An elementary algebraic inequality How do I go about proving that $$\frac{x^2}{yz+2}+\frac{y^2}{xz+2}+\frac{z^2}{xy +2}\ge \frac{x+y+z}{3} $$ for $1\le x,y,z\le 2$ ?  This is straightforward when $x=y=z$, but I don't see where to go from there.
How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$ Prove that for $x,y,z\in [1,2]$ the following inequality holds: $$\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}.$$ I tried to apply the Cauchy-Schwarz inequality or the power mean inequality, but my attempts were unsuccessfull, so I am asking for help.
16.04 VirtualBox VM (from vhd file) hangs at "non blocking pool is initialized" I'm trying to boot Canonical's Azure 16.04 .vhd file in a VirtualBox VM and it hangs at "non blocking pool is initialized".  It works in Hyper-V, so it may only be a bug in virtual box (5.0.20). A map $f:S^n\rightarrow S^n$ satisfying $f(x)=f(-x)$ for all $x\in S^n$ is said to be an even map. Show that if $f:S^n\rightarrow S^n$ is an even map, then $\deg f=0$ when $n$ is even and $\deg f$ is even when $n$ is odd. Moreover show that when $n$ is odd, there exist even maps of any given even degree.   My Try:  So $f=(-I)\circ f$. Then $\deg f=\deg (-I)\times \deg f$. If $n$ is even $\deg (-I)=-1$. Hence, $2\deg f=0$, so $\deg f=0$. But how do I show the results when $n$ is odd. No clue at all. This problem is in Hatchers book (A very complicated text). There is a hint but I do not understand it. Can somebody help me to proceed? Chat Feature-Request: Thumbs-Up Add an additional form of "I agree" besides starring or pinning (since pinning is restricted to a smaller subset to begin with) or something that we can use to not clog up the right hand star bar.  This is because sometimes you want to say "I agree" but you don't want to star it because ... let's face it, that's not star-worthy, and you don't want to clutter the stream with fifty thousand "like totes for sure, hella right!" because then it becomes one big back-patting party and you lose all semblance of conversation.  So I would propose a fourth "flag" for chat, that allows us to say "I {agree with|like|am of the same opinion as} this comment" and that it would only be useful in the live-chat (and maybe but not really in the transcript) and that it have no other useful semantics (no badges, no list of's [unless the list is of comments you've upthumbed]). Explanation of C++ FAQ's unsafe macro? Square of a second derivative is the fourth derivative I have a simple question for you guys, if I have this:  $$\left(\frac{d^2}{{dx}^2}\right)^2$$  Is it equal to this:  $$\frac{d^4}{{dx}^4}$$  Such that if I have an arbitrary function $f(x)$ I can get:  $$\left(\frac{d^2 f(x)}{{dx}^2}\right)^2 = \frac{d^4 f(x)}{{dx}^4}$$  Sorry if it's a pretty simple question, but I was trying to simplify something like this:  $$\left(a \cdot \frac{d^2}{{dx}^2} - f(x)\right)^2 g(x)$$ that's where it came up. Where can the viewport clipping parameters be changed? R: linear regression: very small coefficient and R-squared but significant P values Code   find / -name netcdf   Output  find: `/root/.dbus': Permission denied find: `/root/.gconf': Permission denied find: `/root/.gconfd': Permission denied find: `/root/.gnome': Permission denied find: `/root/.gnome2': Permission denied find: `/root/.gnome2_private': Permission denied Is the CMB rest frame special? Where does it come from? It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. Show that$\displaystyle\sum_{n=1}^\infty\frac{\sin( nz)}{n}$ diverges if $\mathfrak{Im} \,(z)\neq0$ We have to show that$\displaystyle\sum_{n=1}^\infty\frac{sin( nz)}{n}$ diverges if $Im\,(z)\neq0$  My try:$$\sin (nz) = \mathfrak{Im}(e^{nzi})$$  $$\sum_{n=1}^{\infty} \frac{\sin nz}{n} = \mathfrak{Im} \sum_{n=1}^{\infty} \frac{(e^{zi})^n}{n} = - \mathfrak{Im} \ln\left(1-e^{zi}\right). $$ Then how to proceed? Android : Is there any alternative to listFile function (Android java.io.File.listFiles()) Excel-Count number of visible cells containing certain string I have a sheet in which I first apply a filter to a certain column to show "Unique records only". Now I have a column which contains, for e.g., values like:  Applepie Applesauce Crabapple Banana Mango   What I need is the count of all visible cells containing the word "Apple". Note that this should exclude the hidden (filtered out) cells. Now I have found out a formula:  `=SUMPRODUCT(SUBTOTAL(3,OFFSET(Sheet1!A:A,ROW(Sheet1!A:A)-MIN(ROW(Sheet1!A:A)),,1))*(Sheet1!A:A="&lt;Text to search&gt;"))`   However the problem with this one is this formula searches for the entire text. NOT just a part of it. How do I modify this formula to return count of cells containing "Apple" i.e. for this example it should return 3 as the result. Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$? Smart Phone's camera socket I just want ask how the socket in the image is called?   I couldn't really find and the closer I could get is only Raspberry's module. What is the best term to describe the fact that most users have an initial dislike of any change to a UI even though it may really be net better over time?  Note: Question originally appeared I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct? Totally bounded, complete $\implies$ compact How to tenderize large squid? My whole life I've been buying small squid (maybe 4-6" long bodies without the head and tentacles), which are very mildly chewy and tender.  All it takes is sautee them in a little OO and they are good to eat, IOW, no additional tenderizing is needed.  Yesterday I got some large squid, about 1 lb each before cleaning, probably 10"+ long without the head, probably 5-6 times as much meat as the little ones I usually get.  I remember trying to cook that before and they came out very tough and with a bit of unpleasant odor.  Is there a way to marinate and tenderize large squid and to somehow neutralize the odor? I can't boot into a usable system after updates. What should I do? Since some system updates, my machine can no longer boot properly. I am no Ubuntu guru. I do not have a week to spare diagnosing the issue. I just want to retrieve important data from this machine's hard drive and get a working computer again ASAP.  What should I do? Cancel/kill window.setTimeout() before it happens I have a few places where I use the line below to clear out a status, for example.  I have a few of these that hang out for 10 seconds or more and if the user gets clicking around the action can occur at incorrect time intervals.  window.setTimeout(function() { removeStatusIndicator(); }, statusTimeout);   Is it possible to cancel or kill this with some jQuery or JavaScript code, so I don't have this process hanging around?