How can I show this implication for $\frac x{xy+x+1} + \frac y{yz+y+1} +\frac z{zx+z+1} = 1$? $$(xy+x\ne -1 \land yz+y\ne-1 \land zx+z\ne -1 \land xyz=1)\\ \implies \frac x{xy+x+1} + \frac y{yz+y+1} +\frac z{zx+z+1} = 1$$ I tried finding each variables value using $xyz=1$ $(ex.x=1/yz)$ , and then replacing them but didnt really get anything. Then i tried writing the $1$s as $xyz$ and i could simplify some of them but still nothing. Thank you in advance
Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$ If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$  My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$  $=\dfrac y{y+xy+1}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  $=\dfrac{yz}{yz+xyz+z}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  What to do next?
How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them. How do I exclude plugins from getting automatically updated? There is an opt-in filter that allows all plugins on my site to receive automatic updates:  add_filter( 'auto_update_plugin', '__return_true' );  I like this feature, but I don't want all my plugins to be updated automatically. How can I allow some plugins to be updated automatically, while excluding those I want to do manually? What is a "closed" or "duplicate" question? Questions can be closed or marked as duplicates.  What does it mean for a question to be &quot;closed&quot;? What does it mean for a question to be marked as a &quot;duplicate&quot;? Who can close a question? What are the reasons for closing a question ? Is closure the end of the road for a question? When are closed questions eligible for deletion?   Related        For more information, see &quot;&quot; in the . When and why should I use % !TEX TS-program and % !TEX encoding? How do I remove code duplication between similar const and non-const member functions? Let's say I have the following class X where I want to return access to an internal member:  class Z {     // details };  class X {     std::vector&lt;Z&gt; vecZ;  public:     Z&amp; Z(size_t index)     {         // massive amounts of code for validating index          Z&amp; ret = vecZ[index];          // even more code for determining that the Z instance         // at index is *exactly* the right sort of Z (a process         // which involves calculating leap years in which         // religious holidays fall on Tuesdays for         // the next thousand years or so)          return ret;     }     const Z&amp; Z(size_t index) const     {         // identical to non-const X::Z(), except printed in         // a lighter shade of gray since         // we're running low on toner by this point     } };   The two member functions X::Z() and X::Z() const have identical code inside the braces.  This is duplicate code and can cause maintenance problems for long functions with complex logic.    Is there a way to avoid this code duplication? Let $\{X_\alpha\}_{\alpha\in J}$ be a family of topological spaces and let $X=\prod_{\alpha\in J}X_\alpha$ be endowed with the product topology. Show that $X$ is path connected if and only if each $X_\alpha$ is path connected.  I know this is true if path coennected is changed by connection but I am very confused with this problem, I have tried to do the following:  $\Rightarrow $ let $x,y \in X_\alpha$ be then $(x_\alpha), (y_\alpha)\in \prod X_\alpha$ where $x_\alpha=x, y_\alpha=y$ and $x_\beta=0, y_\beta=0$ for any $\beta\neq\alpha$, and since P is path connected then there is continuous $f:[0,1]\rightarrow \prod X_\alpha$ such that $f(0)=(x_\alpha)$ and $f(1)=(y_\alpha)$, then if $\pi_\alpha:\prod X_\alpha \rightarrow X_\alpha$ is the projection, which is continuous, we have that $\pi_\alpha f:[0,1]\rightarrow X_\alpha$ is continuous and $\pi_\alpha (f(0))=x$ and $\pi_\alpha (f(1))=y$.  $\Leftarrow$ I do not know how to do this address, could someone help me please? Thank you very much. Is it possible to experimentally show that observable universe is much larger than ~93B ly? Thought experiment:   Entangle a pair of particles.  Package one half of each pair in a special package, and send it out in a nano-spaceship keep the other half of the pair of entangled particles here on earth accelerate nano-spaceship to 99.99999999999999% of the speed of light using a number of    Due to Lorentz ,  $$ L = L _\text{0} \sqrt {1 - \frac {v^{2}}{c^{2}}} $$  In one years time, the spaceship would have traveled 93B ly away from us (in the spaceships relative frame)  After one year of travel, the spaceship could be preconfigured to detect the CMB in the direction of travel, and then trigger their quantum entangled particle to send a '0' or a '1' depending on whether the CMB was detected.  Repeat this process using any number of spaceships in any number of directions, and soon you would experimentally show how big the universe really is.  Any problems? In a map I'm making the player has to survive while locked in a maze. To stop them from breaking out, they will be in adventure mode. However, I still want them to be able to craft items and use them with specified blocks. I have tried several methods already but they all have a flaw.  One method is to have a scoreboard objective called item and giving certain item entities a specific score using something like:  /scoreboard players set @e[type=Item] item 1 {Item:{id:"minecraft:sapling"}}   Then it is possible to select these items with:  /entitydata @e[type=Item,score_item=1,score_item_min=1] {Item:{tag:{CanPlaceOn:["dirt"],HideFlags:24}}}   This works but it requires the item to be dropped, which isn't a problem for saplings, but for tools and other craftable items, it becomes a hassle.  To fix this, I'm currently clearing tools without damage and giving the player a slightly damaged tool in return:  /clear @a minecraft:stone_axe 0 1   /give @a minecraft:stone_axe 1 1 {CanDestroy:["log","log2","leaves","leaves2"],HideFlags:24}   This doesn't work for items without damage values (e.g. torches), but I don't mind these having to be dropped as there aren't too many of these. However, there is another problem. In the nether, the player will have access to an enchantment table and an anvil. This means they can repair enchanted tools, which will cause them to be cleared and replaced with unenchanted ones as soon as they go back to the overworld.  I also thought about using a stat.craftItem scoreboard objective and giving them the usable item if they craft it. However, this would result in the player having a duplicate of every item they crafted, as I think it is impossible to select an item without NBT-data in the /clear command.  Is there any way to make this work? And if not, would it be better to have the player drop all crafted items or have duplicates of them all? When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The question is:     When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?   The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?  Edit  I should add that $f:\mathbb R\to\mathbb R$ to the question. How to install the latest versions of NodeJS and NPM? I noticed over at the  website that node is currently at v 0.12.0.  Can someone let me know how to install the latest version of node together with npm (terminal commands please)? This is a  about Hairpin NAT (Loopback NAT).   The generic form of this question is:  We have a network with clients, a server, and a NAT Router. There is port forwarding on the router to the server so some of it's services are available externally. We have DNS pointing to the external IP. Local network clients fail to connect, but external work.   Why does this fail? How can I create a unified naming scheme (DNS names which work both locally and externally)?   This question has answeres merged from multiple other questions. They originally referenced FreeBSD, D-Link, Microtik, and other equipment. They're all trying to solve the same problem however. How does atan2 work when getting angle of a vector? I know that atan2 gives me the absolute angle of any vector. But it doesn't give a value from 0 to 360 degrees. Instead, it gives a value (if I'm not mistaken), between 0 and 180, or between 0 and -180. For example, instead of giving angle 270, it will say -90.  This is a little confusing for me. Let me try to describe how I think this works, and please tell me if I'm correct. Actually, I'll draw it.    Is this correct? I'm having a little trouble understanding this. Thanks How do the bonus perks from the Advanced Warfare Center work? Can I have two Schengen visas in one passport? Java: how do I check if a Date is within a certain range? Error while running Quickly Why does pizza cheese seem hotter than the crust? When I eat hot pizza or a melted cheese sandwich, the cheese feels a lot hotter than the crust or bread: in particular, the cheese might scald the roof of my mouth. but the crust will not. Is this   my imagination, or because the crust cools a little faster than the cheese, so has already cooled a bit by the time I eat it, or because the cheese cools a little faster than the crust, so transfers heat to the roof of my mouth a bit more, or what? I was listening to the song  by Eminem. I've listened it before many times but from the time I've started learning English grammar I am getting confused while reading or listening English. This time I got stuck at the line "Too much pride, between you and I Not a jealous man but females lie". The phrase "between you and I" always sounded correct to me. But recently I learned that me is only used as an object and hearing that phrase sparked my head. Then I searched over the net and found on  that it is actually a hypercorrection.  says,      "...according to many grammarians and stylists a pronoun in a prepositional phrase in English is supposed to be in the oblique case...   ...though there is still disagreement on whether the phrase itself in today's language is grammatically correct or not."   My first question is, why is there a disagreement? Don't we always use the object form in the oblique case? If not then what are those cases?  Secondly, I want to know how common the phrase "between you an I" is? Is it used colloquially by common native English speakers. By common speakers I mean who don't bother grammar. I searched on  and got some results, e.g. . I found the phrase in another song -- . Is it a common mistake among native English speakers or is it used to give some special linguistic affect? I think this phrase has some linguistic affect that is why the singer would have given the name of her song "Between You and I". Because no one would like to use incorrect English in the title of a song, it must have some purpose. changing notification emails from WordPress @mydomain.net to something else How do I change notification emails address from WordPress @mydomain.net to something else.  I want to do this because WordPress @mydomain.net ends up getting flagged as junk mail.  Thanks  Daniel Order of items in a context menu