The question is to show $$ xyz&gt;(-x+y+z)(x-y+z)(x+y-z) $$ where $x$, $y$ and $z$ are distinct and strictly positive. My attempt: Let, $$ \begin{aligned} -x+y+z &amp;= a \\ x-y+z &amp;= b \\ x+y-z &amp;= c \end{aligned} $$ Clearly, $$ x=\frac{b+c}{2} \text{ , } y=\frac{c+a}{2} \text{ , } z=\frac{a+b}{2} $$ Therefore, $$ \begin{aligned} xyz=\frac{(a+b)(b+c)(c+a)}{8} &amp;= \frac{a^2(b+c)+b^2(c+a)+c^2(a+b)+2abc}{8} \\ \implies 8xyz-2abc &amp;= a^2(b+c)+b^2(c+a)+c^2(a+b) \end{aligned} $$ Now, for all $a,b,c&gt;0$, we can easily prove that $$ \begin{aligned} a^2(b+c)+b^2(c+a)+c^2(a+b) &amp;&gt; 6abc \\ \implies 8xyz-2abc &amp;&gt; 6abc \\ \implies 8xyz &amp;&gt; 8abc \end{aligned} $$ But it is also possible that at most one of the $a$, $b$ or $c$ could be negative, in which case, it is quite obvious that $xyz&gt;abc$. My question: Is there a more concise way to prove this?
Let $a,b,c\in\mathbb{R}^*_+$, $abc=1$.  How can i show that $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ ?  I got $\left(ab-b+1\right)\left(bc-c+1\right)\left(ca-a+1\right)\leq abc=1$, but i can't go any further ...  If i expand it :   $2-ab-\frac{b}a-c-ac-\frac{a}c-b+2a+\frac{2}a-bc-\frac{c}b-a+2b+\frac{2}b+2c+\frac{2}c-4$  Which gives using $c=\frac{1}{ab}$ :   $5ab-\frac{b}a-\frac{1}{ab}-a^2b+\frac{1}a+b+\frac{1}b-2$
Testing Whether a Binomial Distribution Fits Data I have a sample from a discrete distribution as such:   Type:          0   1   2   3   4   5     Occurrences:  88  12  52  43  21   5   My task is to test whether or not a Binomial Distribution (n=5,p) fits this data.   I understand that I'm meant to use hypothesis testing, and that the chi-square test is a staple for this sort of task. I looked at  but I'm still not sure how to set up the test. How do I do this? combining results of two select statements This is probably easier to show than tell...   On the recent question , I see the following:    As you can see, the first five links render properly, but after that, they stop converting.  Unfortunately, this isn't obvious to the person writing the post, which I discovered when I went to edit the post to fix the "broken" links. In the edit window, it renders all of the links just fine:    I understand that this behavior is expected based on balpha's answer :        Links to questions on the same site are unlimited.   Links to other sites are limited to             a) 5 linked-to sites and     b) N questions per target site, where N is 200 on meta sites and 10 on non-metas (this is about the site that the post containing the link is posted on, not the site that is being linked to)         So if this rule is still going to stay in effect, please fix the edit view to also not automatically convert more than five linked-to sites to help people writing posts know that they need to manually title the links to the other questions.  That said, I'd prefer that (at least on MSE) this limitation be removed or raised. Even in the answer, balpha says:     The restrictions are currently as follows, but note that they do look a bit contrived, resulting from the fact that they have grown over time (from single site to cross-site, from "there are no subsites" to "facebook.SO", etc.), so it's not unlikely that we'll revisit them.   So, if it's not hurting performance, can we please increase the limit or remove it? My question is the same as . I put my stackexchange flair at the bottom of my sites:     And get a &quot;profile for Ninsuo on Stack Exchange, a network of free, community-driven Q&amp;A sites&quot; link instead of my nice flair. By opening Firebug, I can read the following error:  Or as text:  NetworkError: 500 Internal Server Error -   I did not tried to update a new picture - just in case you want to get further first. Good luck! How do I resolve "WILL_NOT_PERFORM" MS AD reply when trying to change password in scala w/ the unboundid LDAP SDK? How is The Wall bypassed? In the movies, the pack of Uruk-Hai capture Merry and Pippin, and are bringing them back to Saruman as quickly as possible.  Why didn't they use Wargs to carry the Hobbits instead of packing them on their backs?    Or, better yet, why didn't the Nazgul fly down and take them on their Fell Beasts? Why aren't the braces showing HTTP 401 twice before HTTP 200 when serving files What Thunderbolt to USB 3.0 cables/adapters exist for OS X and Mac hardware? I have the Late 2013 Macbook Pro, which has two USB 3.0 ports and two Thunderbolt 2 ports. I currently use both USB 3.0 ports and one TB2 port. I want to connect a third USB 3.0 device to the machine, but I don't want to use a USB splitter on one USB port as I'm concerned the devices would draw too much power from the bus. I have no current use for the second TB2 port and I would love to be able to use it for this purpose.   Browsing extensively online, I've found numerous docks, such as , that connect via a Thunderbolt cable. This would work, but it's an expensive, 'bulkier' solution (and not the one I'm looking for).   I'm aware that a USB 3.0 device wouldn't be able to take advantage of Thunderbolt's I/O speeds, but it's certainly compatible, which is all I'd need. After all, Apple even sells a . I also know that Thunderbolt is not only used for digital video purposes (along with the TB > Firewire option, there exists ).   There are almost no Thunderbolt to USB 3.0 cables/adapters on the market. I'd think this hardware would be more prevalent given the prominence of USB devices out there. What adapters are available for OS X? How should I implement hook_menu()? Yesterday I was filming a scene with my phone but suddenly the application crashed and when I try to open the m4v generated it says that is corrupted.  I have downloaded the video files (around 36MB) on my PC and I am trying to open and repair it. Can you suggest me a way to do it?  FFMPEG info  As suggested in comments I try check what FFMPEG says me about that file with this command  ffmpeg.exe -i VID_20120425_144739.m4v -c:v copy -c:a copy out.mp4 2&gt; log.txt   Here it is the result:  ffmpeg version N-40126-ga4b58fd Copyright (c) 2000-2012 the FFmpeg developers   built on Apr 26 2012 03:29:50 with gcc 4.6.3   configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-runtime-cpudetect --enable-avisynth --enable-bzlib --enable-frei0r --enable-libass --enable-libcelt --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libfreetype --enable-libgsm --enable-libmp3lame --enable-libnut --enable-libopenjpeg --enable-librtmp --enable-libschroedinger --enable-libspeex --enable-libtheora --enable-libutvideo --enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx --enable-libx264 --enable-libxavs --enable-libxvid --enable-zlib   libavutil      51. 47.100 / 51. 47.100   libavcodec     54. 15.100 / 54. 15.100   libavformat    54.  3.100 / 54.  3.100   libavdevice    53.  4.100 / 53.  4.100   libavfilter     2. 72.100 /  2. 72.100   libswscale      2.  1.100 /  2.  1.100   libswresample   0. 11.100 /  0. 11.100   libpostproc    52.  0.100 / 52.  0.100 [mov,mp4,m4a,3gp,3g2,mj2 @ 003fc940] moov atom not found VID_20120425_144739.m4v: Invalid data found when processing input   Latest two lines are displayed in red. What is the best backup strategy for macbook pro laptop? I have a lot of data on my macbook pro retina laptop and i am looking for the best backup strategy.  I have previously had a windows desktop with an external harddrive connected all of the time and a backup program that copied to the external harddrive incrementally.  I am trying to figure out the best backup strategy given its a laptop that i move around with and travel with alot (so can't really have a external hard drive always connected)  What is the recommended backup solution for map laptops? Should I put myself last? "me and my friends" vs. "my friends and me" or "my friends and I" I've always been taught to put myself last when referring to myself in the same sentence as others but the usage of "me and..." seems to be everywhere these days. The misuse of the word "me" instead of "I" aside, is there some new rule I haven't heard of? Shouldn't we put ourselves last regardless of the "me"/"I" usage?  Examples of "correct" usage:      My friends and I went for some ice   cream. Did you see my friends and me   at the ice cream stand?   Examples of "incorrect" usage:     Me and my friends went for some ice   cream. Did you see me and my friends   at the ice cream stand?   Note: I was also taught that the only person who could put themselves first was the queen. During the First Wizarding War, why didn't Dumbledore just hunt Voldemort down, as he had done with Grindelwald?  By 'hunting down', I mean finding, defeating and imprisoning him, exactly as in Gridelwald's case, and not, say killing him, something which Dumbledore would not do, as the answers to a previous question (as to him not hunting down Death Eaters) show.  Dumbledore knew that Voldemort had delved into the Dark Arts deeper than anyone else in history and hence "had powers he will never have". But he must have also known well that he could have defeated Voldemort at a direct confrontation, given that he had defeated Grindelwald earlier even against the Elder Wand, and now had that same wand at his disposal. He is seen confident about this throughout the series.  At the Battle of the Department of Mysteries, he did not go beyond outweighing Voldemort and forcing him to flee, probably because, by then, he had reasons to suspect that there was more to Voldemort's survival than his physical body, and thus, destroying that body which had Harry's blood without permanently disposing him off would not have been wise.  But why did he not do that at the First Wizarding War? Is it just that they could not track him down, or did he suspect even then that Voldemort had more sinister things about his soul? is there any canon confirmation of the latter suspicion? Keyboard shortcut for switching between different keyboard layouts in Ubuntu Mate Is there a pre-assigned keyboard shortcut for switching between different keyboard layouts in Ubuntu Mate? How can I block app and game invitations once and for all in Facebook? I want to block app and game invitations once and for all in Facebook. How can I do it so they cannot disturb me anymore? In the radio series, how did Zaphod get into the artificial universe? In the book The Restaurant at the End of the Universe, it's very clearly explained how Zaphod gets into the simulated universe; he's in it from the moment he enters Zarniwop's office, and a big deal is made of him leaving via the window rather than the door so that he stays in it.  In the radio series, however, this doesn't happen. At the end of Fit the Seventh, Zaphod asks "How do we get to this universe, then?" and Roosta replies "I'm afraid it looks like we don't...", pointing out that the building is splitting down the middle, preventing them reaching the office.  Later, in Fit the Twelfth, Zaphod finally meets Zarniwoop in the lounge of the slightly-delayed spaceliner and reacts with astonishment at learning he's been in the artificial universe all this time. "How did we get into it? I mean like where, when?"  Zarniwoop only replies enigmatically: "You didn't notice? Well, I'll let you work it out for yourself."  So, how did he get in? As noted, he never actually entered Zarniwoop's office. My only thought is that it happened during the "body debit" incident with the disco, but there's no particular indication that that should be the case. Is there any other possibility? What is the Taylor series of $$\ln \frac{1-x^2}{1+x^2}$$   ?   I started by evaluating the first derivatives but the more I go, the more complicated they are and I can't identify a pattern. I am currently working on an update of tkz-euclide and new documentation but I just met the following problem. If I compile this code with pdflatex or lualatex and pgf 2.1 (The library babel does not exist with pgf 2.1) then the code compiles fine but with 3.0 pgf I must comment on the call \usetikzlibrary{babel} (pgf doc gives so I recommend to always load this library by saying \usetikzlibrary{babel}).  I'd like to know where the error occurred. Especially I would like to work around this problem.   With \usetikzlibrary{babel} I get the error Latex Error: ./untitled 2.tex:46 Undefined control sequence. l.1 \tkz@line@start  Bug in babel, bug in tikzlibrary{babel} bug in my code. I would like to keep  \usetikzlibrary{babel} but to avoid the error.  %!TEX TS-program = pdflatex \documentclass[a4paper]{article} \usepackage[french]{babel} \usepackage{tikz}  \usetikzlibrary{calc} %\usetikzlibrary{babel}  \makeatletter \pgfkeys{%     tkzdrawline/.cd,     start/.code         =  {\gdef\tkz@line@start{#1}},%     end/.code           =  {\gdef\tkz@line@end{#1}},%     start style/.code   =  {\tikzset{tkzstartstyle/.style={#1}}},%     end style/.code     =  {\tikzset{tkzendstyle/.style={#1}}},%    /tkzdrawline/.unknown/.code ={\let\searchname=\pgfkeyscurrentname\pgfkeysalso{%                          \searchname/.try=#1,/tikz/\searchname/.retry=#1}}}  %&lt;--------------------------------------------------------------–&gt; %                              Draw line %&lt;--------------------------------------------------------------–&gt; \pgfkeys{tkzdrawline/.cd, start/.code       =  {\def\tkz@line@start{#1}},% end/.code         =  {\def\tkz@line@end{#1}}, start style/.code =  {\tikzset{tkzstartstyle/.style={#1}}}, end style/.code   =  {\tikzset{tkzendstyle/.style={#1}}},   /tkzdrawline/.unknown/.code   = {\let\searchname=\pgfkeyscurrentname                                  \pgfkeysalso{\searchname/.try=#1,                                 /tikz/\searchname/.retry=#1}}}%         %&lt;--------------------------------------------------------------–&gt;  \def\tkzDrawLine{\pgfutil@ifnextchar[{\tkz@DrawLine}{%                                   \tkz@DrawLine[]}}%   \def\tkz@DrawLine[#1](#2,#3){%  \begingroup   \pgfkeys{tkzdrawline/.cd,start = {}, end = {}, start style={}, end style={},}   \pgfqkeys{/tkzdrawline}{#1}   \draw[ /tkzdrawline/.cd,#1] (#2) to%     node [at start,left,/tkzdrawline/.cd,tkzstartstyle] {\tkz@line@start}%     node [at end,right,/tkzdrawline/.cd,tkzendstyle] {\tkz@line@end} (#3);  \endgroup  }%   \makeatother   \begin{document}    \begin{tikzpicture}      \node (A) at (0,1) {};       \node (B) at (5,2) {};       \tkzDrawLine[end = {$\alpha$}](A,B)       \end{tikzpicture}  \end{document}