If $a$, $b$ anc $c$ are three positives such that $abc = 1$ then prove that $$ \sum_{cyc}\dfrac{a}{a + b^4 + c^4} \le 1$$   Here's what I did.  $$ \sum_{cyc}\frac{a}{a + b^4 + c^4}$$  $$\le \sum_{cyc}\frac{a(b^2 + c^2 + bc)}{(b^2c + c^2b + abc)^2} = \frac{1}{(a + b + c)^2}\sum_{cyc}\frac{ab^2 + ac^2 + 1}{(bc)^2} \le \frac{1}{9}\sum_{cyc}\left(\frac{a}{b^2} + \frac{a}{c^2} + a^2\right)$$  And this is where I gave up.   Thanks for reading this. Even more thanks if you can solve the problem.  (Oh wait, I did try again.)  $$\sum_{cyc}\frac{a}{a + b^4 + c^4}$$  $$= 3 - \sum_{cyc}\frac{b^4 + c^4}{a + b^4 + c^4} \le 3 - 4\sum_{cyc}\frac{a^4}{b + c + c^4 + 2a^4 + b^4} = 2\sum_{cyc}\frac{b + c + b^4 + c^4}{b + c + c^4 + 2a^4 + b^4} - 3$$  ...and again...  $$ \sum_{cyc}\frac{a}{a + b^4 + c^4}$$  $$= 3 - \sum_{cyc}\frac{b^4 + c^4}{a + b^4 + c^4} \le 3 - \frac{1}{2}\sum_{cyc}\frac{(b^2 + c^2)^2}{a + b^4 + c^4} = \sum_{cyc}\frac{2(a - bc) + (b^4 + c^4)}{a + b^4 + c^4}$$  $$\le \sum_{cyc}\frac{1}{2(a - bc) + (b^4 + c^4)}\frac{[2(a - bc) + (b^4 + c^4)]^2}{a + 2b^2c^2}$$  $$= \frac{1}{2}\sum_{cyc}\dfrac{(a - bc + b^4)^2 + (a - bc + c^4)^2}{[2(a - bc) + (b^4 + c^4)](a + 2b^2c^2)}$$  And I need help. I would be grateful if you could help with this problem.
If $a,b,c \in \mathbb{R+}$ and $abc=1$ Prove That  $$S=\sum \frac{a}{a+b^4+c^4} \le 1$$  My approach:  we have $$S=\sum \frac{\frac{1}{bc}}{\frac{1}{bc}+b^4+c^4}=\sum \frac{1}{1+b^5c+bc^5}$$  Now by $AM \ge GM$ we have  $$\frac{1+b^5c+bc^5}{3} \ge b^2c^2$$ $\implies$  $$\frac{1}{1+b^5c+bc^5} \le \frac{1}{3b^2c^2}=\frac{a^2}{3}$$ Hence  $$S \le \frac{a^2+b^2+c^2}{3}$$  Can we proceed with this?
dropping ticklabel on pgfplot axis We put all of our unit tests in their own projects.  We find that we have to make certain classes public instead of internal just for the unit tests.  Is there anyway to avoid having to do this.  What are the memory implication by making classes public instead of sealed? Getting "'file' is different from the same file on the system" when I attempt to update through synaptic Encountered an error while updating through Synaptic Manager. Subsequent updates gave the error at the bottom of the output. I'm a noob so any help is appreciated.  al@al:~$ sudo apt-get -f upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following NEW packages will be installed:   libjack-jackd2-0:i386 The following packages have been kept back:   linux-headers-generic skype 0 upgraded, 1 newly installed, 0 to remove and 2 not upgraded. Need to get 0 B/202 kB of archives. After this operation, 493 kB of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 310056 files and directories currently installed.) Unpacking libjack-jackd2-0:i386 (from .../libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb (--unpack):  './usr/share/doc/libjack-jackd2-0/buildinfo.gz' is different from the same file on the system dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) Labeling (one-way) streets in QGIS? Literal @YES not working in iOS 5 / Xcode 4.4 How to simulate slow internet connections on the mac What's a good software solution to slow down internet connections on the mac for testing and simulations?  My primary need is when testing code in the iPhone / iOS simulator, so slowing down the connection for specific applications or processes would be great. Remove the most popular tag from the title using JavaScript So, I see that  to fight site scrapers. But, this is harmful to User Experience. With browsers like Google Chrome, all I can see is the first few characters of the title, and it becomes annoying to try to find the right tab.  So, my request: Can you remove the tag from the &lt;title&gt; using JavaScript? Crawlers will still see the SEO-friendly &lt;title&gt;, and users won't have their user experience hampered by it. We all win. Where's my 5 minute grace period gone? I just posted an answer. (Revision history .)  I edited it a few seconds later (within my 5 minute grace period), and it comes up as a new revision!? Why?    ^^You can see 'answered 1 min ago', and then 'edited 1 min ago' - how is that possible?  Additionally, !?    I don't think this is a dupe of . See my comments on AI.E's answer. If you're a mod, it will really help. How to exclude a column from SELECT query? When writing my custom function I need to manually append node name to a taxonomy term link. To get the taxonomy term part done I use:    $link_uri = taxonomy_term_uri($term);   $path_alias = drupal_get_path_alias($link_uri['path']);   This works ok, now - I want to append node name ("/" . $node_name), but instead of raw string, I want the Pathauto transliterated output, as set in the URL alias configuration, for that node context.  What function should I use to mimic that output? I looked at hook_pathauto_alias_alter(&amp;$alias, array $context) but I'm not sure that's what I need. Troubleshooting for biber I'm trying to switch from BibTeX to biber and ran into the problem that biber does not produce any output file. I noticed several questions about problems with biber e.g.,            ,                       But none of them helped me in my particular situation (which was actually just an old version of biber somewhere in the path of my OS). Since I also noticed that several questions regarding problems with biber were closed as too localized since the underlying problem was due to formatting of a bib resource or similar simple issues, I would rather like to ask for a general troubleshooting guideline for using biber (since I saw many hints like "check/delete the cache", "check versions of this and that" etc.).  What steps are to be taken and what things (in particular files) to be checked if biber does not produce an output file (and its output on the command line does not directly give you the advice what to do)? Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| &gt; 1$.  There has been a similar question on MSE here:  Unfortunately, I cannot comment on other user's questions yet, so I made my own question.  I know that the trick is to use conjugation but I can't follow the logic. Can someone elaborate? Especially on the last part of the top answer. I've always managed my iPhone/iPad/iPod Manually, I recently updated to iTunes 11 and now I can see no way to get my music onto my device.  I used to be able to just drag songs from my library to my device, now there is seemingly no way to do this.  I can view the music on my device by clicking on it on the top right corner of iTunes, but I can't seem to get to a state where I can drag music from my computer onto my device.  I'm using iTunes 11.0.1.12 for Windows if that makes a difference. There are a lot of poorly drawn schematics here. A few times people have actually asked for critiques of their schematics. This question is intended as a single repository on schematic drawing rules and guidelines can point people to. The question is  What are the rules and guidelines for drawing good schematics?     Note: This is about schematics themselves, not about the circuits they represent. What is the black thing on the circuit What is the black thing on the below circuit?   Is it used to hide the IC (or other components) that is used on the circuit?   What material is it? Ubuntu installation is slow, why? Increase progress bar each second -- error Prove every element of $G$ has finite order. In another  regarding a difference between short on and short of, the  says the following:     ... But this is not always the case. A person can be short on/of money,   but lacking a specific amount, someone can only be €10 short or short   €10. This construction never uses a preposition.   but is that really true? I can see sentences like "I am short of 10 dollars" quite often. When to use (and not use) the rule of three