'm the model of a student who is probably smart enough but didn't work hard enough.  I want to do a Ph.D. in Political Science, but the response from my applications is looking pretty grim.  Am I permanently out of the running, or is there something I can do for the next few years which will help to counterbalance my unfavorable GPA? Edit Mail rules manually How to switch from Resident visa to Tourist visa status in the Schengen area? Removing files with strange names Possible Duplicate:          Somehow I ended up with a file named "-r".  How do I remove it?  rm -r doesn't work.  I tried 'rm -i `ls -a`' to step through the file names, but it didn't prompt me to delete this one.  Edit A very hacky approach was to use python's os.unlink function.  That worked, but I'm curious to hear other ways.