Let $x,y,z$ be positive integers such that $x^2+y^2+1=xyz$.   Show that $z=3.$
Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks
I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may approach it. Easiest way to alternate row colors in PHP/HTML? Here's a PHP example of mine. Can anyone find a shorter/easier way to do this?  &lt;? foreach($posts as $post){?&gt;     &lt;div class="&lt;?=($c++%2==1)?‘odd’:NULL?&gt;"&gt;         &lt;?=$post?&gt;     &lt;/div&gt; &lt;? }?&gt;  &lt;style&gt;     .odd{background-color:red;} &lt;/style&gt;   Examples in other languages would be fun to see as well. Why Was Peter Pettigrew Sorted To Gryffindor? Started new job. Rollover previous employer 401k to new 401k, IRA or Roth IRA? I recently started a new job and have a 401k from my previous employer that I don't know what to do with.  Should I roll it into my new employer's 401k, a traditional IRA or a Roth IRA? In comics does Batman have a rule against killing? Inspired by  question, I have to admit I don't know a lot about the Comic Book Batman, but I know a fair amount about his other incarnations.   I know for example, that the 1989 movie had him kill the Joker. He vehemently wanted the Joker dead and had no compunctions about taking a life. I also know that he had a rule against killing in The Animated Series that he never broke, and that he very nearly broke it in Batman Beyond.  But as far as the comic books go, when did Batman develop a rule against killing his foes? Does he ever kill them rather than deliver them to the proper authorities? I am trying to use notepad++ and regular expressions to edit a document. I have text in this format  123.45607894.165@abcd;aba 23.456@av1;084 9871.001@fab:9782581afa   And I need to get only the first part of the line , the one before the @ sign. How would I do that to get this:  123.45607894.165 23.456 9871.001 Is there an infinite amount of wavelengths of light? Is the EM spectrum continuous? The electromagnetic spectrum is a continuum of wavelengths of light, and we have labels for some ranges of these and numerical measurements for many.  Question: Is the EM spectrum continuous such that between two given wavelengths (e.g. 200nm and 201nm) there is an infinite number of distincts wavelengths of light?  Or is there some cut-off of precision with which light might exist (e.g. can light only have wavelengths of whole number when measured in nanometers, etc.)? Too much space between links in post menu Strip whitespace from jsp output How can I strip out extra whitespace from jsp pages' output?  Is there a switch I can flip on my web.xml?  Is there a Tomcat specific setting? What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for all.  $$\frac{\partial y}{\partial x}, \frac{\delta y}{\delta x}, \frac{\mathrm{d} y}{\mathrm{d} x}$$   For what I know, $\mathrm{d}$ is used as a small infinitisemal change (and I guess the straight-up letter $\mathrm{d}$ is usual notation instead of italic $d$, simply to tell the difference from a variable). Of course we also have the big delta $\Delta$ to describe a finite (non-negligible) difference. And I have some vague idea that $\partial$ is used for partial derivatives in case of e.g. three-dimensional variables. Same goes for $\delta$, which I would have sworn was the same as $\partial$ until reading this answer on Math.SE:    Then to make the confusion total I noticed an equation like $\delta Q=\mathrm{d}U+\delta W$ and read in a physics text book that:     The fact that the amount of heat [added between two states] is dependent on the path is indicated by the symbol $\delta$...   So it seems $\delta$ means something more? The text book continues and says that:     a function [like the change in internal energy] is called a state function and its change is indicated by the symbol $\mathrm{d}$...   Here I am unsure of exactly why a $\mathrm d$ refers to a state function.   So to sum it up: down to the bone of it, what is $\delta$, $\partial$ and $\mathrm{d}$ exactly, when we are talking derivatives in physics.  Addition  Especially when reading a mathematical process on a physical equation like this procedure:  $$\delta Q=\mathrm{d}U+p\mathrm{d}V \Rightarrow\\ Q=\Delta U+\int_1^2 p \mathrm{d}V$$  It appears that $\delta$ and $\mathrm{d}$ are the same thing. An integral operation handles it the same way apparently? Let $f:\Bbb{R}\to\Bbb{R}$ be a function given by  $$f(x)=\begin{cases} \exp\left(\frac{1}{x^2-1}\right) &amp; \text{if }\vert x\vert\lt 1\\  0 &amp; \text{if }\vert x\vert\geqslant 1  \end{cases}$$  I would like to prove that $f\in C^\infty$, that is, $f\in C^k$ for all $k\in \mathbb{N}$. I think that it can be done by induction on $k$. If $\vert x\vert\gt1$, the problem is trivial. On other points, the base case is the simplest and the only that I'm be able to do. Can someone help me?  Thanks. In a theoretical physics homework problem, I have to show the following: $$\partial_\nu T^{\mu\nu} = - j_\nu F^{\mu\nu}$$  Where $T$ is the Energy-Momentum-Tensor, $j$ the generalized current and $F$ the Field-Tensor. We use the $g$ for the metric tensor, I think in English the $\eta$ is more common.  I know the following relationships:   Current and magnetic potential with Lorenz gauge condition: $$\mathop\Box A^\mu = \mu_0 j^\mu$$ Energy-Momentum-Tensor: $$T^{\mu\nu} = \frac1{\mu_0} g^{\mu\alpha} F_{\alpha\beta} F^{\beta\nu} + \frac1{4\mu_0} g^{\mu\nu} F_{\kappa\lambda} F^{\kappa\lambda}$$ Field-Tensor: $$F^{\mu\nu} = 2 \partial^{[\mu} A^{\nu]} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}$$ d'Alembert operator: $$\mathop\Box = \partial_\mu \partial^\mu$$ Bianchi identity: $$\partial^{[\mu} F^{\nu\alpha]} = 0$$   So far I have set all the definitions into the formula I have to show, but I only end up a lot of terms from antisymmetrisation and product rule. I also drew all what I have in , but I still cannot see how to tackle this problem.  Could somebody please give me a hint into the right direction? What happens to water level when ice XII melts? There exists this famous idea that if all floating icebergs melt, water level will stay the same (because the water replaced by ice is the volume of the melted ice). Now,   Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ? Is this always so on/in earth ?   EDIT:  I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. "Ice" is always floating or sunken(if possible).  To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)?  EDIT 2:  Wikipedia: "Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)".   And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the "lake" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen? I need a counsel about my websites' domain.  Let's say I have just bought two domains: xyz.com and xyzphotography.com.  As we can see, the first one is shorter but way less meaningful than the second one. The second one contains a photography keyword but it's longer.  I have only one website which is my photography portfolio. Can you advise me how to set those domains to make it all as SEO-friendly as possible? Which one should be the main one, does the rest should work on their own or have some kind of SEO-friendly redirect to the main domain? How can I repair a ceiling fan that hums, but doesn't turn? I was wondering if there was any was I could right click inside any instance of a "Finder" window such that I have an option that says "Open Terminal Here". It would be really helpful. How do I fix a "Problem with MergeList" or "status file could not be parsed" error when trying to do an update? How do you install an APK file in the Android emulator? Festival stores voicepack data in the following example directory structure:  /usr/share/festival/voices/&lt;language&gt;/&lt;voicepack name&gt;  What is the simplest one-liner (preferably using ls) to print out just the &lt;voicepack name&gt;'s, in all the potentially numerous &lt;language&gt; subdirectories? How can I show or hide boot messages when Ubuntu starts?