Note: we have $a&gt;0,b&gt;0,c&gt;0$.  Solve $(1-a)^2+(1-b)^2+(1-c)^2 \ge \frac{c^2(1-a^2)(1-b^2)}{(ab+c)^2}+\frac{b^2(1-a^2)(1-c^2)}{(ac+b)^2}+\frac{a^2(1-c^2)(1-b^2)}{(bc+a)^2}$ where $a&gt;0,b&gt;0,c&gt;0$. Elementary (high-school) methods are preferred.   I'm not an expert in inequalities. I only know GM-AM and it does not appear to be working. It is not clear to me how to take advantage of the symmetry here.
Let $x,y,z&gt;0$. Show that $$\sum_{cyc}(1-x)^2\ge \sum_{cyc}\dfrac{z^2(1-x^2)(1-y^2)}{(xy+z)^2}.$$ Here's what I have done. The expression $$\sum_{cyc}(1-x)\left((1-x)-\dfrac{z^2(1+x)(1-y^2)}{(xy+z)^2}\right)\ge 0$$ or $$\sum_{cyc}(1-x)\cdot\dfrac{x^2y^2+y^2z^2+xy^2z^2+2xyz-2xz^2-2x^2yz-x^3y^2}{(xy+z)^2}\ge 0.$$ But this way does not help for the starting inequality.
This is a repost with additional info of an older question that was never answered. I have stumbled upon this old question while also trying to find the name of the book series myself, as I dearly wish to read it again but for the life of me, I can't remember the name of it.      This is a long shot but I'm really in need of help for this because      it was an amazing book. It was either a kid's or YA book. A guess      from what year it was published would be 90s-mid 2000s, as I read it      around 2008-2011.         The book cover was in first person view, with the main character   holding a weapon, and was a 3D rendered image inside a dungeon or   cave (that was very reminiscent of early 90s CGI animation), though   there may be variants.    It would have been set around when computer   games were mainstream (maybe around the same time as when it might've   been published) It's about a teenage boy who is bored at home, so he   looks through an old family computer to play a game and finds one   that he has never seen before. When he plays it he gets transported   into the game which is a fantasy-medieval setting.    All that I can   remember about the game and story is that the main character is on a   quest, he rides horses a lot, (to the point where he can ride horses   perfectly in real life), at one point he eats food at an ingame   tavern/restaurant? and he tosses the ingame coin currency at the   server because he's in a hurry, (don't know why I remember that bit   specifically haha), and he meets a gnome/dwarf character who is   helping him on his journey, who has a home in an ingame mountain, and   that time goes faster in the game. (For example, if he spent 5   minutes in the real world, 3 in-game days would have passed)   Sometimes as he's playing something stops him from being able the   play the game for a few weeks, maybe a concerned parent because he's   playing too much or there's a family vacation. I remember during the   time when he's not playing the game he's on a farm and knew how to   ride the horses there.    There was a character in the real world who   was the main character's sibling, relative or friend. They were like   a rival and was also very whiny (to some extent). They were on the   farm as well with him and questioned how the main character knew how   to ride a horse when he had never done it in real life.   This is all I can remember but if you need more specific information I can try and answer. It is definitely not Space Demons or Demons Don't Dream. If anyone knows anything remotely similar please mention it, I would really appreciate it. Thank you!!            Additional info:    It is not an online game or virtual reality game. It was a normal PC game, and all other characters are (I think), just NPCs.  He woke up for the first time in a field/farm and was super confused about what happened and where he was, as he had never played the game before. He is asked a super serious question before starting the game up for the first time, like "are you sure you wish to play ___?", and then he is transported into the game.  He frequently exited the game, to continue his IRL life. And sometimes, he would lose access to the family desktop PC for a couple of days at a time, and when he went back everyone in the game (NPCs), thought he had disappeared because he was gone for ages. I think once, his family when on holidays and he couldn't take the PC, for example.  It was a book series, not a single book. In later books he would play the game again months later in the real world, and years and years would have passed in the game world.  The plot involved 2 main warring sides/factions/countries of humans, but there were fantasy/magical races and characters involved. I vaguely remember something about each side having official colours, one red and one blue. I think the main character was on the blue side.   The original question was never correctly answered, so I'm posting again to see if anyone can remember.   Books that were ruled out in original post: Eden's Gate &amp; Heir Apparent. How to install Tor? I tried to install Tor on my new version of Ubuntu, but it seems that Tor is not available in the repositories.  How else can I install and use Tor? Is spacetime absolute? As I understand it Newton's Laws imply that space is relative, as the laws of physics are the same in all inertial frames and as such there is no way, even in principle, to distinguish a frame that is truly at rest (absolute space). Hence the concept is physically meaningless, and the positions of (and distances between) physical objects, events, etc. are relative to the frame in which one is observing them from. Advancing on to Special Relativity, by postulating that the speed of light (in vacuum) is constant in all frames of reference, we are forced to conclude that time is relative also (as if it weren't then different observers would observe a different speed of light (in vacuum)). This leads to the result that the concepts of space and time are no longer completely separate and independent of one another and instead intertwined (as an event that occurs at rest over some time period in one frame, will occur over some spatial interval and a different time interval in another frame). Hence they should be considered as should be considered as a single entity, called spacetime.  Sorry for the waffling so far, just want to check that my understanding is correct up to this point?!  My main question is, given that space and time are (individually) relative quantities, is spacetime itself relative, or can it be considered absolute (as after all, it is the mathematical space of all possible events and exists independently of the physical events that occur within it)? Explanation of C++ FAQ's unsafe macro? How to add new element to Varargs? Fontspec causes long underscores My collaborator added \usepackage[no-math]{fontspec} to a document we're working on. The result is that \_ underscores become v. long: cf. pictures below.   \documentclass{article} \usepackage[no-math]{fontspec} \pagestyle{empty}  \begin{document} \textrm{metric\_space} \end{document}   With fontspec:    Without fontspec:   The underscores are about twice as long with fontspec, and also much thinner. My question is, how do I get back to the original width? (Ideally I would like to preserve the thinness -- it looks much nicer than the original.)  In case someone asks, "why are you using \_?", the short answer is that I need to typeset identifiers involving an underscore. See  for an example in context. I'd be happy with using something other than \textrm, if it looks decent in the context of that example. I am writing a small KOMA-Script scrartcl document with multiple sections. In the table of contents, LaTeX fills the current line with dots between the end of the section name and the page number. Is there any way to customize this behavior? I would like to reduce the space between the single dots (i.e. use more dots in the same space). Remove files with 0 length inside bash I want to create a bash script inside Linux terminal to go into a file and then delete it if it's empty:  #!/bin/bash echo " Enter name of a directory" read Dir for Directory in $( ls $HOME ) do    test -d $HOME/$dir   if [ $# -eq 0 ]   then      for filename in $( ls $HOME/$Dir )     do        test -d $HOME/$dir/$filename       if [ ! -s "$filename"  ]       then         rm $HOME/$dir/$filename       else          echo " not empty"       fi      done    else      echo " $Dir is not a directory"    fi  done   This is the code I am using but it does not work. It recognizes the file created but doesn't delete it even when empty. It Salah displays there is no such file or directory Should Dalvik Cache be wiped in Android OS that supports ART? When switching custom ROMs (in ART), is it necessary to wipe the Dalvik Cache?  Wiping the Cache is necessary, but should wiping Dalvik (which doesn't exist in Android Lollipop and afterward) be necessary?  What impact will it cause if we wipe Dalvik (something not present)?  In ART, what does the Dalvik cache contains? (is it an empty folder?) Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$ Prove that  $(x_1+\dots+x_n)^2 \leq n(x_1^2 + \dots+x_n^2)$  for all positive integers n and all real numbers $x_1,....,x_n$  I am attempting a proof by induction but wasn't sure if i need the Cauchy-Schwarz Inequality or perhaps another way other than induction to prove this.  Proof   $n=1$ true  assume true for $n=k$  Now for $n= k+1$  $(x_1 + \dots +x_k + x_{k+1})^2 \leq \dots$ Pronoun question: referring to inanimate objects as 'he' or 'she' I read the following claim concerning pronouns referring to inanimate objects:      Anything that is meant to contain you, protect you or provide you with something beneficial is [often referred to as] a she; anything that is a perceived threat is a he. That's why cars, boats and some countries are she.    Is this really the case, or is it just a subjective claim? (According to the  using he/she for inanimate objects is not recommended and it should be used instead.) Is there a 'rule' for determining whether to use he, she or it based on the impression one would like to express? (I only know the 'rule' for animals: In the case where you know their gender and they are important to you, you refer to them using he/she. E.g. A dog attacked me in the street. It bit me. versus This is my dog Roger. He is 7 years old.) Have these practices changed over time?   Note 1: There is already a similar question (), but that considers only the feminine case and none of the answers offers an objective discussion of the matter.  Note 2: My native language is Czech, where the gender of the pronoun is based only on the grammatical gender of the noun. E.g. a cat (kočka) is always she, a dog (pes) is always he, a boat (loď) is always she and a car (auto) is always it. What should the standard spelling be - British or US? Trouble downloading packages list due to a "Hash sum mismatch" error Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$. The statement I am trying to prove:  Let $\{ f_n \}$ be a sequence of equicontinuous, real valued, uniformly bounded   continuous functions on $\mathbb{R}$. Show that $\{ f_n \}$ has a convergent subsequence   which converges uniformly on any bounded subset of $\mathbb{R}$ and pointwise on all   of $\mathbb{R}$ to a continuous function. (Royden pg 210, problem 9).  I believe the first portion follows directly by Arzela Ascoli, once it is noted that if we are given some bounded subset $E$, we can look at $\overline{E}$ a compact set on which the family is equicontinuous and uniformly bounded.  For pointwise convergence on all of $\mathbb{R}$, the question seems to require some sort of diagonalization argument: Take some dense sequence of $\{x_i\}$ (the reals are separable) and write columns of $f_n$ applied to each individual point. There will be a convergent subsequence by Bolzano-Weirstrass and since the outputs are real and bounded by the uniform boundedness of the family. I am a bit fuzzy on this part: Repeating this literally pointwise and moving far enough diagonally, we can correct for different convergence "speeds" pluck out a single convergent subsequence for any given epsilon and have $\sup|f_n(x)-f(x)|&lt;\epsilon$ for any given epsilon. Finally, continuity should follow from an $\epsilon/3$ argument and equicontinuity of the family.  However, in formalizing the above, I am getting caught up in the notation of that sort of argument and am hoping for a more elegant solution. Is there a better way? Am I just generally confused and that's why I'm having trouble writing things down formally?   Thanks! Why didn't Gollum grow old and die after he lost the One Ring to Bilbo? A comparison between the verbs (come in / by), (drop in / by), (check in / by) (step in / by) and come over Are there benefits of passing by pointer over passing by reference in C++? What are the benefits of passing by pointer over passing by reference in C++?  Lately, I have seen a number of examples that chose passing function arguments by pointers instead of passing by reference.  Are there benefits to doing this?  Example:  func(SPRITE *x);   with a call of  func(&amp;mySprite);   vs.  func(SPRITE &amp;x);   with a call of  func(mySprite); Does an unplugged hard drive used for data archival deteriorate? If I were to archive data on a hard drive, unplug it, and set it on a (not dusty, temperature-controlled) shelf somewhere, would that drive deteriorate much?  How does the data retention of an unplugged hard drive compare to tapes? Need to print out iPhone text messages I have a court case and have text messages on my iPhone I need to transfer to my iPad email for evidence.  Every time I try it doesn't work, it will send them to my iMessage but not my email.  I need to print them out How do I do this? How to force a web browser NOT to cache images Background  I am writing and using a very simple CGI-based (Perl) content management tool for two pro-bono websites. It provides the website administrator with HTML forms for events where they fill the fields (date, place, title, description, links, etc.) and save it. On that form I allow the administrator to upload an image related to the event. On the HTML page displaying the form, I am also showing a preview of the picture uploaded (HTML img tag).  The Problem  The problem happens when the administrator wants to change the picture. He would just have to hit the "browse" button, pick a new picture and press ok. And this works fine.  Once the image is uploaded, my back-end CGI handles the upload and reloads the form properly.  The problem is that the image shown does not get refreshed. The old image is still shown, even though the database holds the right image. I have narrowed it down to the fact that the IMAGE IS CACHED in the web browser. If the administrator hits the RELOAD button in Firefox/Explorer/Safari, everything gets refreshed fine and the new image just appears.  My Solution - Not Working  I am trying to control the cache by writing a HTTP Expires instruction with a date very far in the past.  Expires: Mon, 15 Sep 2003 1:00:00 GMT   Remember that I am on the administrative side and I don't really care if the pages takes a little longer to load because they are always expired.  But, this does not work either.  Notes  When uploading an image, its filename is not kept in the database. It is renamed as Image.jpg (to simply things out when using it). When replacing the existing image with a new one, the name doesn't change either. Just the content of the image file changes.  The webserver is provided by the hosting service/ISP. It uses Apache.  Question  Is there a way to force the web browser to NOT cache things from this page, not even images?  I am juggling with the option to actually "save the filename" with the database. This way, if the image is changed, the src of the IMG tag will also change. However, this requires a lot of changes throughout the site and I rather not do it if I have a better solution. Also, this will still not work if the new image uploaded has the same name (say the image is photoshopped a bit and re-uploaded).