Let a, b, c $\in \mathbb{Z}$ . If $\frac{ab}{c} + \frac{bc}{a} + \frac{ac}{b}$ is an integer, prove that each of $\frac{ab}{c}, \ \frac{bc}{a}, \ \frac{ac}{b}$ is an integer. I've tried to solve this problem but still got no solution. All I think is divisibility and GCD $\frac{ab}{c} + \frac{bc}{a} + \frac{ac}{b} \\ = \frac{a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2}}{abc}$ Note that $2a^{2}bc + 2ab^{2}c + 2abc^{2}$ is divisible by abc. Put those in, we get: $\frac{a^{2}b^{2} + b^{2}c^{2} + a^{2}c^{2} + 2a^{2}bc + 2ab^{2}c + 2abc^{2}}{abc} \\ = \frac{(ab + bc + ac)^{2}}{abc}$ Because it's an integer, thus $abc \mid (ab + bc + ac)^{2}$ Assume $GCD(ab + bc + ac, abc) = d$, then $ab + bc + ac = dk_1$ and $abc = dk_2$ for an integer d where $GCD(k_1, k_2) = 1$ $\frac{(ab + bc + ac)^{2}}{abc} = \frac{d^{2}{k_1}^{2}}{dk_2} = \frac{d{k_1}^2}{k_2}$ Because $GCD(k_1, k_2) = 1$, thus the only possibility is $k_2 \mid d$. Let d = $k_{2}p$ where p is an integer, thus it implies that $abc = dk_2 = {k_2}^{2}p$ I got stuck here, I probably used the wrong method to solve this problem, does anyone know how to solve this?
Let $p,q,r$ be integers such that the symmetric sum of fractions $\dfrac{pq}{r}+\dfrac{qr}{p}+\dfrac{pr}{q}$ is an integer. Prove that each of the numbers: $\dfrac{pq}{r},\dfrac{qr}{p},\dfrac{pr}{q}$ is an integer. How to do this? Someone posted but deleted soon.
About two weeks ago I had a problem with my PC that my brother wrote a question about at .  I still haven't solved the problem, so my PC is unusable now.  Now I think that it would be simpler to re-install both Win7 &amp; Ubuntu than repair them, but there're some data on their partitions. I want to recover this data.  These partitions are not mountable for some reason. I don't know why.  sudo fdisk -l about the problematic partitions:     Device    Boot      Start         End      Blocks   Id  System /dev/sda1            2048     147912703    73955328   83 Linux /dev/sdb2            206848   174079999    86936576    7  HPFS/NTFS/exFAT   And in GParted:      And the error of the mount command:  Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   And the exact same message for /dev/sda1  How can I recover my data from those HDDs?  EDIT     I have tried to specify the type of the drive:  $ sudo mount -t ext4 /dev/sda1 /olddisk mount: wrong fs type, bad option, bad superblock on /dev/sda1,        missing codepage or helper program, or other error        In some cases useful info is found in syslog - try        dmesg | tail  or so $ sudo mount -t ntfs /dev/sdb2 /olddisk NTFS signature is missing. Failed to mount '/dev/sdb2': Invalid argument The device '/dev/sdb2' doesn't seem to have a valid NTFS. Maybe the wrong device is used? Or the whole disk instead of a partition (e.g. /dev/sda, not /dev/sda1)? Or the other way around?   EDIT     sudo ntfsfix /dev/sdb2 returns:  Mounting volume... NTFS signature is missing. FAILED Attempting to correct errors... NTFS signature is missing. FAILED Failed to startup volume: Invalid argument NTFS signature is missing. Trying the alternate boot sector Unrecoverable error Volume is corrupt. You should run chkdsk. Unicoin Bug: Automining Returns Diminishing I'm using the auto-mining script from one of the meta posts here and it doesn't works!  At first I got steady returns but the returns seem to diminish over time. Using Fiddler I confirmed that the server seems to release less and less coins.  This is unacceptable! I am a loyal customer paying nothing and I demand that my auto-mining script continues to yield returns forever!  If you cannot deliver on that please refund what I have payed for using this site.  Plz advise!    In case you didn't notice: This is a parody of over-demanding, leeching askers on Stack Overflow. Is there a way to stop the gmail address field (on the login page) from displaying all the gmail addresses that you have ever typed in it?   When logging in with Google Chrome or trying to set up a new account, anything you type which starts with the same letter as the one that you are trying to use currently will show up in the auto-complete field, I don't want that.    I understand that this is a convenient feature in some cases, but the thing is--I know this shouldn't've happened, but it has actually happened more than once--I sometimes mistyped my password in the Gmail address field right after the address, mistakingly assuming that the prompt had already been moved to the password field, and then hit "Enter" without looking first. The next time I tried to enter my Gmail address, it popped up the address followed by my password as one of the entries on the list of email addresses that shows up, and I couldn't get rid of the entry.   Is there a way to remove these? The autofill feature in Chrome is a time saver, and so I don't really want to turn it off. It would be helpful to be able to edit out some of the typed info, though, so it won't show up again -- is this possible to do? Can I get back my saved data? Continuity of a convex function I'm trying to solve the following problem:  Let $f:K\rightarrow \mathbb{R} $, $f$ convex and $K \subseteq \mathbb{R}^n$ convex. Then $f$ is continuous on $K$.   I have proved the only case $n=1$, but for an arbitrary $n$?? Touching the front wheel with my foot when turning Proving $\lim_{x\to 1} x^3=1$ with $\epsilon$-$\delta$ definition Problem: I need to formally prove that $$\lim_{x\to 1} x^3 = 1.$$ My work: This is what I have so far and I'm generally a bit stuck with these proofs from here onwards. Because$$-\epsilon &lt; x^3-1 &lt; \epsilon  =  | x^3-1 | &lt; \epsilon,$$ then $$ -\epsilon &lt; x^3-1 &lt; \epsilon$$ $$-\epsilon+1 &lt; x^3 &lt; \epsilon +1$$ $$ \sqrt[3]{-\epsilon+1}&lt; x &lt; \sqrt[3]{\epsilon+1}.$$ Hoping that what I have done so far is correct. Am I right in thinking that $$ \sqrt[3]{-\epsilon+1}&lt; x &lt; \sqrt[3]{\epsilon+1}$$ is giving me an interval where $x$ is going to give me a $f(x)$ value that falls within the distance $\epsilon$ from the limit on the $y$-axis ? Or is this interval smaller than the $\epsilon$-distance on the $y$-axis? Database design help. many-to-many relationship For instance if me and my friend had a Pidgeot and we powered them all the way up. Could mine end up with like 525 and his at like 500? Execute a program at login I'm running Arch-Linux ARM on a Raspberry Pi.  My project goal is to have a barcode scanner attached to the Pi that forwards the scanned messages to a server in a LAN.  To harden the system against power outages etc. and minimize required user interaction, I wish to set it up so that a certain user is logged in automatically at startup (done and works) and a specific program is to be run at startup (done and works but not as intended).  I've done this so far by adding  exec mono scannerSoftware.exe 127.0.0.1 1234 randomString   to /home/certainUser/.bash_profile so it gets executed only once even when I'm switching back and forth to root (in contrast to bashrc).    The particular problem I am having: When I launch the system, it works as intended (login as certainUser, starting scannerSoftware.exe); BUT if I want to switch users and hit Ctrl+C, it restarts the login process, logging certainUser in again and running the program again => back to where I came from.  My guess as to why this happens is that the sys detects "login not complete, better start it again".  I could circumvent this by spamming Ctrl+C right around the login so that I practically interrupted it. Of course, this is not good practice. Suffice it to say it is not practical at all.  My question: How can I set it up so the program gets executed AFTER login is complete so that Ctrl+C'ing out brings me to a normal bash prompt as certainUser instead of beginning the endless login-cycle?  EDIT: I'm not using any graphical interface or such, only commandline (bash on tty), because when done the system won't even have a display. how do we know the BBP formula for $\pi$ is valid? I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?  Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind? So, after a few sessions, the new player I'd invited has been a bit of a disaster. He doesn't get on well with any of the other players, laughed when someone's character died (in our campaign that's permanent at the moment), and generally has been a nuisance. However, he doesn't seem to notice this. I've talked to him about his behaviour, but it hasn't helped. Several other players have also approached me on their own to register concerns.  The long and short of it is, he's not a good fit for our game or social group, due to his abrasive personality/playstyle (as I and the other players perceive it). He's also not interested in roleplaying (something my group does heavily, at the moment) and continually pushes to place himself in the limelight - which he does by instigating combat and trying to skip over what he sees as 'boring' scenes. His character is a Minotaur barbarian, and the most exotic race in the game, but he only uses it for the stat bonuses, and doesn't even have a backstory - he simply says he doesn't need one. However, I don't want to be rude or otherwise unkind to him, as I will still see him around after he leaves the group, and so will the other players.  How can I politely ask him to leave the group, without being dishonest but avoiding embarrassment for anyone? I also want to avoid seeming to 'abandon' him, or 'go behind his back'.  Note: There are few/no in-game problems with adding or removing characters, and I don't need advice for that. This is strictly about handling the player. The system (not that it's all that relevant to the problem) is Can \includegraphics be used to change an image color? I'm including a number of PDF images in my document using \includegraphics.  I'm wondering if there's a way for LaTeX to adjust their color profile, hopefully the same way it can flip images upside down, resize, and crop, as here:   Let me detail a little more my desired outcome, in case there's another way to achieve this: The images are brightly colored---I'd like to reduce color levels, or possibly render them grayscale, and I'd prefer to not have to do external pre-processing, if this is possible.  I'm using XeTeX. Why do we need to discard one dummy variable? I have learned that, for creating a regression model, we have to take care of categorical variables by converting them into dummy variables. As an example, if, in our data set, there is a variable like location:   Location  ---------- Californian NY Florida   We have to convert them like:  1  0  0 0  1  0 0  0  1   However, it was suggested that we have to discard one dummy variable, no matter how many dummy variables are there.   Why do we need to discard one dummy variable? Visa for Istanbul trip during layover I am an Indian (Indian Passport) presently in Kano, Nigeria on a business visa. I am going back to Mumbai on 22 Sept by Turkish Airlines with a 12 hour layover time in Istanbul.  We plan to see the city during this time.  Can I get a visa on arrival? How can I suppress all output from a command using Bash? I have a Bash script that runs a program with parameters.  That program outputs some status (doing this, doing that...). There isn't any option for this program to be quiet. How can I prevent the script from displaying anything?  I am looking for something like Windows' . CSS to make HTML page footer stay at bottom of the page with a minimum height, but not overlap the page Google Sheets has several import functions   IMPORTDATA IMPORTFEED IMPORTHTML IMPORTXML   Sometimes the above functions returns errors like #N/A Imported content is empty and would like to be sure that there isn't a problem with the resource content to be imported.  How do I know if these functions are able to get the content that I looking to import?  I know that there also exist IMPORTRANGE but that function only is able to import content from Google Sheets spreadsheets How to express a polar coordinate for a macro that only accepts space-separated values? \psElectricfield only accepts a coordinate in space-separated values as follows. For example, a charge of -0.1 unit at (-1,0) is written as [-0.1 -1 0]. I have attempted to write in PS notation but it still does not work.  \documentclass{article} \pagestyle{empty}  \usepackage{pst-electricfield,multido} \def\M{2}  \begin{document} \multido{\i=0+30}{12}{% \begin{pspicture*}(-\M,-\M)(\M,\M)     \psElectricfield     [         Q={[-0.1 -1 0][-0.1 1 0][0.1 !\i sin 2 mul \i cos 2 mul]}     ] \end{pspicture*}\newpage} \end{document}   How to express  a polar coordinate for a macro that only accepts space-separated values? How do I get the MAX row with a GROUP BY in LINQ query?