If $x,y,z\in [0,\infty )$ such that $x+y+z=3$ prove that $27\leq \left( x^{2}+2\right) \left( y^{2}+2\right) \left( z^{2}+2\right) \leq 44$. I tried to use the relation $x^{2}+2\leq x^{2}+3x+2,x\geq 0$, but without success.
This problem is from Romanian G.M. and although it is very short, it is also (I think) very hard. Let $x, y, z$ real non-negative numbers such that $x+y+z=3$. Prove that   $$27 \leq (x^2+2)(y^2+2)(z^2+2) \leq 44 $$  I didn't succeed in any of the inequalities, I only managed to find the equality cases ($x=y=z=1$ for the first one and one of them $3$ and the others $0$). Any hint/idea/solution is welcome.
How do I tunnel and browse the server webpage on my laptop? I run a web app in one of my lab servers, and I have already setup X11 Forwarding on those machines. Other lab members can tunnel through SSH, and browse that web app on their local browser at home.  I can't. Last time I checked with them, there is almost nothing I need to do.  When I type 192.168.1.113/webapp I get nothing.   Any tips?  THanks.    I ended up using the second method :)  ssh -L 8080:&lt;server-ip-address&gt;:80 &lt;username&gt;@&lt;remote-addr&gt; -N No error message if the URL is very long I entered a URL with a very long parameter on Stack Exchange.  I have type this url    It shows a blank screen , does not shown any messages.    We need to show a “Page Not Found” message, same as what you did on other Stack Exchange pages. Generating a separate pdf file of tables and figures from a latex file? Is it possible to generate a PDF of figures and tables of a paper from LaTeX itself?  I saw Journals and Conferences asking for a separate file of tables and figures.  In  question, questioner proposed a method that uses \usepackage{endfloat} to push all the figures and tables to the last pages and then use a PDF editor to cut the last pages of PDF. This is a solution but not a good one. LaTeX should be able to do this by its own.  What I have in mind is to compile the LaTeX file and it generates a complete PDF output and beside that a separate file that contains all the figures and tables of former PDF. Normality assumption in linear regression Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$ I want to turn my system on automatically every day. So I use the below code in my Python script, but sudo asks me for a password every time:  os.system('sudo sh -c "echo date \'+%s\' -d \'+ \        24 hours\' &gt; /sys/class/rtc/rtc0/wakealarm"')   How can I run this script without sudo asking for the password every time? 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The $n$th summand is $\frac{1}{(2n)(2n + 1)(2n+2)} = \frac{1}{4} \frac{1}{n(2n+1)(n+1)}$.  I have tried expressing this as a telescoping sum, or as the limit of Riemann sums of a partition (the usual methods I normally try when doing this type of question- what are some other strategies?) I was looking at coreutils and found this as one of the files included as part of coreutils:  /usr/bin/[. What is [ and what does it do? It is an executable. I just don't know what it does or how to use it. $ file /usr/bin/[   /usr/bin/[: ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), dynamically linked (uses shared  libs), for GNU/Linux 2.6.15, stripped    When I try to run it, I think it is defaulting to the bash built in line expansion. 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Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1):  Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta=\varepsilon\gamma^n$, for $\alpha,\beta$ relatively prime numbers and $\varepsilon$ a unit, implies $\alpha=\varepsilon^{\prime}\xi^n$ and $\beta=\varepsilon^{\prime\prime}\eta^n$, with $\varepsilon^{\prime},\varepsilon^{\prime\prime}$ units.  My attempt: once $\alpha$ and $\beta$ are relatively prime, $\alpha\nmid\beta$ and $\beta\nmid\alpha$. Therefore $N(\alpha)$ and $N(\beta)$ are relatively prime too (where $N(a+ib)=a^2+b^2$). Therefore,  $$N(\alpha\beta)=N(\alpha)N(\beta)=N(\varepsilon)N(\gamma^n)=N(\gamma)^n=(p_1\cdot\ldots\cdot p_r)^n=p_1^n\cdot\ldots\cdot p_r^n$$  And once $N(\alpha)$ and $N(\beta)$ are relatively prime, we have (if necessary, we reorder the $p_i$'s):  $$N(\alpha)=p_1^n\cdot\ldots\cdot p_k^n=(p_1\cdot\ldots\cdot p_k)^n=N(\xi)^n=N(\xi^n)$$ and $$N(\beta)=p_{k+1}^n\cdot\ldots\cdot p_r^n=(p_{k+1}\cdot\ldots\cdot p_r)^n=N(\eta)^n=N(\eta^n)$$  And the result follows.  But I'm not sure about one step. It is $\alpha$ and $\beta$ relatively prime $\Rightarrow$ $N(\alpha)$ and $N(\beta)$ relatively prime. I know that if $\alpha|\beta$ then $N(\alpha)|N(\beta)$, but the previous assertion is not necesseraly true, and if it is, and don't know how to prove it. If it is true, the result follows, unless I made something wrong in the rest of the proof.  That's it. If you know another way of doing the exercise, please show me. As figured out in  localized SE sites must not be displayed in the HNQ's. 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To solve this, I already tried the following, without success:   Disabling and uninstalling all non-essential drivers. Installing newer drivers than the ones that were automagically installed (if available). Disabling/enabling fast boot option. Disabling all of the non-essential services. Sysprep. Resetting BIOS to defaults and various combinations of settings. Flashing BIOS to the latest available version. Clean install from the same media that I use for other PCs. Installing all of the updates offered in Windows Update to this day. Windows Performance Recorder / Analyzer.   I'm not very familiar with Windows Performance Analyzer, so I'm hoping someone here can point me in the right direction - what exactly should I look for, to figure out which device/driver is the culprit. Or, if there's any other approach to figuring out this problem?  For the brave souls, here's my  and a screenshot of the problem: As a follow-up to :  From what I've seen, this should work as expected:  void greet(){   char c[] = "Hello";   greetWith(c);   return; }   but this will cause undefined behavior:  char *greet(){    char c[] = "Hello";   return c; }   If I'm right, what's the best way to fix the second greet function? In an embedded environment? On a desktop?