Prove that if $a, b, c\ge 0$, then $(a + b)(b + c)(a + c)\ge 8abc$.
$a, b, c$ are positive real numbers.   Prove that $$8abc\le (a+b)(b+c)(c+a)$$  I haven't really gotten anywhere, just tried to open up $(a+b)(b+c)(c+a)$ and got null.
Embedding ordinals in $\mathbb{Q}$ How to change the desktop wallpaper to a photo of my own selection in Ubuntu 18.04? I'm not interested in downloading some wallpaper changer application, nor am I interested in coding something up. There has to be an easy way to select a photo as a custom wallpaper in Ubuntu 18.04. Why does my image have space underneath? How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something SQL Servers has Traces and XEvents. These are used to capture and analyze what is going on with our SQL Server Instances. Events are stored in the stack for later analysis. For example,  If I decide to monitor any dead lock in the database, I just query the trace file to see the history of deadlock for a period of time. Here is my question:  While events occur, in our example deadlock event, is there a way to get an email notification using msdb.dbo.xp_send_dbmail? My Ubuntu version 18.04 brings me a GNU GRUB version 2.02 and I cannot access my entry password panel? How to Sync OneDrive Business /Office 365 on Linux? Microsoft OneDrive Business/Office 365 Pro offers 1 TB of cloud storage but unfortunately there has never been a method to sync it on Linux. Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$g(X) = 2$ and deg$h(X) = 3$) Recall that $\bar{a}$ is shorthand for the coset $a + 2\mathbb{Z} \space $in$ \space (\mathbb{Z}/2\mathbb{Z})[X]$ (for any $a \in \mathbb{Z}$)  This question is worth $7$ marks out of a possible $75$, this is my solution..  $f(X) = X^5 + X^3 + \bar{1}$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$ iff it has no roots in $\mathbb{Z}/2\mathbb{Z}$. Since $f(\bar{0}) = \bar{1}$ and $f(\bar{1}) = \bar{1}$, it follows that $f(x)$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$.  Would this be a sufficient answer? am I missing something? “have found” vs. “found” Does this polynomial evaluate to prime number whenever $x$ is a natural number? ffmpeg - how to apply filters without losing quality Is leaving airport at a stop before the final destination illegal? There are tricks such as  or  which involve booking a ticket composed of several connecting flights, then leaving the airport at a stop, possibly even "hopping off" a plane (intermediate landing). Rationale: Sometimes these tickets are cheaper than a ticket to one of the stops. So one adds an additional leg and discards it.  Some people say that when using such tricks, one enters a legal gray area. But I wonder:  Could leaving the airport at a stop really be illegal, i.e. violating an airline's or a travel agency's terms?  Also there are suggestions that one should not request an upgrade, because then the trick may be discovered by the airline. Here I wonder:  How can they know that one will throw away the last leg? My Applications are not showing after upgrading from Ubuntu 18.04 to 20.04 Android WebView for Facebook Like Button How to prove that a nilpotent operator has a basis representation that is strictly upper triangular? How to tell a lambda function to capture a copy instead of a reference in C#? I've been learning C#, and I'm trying to understand lambdas.  In this sample below, it prints out 10 ten times.  class Program {     delegate void Action();     static void Main(string[] args)     {         List&lt;Action&gt; actions = new List&lt;Action&gt;();          for (int i = 0; i &lt; 10; ++i )             actions.Add(()=&gt;Console.WriteLine(i));          foreach (Action a in actions)             a();     } }   Obviously, the generated class behind the lambda is storing a reference or pointer to the int i variable, and is assigning a new value to the same reference every time the loop iterates.  Is there a way to force the lamda to grab a copy instead, like the C++0x syntax  [&amp;](){ ... } // Capture by reference   vs.  [=](){ ... } // Capture copies Aliased edge around light source in cycles Often when a mesh light is visible in the camera, the edges are not anti-aliased.  No matter how many samples I do, the lamp border always looks jagged and pixelated.  Is this a rendering error? How can I get around this issue?    And zoomed in 4x: Why do we say "eat healthy" instead of "eat healthily"? Why do we say "eat healthy" instead of "eat healthily", even though the latter is the only "correct" one, according to the comments in      What came first: ads saying "eat healthy", or the grammatical acceptability of such a phrase?    Merriam-Webster does not consider "healthy" to be an adverb, unlike "quick": , and therefore this question is not a duplicate of I created a simple service layer polygon in network analyst (ArcGIS Desktop 10.2) with no hierarchies set in the street network. Now I don't understand why is this polygon created along some particular streets?   Why isn't is properly spread out? For example, why doesn't the polygon cover the areas/streets I have marked in red? Obviously those marked areas are also within 1 km.  The impedance is 1 km and 2 km respectively.   The streets are also well connected individual line elements as you can see the junctions. How many Marvel Earths (Universes) are there? I know that there are at least 3 Marvel Universes:  Marvel Cinematic Universe -    Has all of the main ongoing Marvel movies    (1, 2)  (1, 2, 3)  (1, Age of Ultron) etc    Marvel's Ultimate Universe -    Has the other Marvel movies    (1, 2, 3, Origins: Wolverine, First Class, The Wolverine, Days of Future Past)  (1, 2)  etc       Main comic book continuity This is the big one, it contains         etc     Then there are apparently off-shoot Earths, continuities which deviate from the main continuity (Earth 616). How many Marvel Earths (Universes) are there?   Further Reading: