If $x,y,z&gt;0$ and $x+y+z=60$. Then minimum value of $\displaystyle \frac{x^2+y^2}{x+y}+\frac{y^2+z^2}{y+z}+\frac{z^2+x^2}{z+x}$   Try: $$\frac{(x+y)^2-2xy}{x+y}+\frac{(y+z)^2-2yz}{y+z}+\frac{(z+x)^2-2zx}{z+x}$$  So $$2(x+y+z)-2\bigg[\frac{xy}{x+y}+\frac{yz}{y+z}+\frac{zx}{z+x}\bigg]$$  Now Using arithmetic geometric inequality  $x+y\geq 2\sqrt{xy}$ and $y+z\geq 2\sqrt{yz}$ and $z+x\geq 2\sqrt{zx}$  Could some help me how to solve further, Thanks
If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$  Trial: Here  I proceed in this way $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge  \dfrac{2bc}{b+c}+\dfrac{2ca}{c+a}+\dfrac{2ab}{a+b}$$ then how I proceed. Please help.
The number of 5-digit numbers such that their sum of digits is even is? I tried this problem by counting the number of 5-digit numbers and then dividing it by 2 just like we do for three digits. Please help. Thank you. Join matrix based on conditions How to find Windows 7 File Transfer History? Is it possible to find out which files are transferred to an external usb flash drive in certain period of time?   To be more clear. If I/someone have a flash drive and take some data from me. How do I know which files are copied? Finding out file copy hostory could be also a work around. I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics  in general.  What are some freely available great physics books on the Internet? edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.  As a starter:         Table of contents sorted by field (in alphabetical order): Factors for max CP at a particular trainer level? What is the role of NS records at the apex of a DNS domain? $ORIGIN example.com. ; not necessary, using this to self-document  $TTL 3600 @        IN     SOA   ns1.example.com. admin.example.com. (                       1970010100 7200 1800 1209600 300)  @        IN     NS   ns1.example.com. @        IN     NS   ns2.example.com.  @        IN     A    198.51.100.1 ns1      IN     A    198.51.100.2 ns2      IN     A    198.51.100.3  sub1     IN     NS   ns1.example.edu.  sub2     IN     NS   ns1.sub2 ns1.sub2 IN     A    203.0.113.1 ; inline glue record   The role of a NS record beneath the apex of a domain is well-understood; they exist to delegate authority for a subdomain to another nameserver. Examples of this above would include the NS records for sub1 and sub2. These allow the nameserver to hand out referrals for portions of the domain that it does not consider itself authoritative for.  The purpose of the NS records at the apex of a domain, ns1 and ns2 in this case, seem to be less understood by the internet at large. My understanding (which may not be holistic) is as follows:   They are not used by caching DNS servers in order to determine the authoritative servers for the domain. This is handled by , which is defined at the registrar level. The registrar never uses this information to generate the glue records. They  used to delegate authority for the entire domain to other nameservers. Trying to do so with software such as ISC BIND will not result in the "expected" referral behavior at all, as the nameserver will continue to consider itself authoritative for the zone. They are not used by the nameserver to determine whether it should return authoritative responses (AA flag set) or not; that behavior is defined by whether the software is told to be a master or a slave for the zone. Most nameserver software will quite happily serve apex NS records that disagree with the information contained by the upstream glue records, which will in turn cause well-known DNS validation websites to generate warnings for the domain.   With this being the case, what are we left with? Why are we defining this information if it doesn't appear to be consumed by caching DNS servers on the internet at large? How can I return NULL from a generic method in C#? How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the three conditions happens. Editing polygons simultaneously in ArcMap? I have two polygons:    How can I reshape both Polygon 1 and Polygon 2 at the same time?  Will the area of each polygon change automatically in the table after reshaping?  I'm using ArcMap 10.1. How did we 'discover' dark matter? I'm an astrophysics student and I've been researching this topic and there is one point that keeps eluding me.  How did the scientific community realize that there had to be  in the Universe? $f_n$ converges uniformly on $[a,b]$ to some function $f$ jQuery/JavaScript to replace broken images I have a web page that includes a bunch of images. Sometimes the image isn't available, so a broken image is displayed in the client's browser.  How do I use jQuery to get the set of images, filter it to broken images then replace the src?    --I thought it would be easier to do this with jQuery, but it turned out much easier to just use a pure JavaScript solution, that is, the one provided by Prestaul. What is the metal sheath surrounding a USB connector, and is it commonly connected? What are the blue dots next to App names in iOS and macOS? They disappear once you open the apps and then return to the home screen, but I can't tell what they indicate.  Are they apps that haven't been opened since an app update? Or that it thinks have never been opened (not accurate in my case, but that could be a function of a "restore" confusing the phone). How to change default window background color in Windows 10? How could a gap between array modifier created objects be closed? Why are the Degrees of Freedom for multiple regression n - k - 1? For linear regression, why is it n - 2? I'm reading a textbook and I see this question:    So there are 200 women, and the DF is 196, implying that the equation for DF is $n - k - 1$. There are 3 variables: bp, age, and type so $k == 3$. What's the intuition behind this?  Also, why is the degrees of freedom for linear regression n - 2? Top Answers in network profile shows incorrect score post of mine has a score of 13, but the  shows that it has 14:  This seems to be a recur of  bug; @Catija's  says it's been fixed.  Can we fix that (again)? Installing Teamspeak and creating TeamSpeak3 .desktop file I am very new to Ubuntu and I want to create desktop launcher.  I have this in my TeamSpeak3.desktop in   [Desktop Entry] Name=TeamSpeak 3 Comment=TeamSpeak 3 VoIP Communicator Exec=/opt/TeamSpeak3-Client-linux_amd64/ts3client_runscript.sh Terminal=false Type=Application Categories=Network;Application; Icon=/opt/TeamSpeak3-Client-linux_amd64/styles/default/logo-128x128.png   in my ~/.local/share/applications/TeamSpeak3.desktop But I dont see it in my launcher and when I double click on it files, it just opens the text file. I even made it executable with     chmod +x TeamSpeak3.desktop  drwx------ 15 hung hung     4096 srp 16 12:29 . drwxr-xr-x  3 root root     4096 pro  9 19:22 .. -rw-------  1 hung hung   174627 srp 16 12:29 CHANGELOG -rwx------  1 hung hung   227136 srp 16 12:29 error_report drwx------  2 hung hung     4096 srp 16 12:29 gfx drwx------  3 hung hung     4096 srp 16 12:29 html drwx------  2 hung hung     4096 srp 16 12:29 iconengines drwx------  2 hung hung     4096 srp 16 12:29 imageformats -rwx------  1 hung hung   334520 srp 16 12:29 libc++abi.so.1 -rwx------  1 hung hung  1934744 srp 16 12:29 libcrypto.so.1.0.0 -rwx------  1 hung hung   746960 srp 16 12:29 libc++.so.1 -rwx------  1 hung hung  5842312 srp 16 12:29 libQt5Core.so.5 -rwx------  1 hung hung   484832 srp 16 12:29 libQt5DBus.so.5 -rwx------  1 hung hung  5683224 srp 16 12:29 libQt5Gui.so.5 -rwx------  1 hung hung  1335584 srp 16 12:29 libQt5Network.so.5 -rwx------  1 hung hung   290008 srp 16 12:29 libQt5Positioning.so.5 -rwx------  1 hung hung  3829280 srp 16 12:29 libQt5Qml.so.5 -rwx------  1 hung hung  3831288 srp 16 12:29 libQt5Quick.so.5 -rwx------  1 hung hung   234552 srp 16 12:29 libQt5Sql.so.5 -rwx------  1 hung hung   331232 srp 16 12:29 libQt5Svg.so.5 -rwx------  1 hung hung   117104 srp 16 12:29 libQt5WebChannel.so.5 -rwx------  1 hung hung 67570152 srp 16 12:29 libQt5WebEngineCore.so.5 -rwx------  1 hung hung   208344 srp 16 12:29 libQt5WebEngineWidgets.so.5 -rwx------  1 hung hung  6439120 srp 16 12:29 libQt5Widgets.so.5 -rwx------  1 hung hung  1334528 srp 16 12:29 libQt5XcbQpa.so.5 -rwx------  1 hung hung   180248 srp 16 12:29 libquazip.so -rwx------  1 hung hung    22936 srp 16 12:29 libsnappy.so.1 -rwx------  1 hung hung   111888 srp 16 12:29 libsrtp.so.0 -rwx------  1 hung hung   383072 srp 16 12:29 libssl.so.1.0.0 -rwx------  1 hung hung    51080 srp 16 12:29 libudev.so.0 -rw-------  1 hung hung     4340 srp 16 12:29 openglblacklist.json -rwx------  1 hung hung   260232 srp 16 12:29 package_inst drwx------  2 hung hung     4096 srp 16 12:29 platforms -rw-------  1 hung hung       26 srp 16 12:29 qt.conf drwx------  2 hung hung     4096 srp 16 12:29 qtwebengine_locales -rwx------  1 hung hung    19720 srp 16 12:29 QtWebEngineProcess drwx------  2 hung hung     4096 srp 16 12:29 resources drwx------  6 hung hung     4096 srp 16 12:29 sound drwx------  2 hung hung     4096 srp 16 12:29 soundbackends drwx------  2 hung hung     4096 srp 16 12:29 sqldrivers drwx------  3 hung hung     4096 srp 16 12:29 styles drwx------  2 hung hung     4096 srp 16 12:29 translations -rwx------  1 hung hung 18987008 srp 16 12:29 ts3client_linux_amd64 -rwx------  1 hung hung     1364 srp 16 12:29 ts3client_runscript.sh -rwx------  1 hung hung  2206064 srp 16 12:29 update drwx------  2 hung hung     4096 srp 16 12:29 xcbglintegrations Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$?   Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using the normal core of $H$, one can find an $m\ge 1$ such that $x^m\in H$ for all $x\in G$.