Find the  positive integral values $n$  can take for which  :${(x+y+z)}^2=nxyz$ is solvable for positive integers.  My try: Due to symmetry WLOG let $x\ge y\ge z$. Now ,making it in a quadratic in $x+y$,we have $$nxyz={(x+y)}^2+2(x+y)z+z^2$$ or $z| {(x+y)}^2$. But I don't know what to do with this. Also I tried playing with inequalities to get $$nyz\le 9x$$ and the equality condition $x=y=z$, yields $$nx=9$$ or $n$ can take the values $(1,3,9)$ but I can't find more. I know this question has been asked before but I have not learned that much. I also tried using a little bit of calculus but nothing promising came  up.
Find all $n\in\mathbb{N}$ so that there exist $x,y,z\in \mathbb{N}$ that solve:   $$(x+y+z)^2=nxyz$$   I tried to attack it finding solutions, but all the solutions doesn't seem to have something in common. For example:  $$ (x,y,z,n)=(1,1,1,9)$$ $$ (x,y,z,n)=(1,2,3,6)$$ $$ (x,y,z,n)=(1,4,5,5)$$ $$ (x,y,z,n)=(2,2,4,4)$$ $$ (x,y,z,n)=(8,8,16,1)$$  I was trying to solve this problem, but someone told me that a hint to solve it is using Vieta Jumping. I don't know how to use Vieta Jumping, so can anyone help me to understand how to apply Vieta Jumping and solve the problem?
How do I show that:  $$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$  This is actually problem B $4371$ given at this . Looks like a very interesting problem.   My attempts: Well, I have been thinking about this for the whole day, and I have got some insights. I don't believe my insights will lead me to a $\text{complete}$ solution.   First, I wrote $\sin\frac{5\pi}{14}$ as $\sin\frac{9 \pi}{14}$ so that if I put $A = \frac{\pi}{14}$ so that the given equation becomes, $$\frac{1}{\sin^{2}{A}} + \frac{1}{\sin^{2}{3A}} + \frac{1}{\sin^{2}{9A}} =24$$ Then I tried working with this by taking $\text{lcm}$ and multiplying and doing something, which appeared futile. Next, I actually didn't work it out, but I think we have to look for a equation which has roots as $\sin$ and then use $\text{sum of roots}$ formulas to get $24$. I think I haven't explained this clearly.         $\text{Thirdly, is there a trick proving such type of identities using Gauss sums ?}$ One post related to this is:  I don't know how this will help as I haven't studied anything yet regarding Gauss sums. Selectively coloring text in RichTextBox How can I paint in red every time I meet the letter "A" in RichTextBox? Why are we still using CPUs instead of GPUs? It seems to me that these days lots of calculations are done on the GPU. Obviously graphics are done there, but using CUDA and the like, AI, hashing algorithms (think bitcoins) and others are also done on the GPU. Why can't we just get rid of the CPU and use the GPU on its own? What makes the GPU so much faster than the CPU? How to remove duplicate lines inside a text file? A huge (up to 2 GiB) text file of mine contains about 100 exact duplicates of every line in it (useless in my case, as the file is a CSV-like data table).  What I need is to remove all the repetitions while (preferably, but this can be sacrificed for a significant performance boost) maintaining the original sequence order. In the result each line is to be unique. If there were 100 equal lines (usually the duplicates are spread across the file and won't be neighbours) there is to be only one of the kind left.  I have written a program in Scala (consider it Java if you don't know about Scala) to implement this. But maybe there are faster C-written native tools able to do this faster?  UPDATE: the awk '!seen[$0]++' filename solution seemed working just fine for me as long as the files were near 2 GiB or smaller but now as I am to clean-up a 8 GiB file it doesn't work any more. It seems taking infinity on a Mac with 4 GiB RAM and a 64-bit Windows 7 PC with 4 GiB RAM and 6 GiB swap just runs out of memory. And I don't feel enthusiastic about trying it on Linux with 4 GiB RAM given this experience. 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I could increase the max heap size using command line option to java, but that would require figuring out available RAM and writing some launching program or script. Besides, increasing to some finite max does not ultimately get rid of the issue.  I could rewrite some of my code to persist objects to file system frequently (using database is the same thing) to free up the memory. It could work, but it's probably a lot work too.  If you could point me to details of above ideas or some alternatives like automatic virtual memory, extending heap size dynamically, that will be great. Commutator Subgroup of Direct Product equals the Direct Product of the Commutator Subgroups I need to prove that $(G_{1}\times G_{2})^{\prime} = G_{1}^{\prime} \times G_{2}^{\prime}$, where $G^{\prime}$ denotes the commutator subgroup of $G$ - i.e., $G^{\prime}=[G,G]$, the subgroup generated by all commutators of elements of $G$.   Recall that the commutator of two elements $x$ and $y$ in a group $G$, denoted $[x,y]$ is defined to equal $x^{-1}y^{-1}xy$.  I have posted my proof as an answer below. Could somebody please take a look at it and let me know if it's okay? If not, please let me know what I need to do in order to fix it. Thank you. :)    Edit: I have been informed that my answer given below actually shows only equality of the set of commutators, and that $G^{\prime}$ consists of products of commutators. So, this question is no longer a proof check, but I am asking specifically how to fix what I have written below in order to make it actually answer the thing I set out to prove. I am having a bit of trouble understanding some of the hints given me thus far, and am going to need more detail in any answers given in order for them to be helpful. How to carry audio over SSH? Is there a way to carry audio over ssh?   Say after connecting to remote machine through ssh -X I opened a browser and want to watch a youtube video. Currently I could only able to watch the video without any audio. But I want to hear the audio too. How to do that? How to install Ubuntu 18.04 alongside Windows 10 Correct usage of will and would DES-X , computation load and storage Do people sometimes get blocked by a webservice for things someone else has done with their dynamic IP-Address It is common practice for operators of web services (e.g. web sites) to block IP addresses when there was malicious traffic originating from that address.  It is also common practice that internet users don't have a static IP address but get one assigned by their ISP via DHCP.  Now I would expect that it sometimes happens that a specific IP address gets banned from a service and later that address gets assigned to another user, leaving the new user (temporarily) banned from the service for no apparent reason. However, I never had this problem or heard of someone having that problem.  Does this sometimes happen? If not, what prevents this from happening? Show that $$\sum_{k=0}^n(-1)^k\binom nk=0$$ So for odd $n$ we have an even number of terms. So $\binom nk=\binom n{n-k}$ which have opposite signs. Thus the sum is 0.  For even $n$ we have that $$\sum_{k=0}^n(-1)^k\binom nk= \binom n0+\sum_{k=1}^{n-1}(-1)^k\binom nk+\binom nn$$ Now $$\sum_{k=1}^{n-1}(-1)^k\binom nk= \sum_{k=1}^{n-1}(-1)^k\left[\binom{n-1}k+\binom{n-1}{k-1}\right]$$ What would that sum be in the square brackets? What happens when you need to tear-down a hotel but no houses are in the bank in Monopoly? Greek pdf bookmarks I am puzzled with sections titled with Greek characters. They do appear in the pdf's Content list, though not correctly.  For example, in my thesis--written in English with LyX, based on the KOMA-script--I choose to include a translation of the summary in Greek (=Περίληψη). The "raw" code for the (unnumbered) chapter title reads: \addchap{\textgreek{Per\char208lhyh}}. In the final .pdf file (using pdflatex) the respective entry appears as Per208lhyh. Using the unicode option for the hyperref package does not fix the problem.  Update detail: the document (should) not only contain Chapter or Section titles in Greek, but normal text as well.  I have found some relevant references within tex.stackexchange and elsewhere (e.g., , the manual of the textgreek package, , ). However, I cannot extract a concrete answer.  How can Greek characters appear correctly as bookmarks in the .pdf's Content list?  Note, the information of this Q&amp;A applies to/under LyX as well. For example an electron radiates when accelerated. So does a positron. But is the radiation emitted by accelerated positronium the sum of the radiation emitted by each separately? If not, why not? If so, does this provide a way of testing whether a given neutral particle is composite? For example, does a neutron bremsstrahlung when decelerated? WPF Get Element(s) under mouse