Let $n$ be a natural number. Show that   $$\left[ \frac{n}{1} \right ] + \left[ \frac{n}{2} \right ] + \left[ \frac{n}{3} \right ] + \cdots + \left[ \frac{n}{n} \right ] + [\sqrt{n}]$$ is even. Here $[x]$ means the greatest integer less than or equal to $x$.    Let $$f(n) = [\sqrt{n}] + \sum_{k=1}^{n} \left[ \frac{n}{k} \right]$$  $$f(1) = [1] + [1] = 2 \checkmark$$  The statement for $f(n+1)$ is to be proved:  $$f(n+1) = [\sqrt{n+1}] + \sum_{k=1}^{n+1} \left[\frac{n+1}{k} \right]$$  I just need some hints from now?
Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$   is even.   I used induction and the inequality $$x-1&lt;\lfloor{x}\rfloor\le{x}$$ to prove it.  Is there any other, nicer way to do it?
How do I enable Remote Desktop connection on Xubuntu 11.10? Converting from shapefile or raster formats to LAS? Recursive matching with regular expressions in Javascript Apply empty style to the entire bibliography So I just got this Nikon D3100 Camera.  And I am trying to take a picture using autofocus with the kit lens(18-55mm).    The system is in Auto mode, AF-A, single frame and the lens A-M switch is on "A",  VR is "ON".  When I press shutter half way down, focus confirmation light in lower left corner of viewfinder neither lit nor blinks(hear clicking). When I press the shutter all the way down, the shutter seems locked (hear clicking, but shutter not released).    The manual says the shutter will lock itself if it can't get focus. But the place is well lit and have distinct color and I can easily focus using my old Canon.    The camera works fine in manual focusing, but I can't get it to take picture under auto-focusing.  I wonder is there something wrong with the setting that I don't know or is that I got a lemon?  Thanks I wanted to create a WP plugin. So I requested and my request was approved by WordPress.org But I don't want to develop that plugin anymore. How can I delete the repository forever or request its deletion. No surname in my passport Give a sequence such that root test works while ratio test fails NSWindow Mac App Store like Title Bar Current situation: I'm required to cite "normative" documents (read: standards and manuals) separately from articles and books etc. Using biblatex and biber with the numeric style I'm doing that by issuing multiple printbibliography commands like so:  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} %... lots of stuff \printbibliography[type=standard,title=Standards \&amp; Manuals] \vskip\baselineskip \printbibliography[type=manual,heading=none] \printbibliography[nottype=standard,nottype=manual]   which gives a heading Standards &amp; Manuals, prints a list of manuals, leaves a bit of space then the list of standards and finally, under a new heading Bibliography gives books and articles etc. Thus far everything is fine, enter:  The Problem: I'm using the shorthand field in my database for the standards and manuals which works perfect as far as output within the text is concerned and sets the shorthand as the label in the bibliography and prints the details to the right of it - problem is: my shorthands are considerably longer than the labels generated by the numeric style, but even though they're in three different \printbibliography commands, everything gets indented to the maximum width of the label, making the part for books and articles look quite ridiculous.  The following shows the problem with only two separate \printbibliography (and an obviously exaggeratedly long shorthand):   Now  what I would obviously want is for each of the \printbibliography commands to only use the indentation actually necessary to accommodate the largest label appearing within, i.e. rendering the Manuals section as above, but the Bibliography section like follows:   Can anyone help me to achieve this? Or give me some pointer where to look?  The above was produced with the following MWE on an up-to-date TeXLive 2014:  \documentclass[10pt,british]{scrbook}  \usepackage{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern}  \usepackage{filecontents} \begin{filecontents*}{\jobname.bib} @manual{HowTo,     title = {How to Make the World Go Round},     shorthand = {HowToMakeTheWorlGoRound},     publisher = {The Publisher},     year = {2000} }  @article{Article,     author = {Author, A. and Buthor, B.},     journal = {Journal Title},     pages = {113--126},     title = {The Article},     volume = {65},     year = {1900} } \end{filecontents*}  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} \addbibresource{\jobname.bib}  \usepackage[hidelinks]{hyperref}  \begin{document} \let\cleardoublepage\relax \chapter{The Document} Nothing to see here, but we refer to the \parencite{HowTo} based on a principle discovered by \textcite{Article}.  \printbibliography[type=manual,title=Manuals] \printbibliography[nottype=manual] \end{document} Where are packages stored installed with APT in Ubuntu 16.04? I know that packages installed with apt-get are stored in /var/cache/apt/archives.  I just installed mariadb server in Ubuntu 16.04 with following command:  sudo apt install mariadb-server   When I tried to find packages in /var/cache/apt/archives, there are no packages. Where are they stored if it's not in this directory? Turning off auto-maximizing of windows in 14.04 I know this has been asked for other Ubuntu releases, but the solutions posted are not working for me in 14.04. I've installed CCSM and turned off the grid option, but the windows still maximize when dragging to the top of the screen. I've dealt with it for a while but it's getting rather obnoxious. Anyone else had this problem and come across a solution?  EDIT:  This is happening in Gnome Classic, but not Gnome 3 What is the strict aliasing rule? When asking about , people sometimes refer to the strict aliasing rule. What are they talking about? How can I completely disable internal Mic and Webcam I want to disable the internal Microphone on my laptop and the internal Webcam completely that they're no longer selectable, and no longer available?  I will only use my external Webcam, and my external (USB) headset.  The internal audio should still be available for playback. When should N-Gons be used, and when shouldn't they? Support for N-Gons (polygons consisting of more than 4 vertices) has been in Blender for a while now, but many of us still may not know when to use them, and how to do so effectively.  In what situations do N-Gons really come in handy? When should they be just plain left out? Do they have any place in finished models? Quieting down a Dell PowerEdge r710 I run a Dell PowerEdge r710 server out of my house and I'm moving in with roommates in a couple weeks. Unfortunately I don't think they'll appreciate the ~70dBA it generates and I'm looking for ways to make it quieter.  I looked into replacing the fans with something less noisy, but unfortunately none of the quieter options move nearly enough air. The best one I found has been the , but at less than half the CFM (30.3 vs 67.85 on the stock  fans) it doesn't seem powerful enough.  I've also thought about putting  around the server to quiet it down, but acoustic panels also have the downside of acting like insulation, so heat could turn into an issue. If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. Grant permission for new created files inside folder automatically I have a folder which permission is set to 777 but when I add any files or folder unziping in that folder. Extracted files or folder permission never change. Destination folder permission is 777 and I want what I will add in that folder will automatically get permission of 777.   When I add any files or folder by extracting .zip in that folder, extracted folder/files permission won't change automatically. I always have to chmod for newly add file/folder! Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric The relativistic energy-momentum equation is:  $$E^2 = (pc)^2 + (mc^2)^2.$$ Also, we have $pc = Ev/c$, so we get: $$E = mc^2/(1-v^2/c^2)^{1/2}.$$  Now, accelerating a proton to near the speed of light, I get the following results for the energy of proton:  0.990000000000000   c =&gt;    0.0000000011    J =      0.01    TeV  0.999000000000000   c =&gt;    0.0000000034    J =      0.02    TeV  0.999900000000000   c =&gt;    0.0000000106    J =      0.07    TeV  0.999990000000000   c =&gt;    0.0000000336    J =      0.21    TeV  0.999999000000000   c =&gt;    0.0000001063    J =      0.66    TeV  0.999999900000000   c =&gt;    0.0000003361    J =      2.10    TeV  0.999999990000000   c =&gt;    0.0000010630    J =      6.64    TeV 0.999999999000000   c =&gt;    0.0000033614    J =      20.98   TeV  0.999999999900000   c =&gt;    0.0000106298    J =      66.35   TeV  0.999999999990000   c =&gt;    0.0000336143    J =      209.83      TeV  0.999999999999000   c =&gt;    0.0001062989    J =      663.54      TeV  0.999999999999900   c =&gt;    0.0003360908    J =      2,097.94    TeV  0.999999999999990   c =&gt;    0.0010634026    J =      6,637.97    TeV  0.999999999999999   c =&gt;    0.0033627744    J =      20,991.10   TeV    If the LHC is accelerating protons to $7 TeV$ it means they're traveling with a speed of $0.99999999c$.  Is everything above correct?