Finding all real  number $n$ in $$5^n+7^n+11^n=6^n+8^n+9^n$$  Try: From given equation $n=0,1$ are the solution But i did not understand any other solution exists or not Although i have tried like this way $$\bigg(\frac{5}{9}\bigg)^n+\bigg(\frac{7}{9}\bigg)^n+\bigg(\frac{11}{9}\bigg)^n = \bigg(\frac{6}{9}\bigg)^n+\bigg(\frac{8}{9}\bigg)^n+1$$ Right side is strictly increasing function. but i have a confusion whether left side is strictly increasing or not could some help me how to solve it, thanks
Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$   It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that?  I've tried to write them either:  $$5^x+7^x+11^x=2^x*3^x+2^{3x}+3^{2x}$$  or  $$5^x+7^x+11^x=(5+1)^x+(7+1)^x+(11-2)^x$$  and tried to think of some AM-GM mean inequality or to divide everything by $11^x$, but those don't seem like the way to go. Any hints?
The word or phrase I am looking for is quite opposite to that of the situation when people, such as students or sports players, sing their national anthem with the same tone and words.  I want a single word or even phrase that can describe the situation when two or more people speak at the same time with different tone and words . . . and at the end, the listener cannot get what they are saying. This situation can be found while debating, telling a news about any incident or even while complaining.     Due to _____ , the teacher could not get what they are saying.   If it's not feasible to find a noun or phrase for such situation, then here I request to have an adverb standing alone to describe how they are speaking, for instance, what is the adverb for "at the same time"?.  Here is how, I want to convert sentence 1 into sentence 2.        They spoke at the same time, and the teacher could not get what they are saying.   They spoke _____ , and the teacher could not get what they are saying. Why can't I press the Install button when installing applications from unknown sources? Does a wireless-N (802.11n) network have poor performance when in b/g "mixed" mode? It seems to be an "old" wives tale that, when using a wireless-N router in "mixed mode" to support legacy 802.11b or 802.11g devices, the performance of 802.11n clients will suffer.      claim that when running in mixed-mode, all (some?) N clients run at G speeds.   make the same claim, but say it only happens when a G client is connected.    say N clients run faster, but still run about 30% slower than if the router were in N-only mode, even if there are no legacy B/G clients connected.    claim there is no speed drop for N clients when running on a mixed-mode network.  They say the only issue is that overall network throughput will be lower, because only one client can be transmitting at any time, so some of that transmission time must be shared with the legacy B/G clients running at lower speeds, reducing the overall throughput from what it would be if there were only N clients connected.    So, which is it?  Will running in mixed-mode slow down my network, even if there are no B/G clients?  If I'm running N, will having another client connected at B/G slow me down substantially vs. if they were running N? I have a Canon 550D which I've been using for around a year now. I really love it and have learned a great deal with this and the 18 - 135mm kit lens. But now with experience I feel the need to get some better glass. I won't be buying a new body any time soon because I figure investing in some good glass is more important.   As I have looked around, I see quite a few good used L lenses I can get in ebay for good bargain prices. But I know that the EF lenses don't have the same focal length in the cropped sensors. Is it a wise decision to buy these EF lenses or buy the cropped sensor alternatives?  I have the following lenses in mind that I am looking to buy down the line from the canon range over the next several months.  16-35mm f/2.8L, 24-70mm f/2.8L, 70-200 f/4L.  Are these apt choices for cropped sensors? I know most pros with FFs use these but I am a bit skeptic that they will serve the same purpose for me. If so what are the alternatives for this that have almost as good image quality? how much money it takes for a day without including accommodation In Uk I ran dolphin like this:  dolphin . &amp;   I see messages in the console like these:  QPixmap::scaled: Pixmap is a null pixmap   How can I keep the console clean or silent? When an account is deleted, don't auto delete its negatively scored posts on the meta site Meta is different. Negatively scored questions are not always bad questions, and sometimes they have merit and draw good answers, especially discussions.  When an account (in the main site) is deleted, either by self deletion or by a moderator, all questions with score of -1 or less are being deleted regardless of their answers, both in the main and meta site.  Can you please change this behavior so that the questions on the per-site meta will survive and won't be auto-deleted? PHP calculating number of days between 2 dates Parallel.ForEach Slower than ForEach Can I disable full-disk encryption? I recently installed Ubuntu 12.10 and it requires a passphrase to boot up (I installed it with an encrypted file system).  Do I have to reinstall to change to a standard unencrypted file system? An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm From reading the rules it would appear there are two kinds of damage and then straight loss of life:     118.3 If an effect causes a player to gain or lose life, that player's life total is adjusted accordingly.   From reading that I would guess that effects like Extort are not sources of damage. You simply lose the life.      119.2a Damage may be dealt as a result of combat. Each attacking and blocking creature deals combat damage equal to its power during the combat damage step.   So damage from creatures during combat is combat damage.     119.2b Damage may be dealt as an effect of a spell or ability. The spell or ability will specify which object deals that damage.   And that is direct damage from an object.  So the arguments are these:  Does loss of life as outlined by rule 118.3 (at top) count as being dealt damage? Do triggered abilities that redirect or reduce damage effect loss of life?  My second related question: You put  on a  and tap her to deal one damage to your opponent. Am I right in saying that because the damage was direct damage (not dealt during combat) then the triggered ability of preventing the combat phase of the damaged player is not triggered? I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$. Why don't applications update by calculating difference? What happened to these edits (and the associated reputation)? My reputation rating on SO took this hit an hour ago, and it is due to what appears to be the simultaneous deletion of multiple questions.  I am uncertain why it happened, am concerned, and would appreciate an explanation.  Going to  shows that it was removed due to moderation.  Was a script run deliberately, or were we hit with a question deletion attack; what happened?  -12     today -2  1 hour ago  removed     How to send a 2d Vector over UDP in c++ -2  1 hour ago  removed     How do I use the Eclipse debugger to debug a Java application? -2  1 hour ago  removed     PHP string error; inconsistency with local and server -2  1 hour ago  removed     Unable to start the MySQL service after a server migration -2  1 hour ago  removed     How to use Java to print a text file on a ticket printer? -2  1 hour ago  removed     How can all of the ZBrush Obj files be imported to make a complete model? Why is Linux reporting "free" memory strangely? This is a  about how Unix operating systems report memory usage.   Similar Questions:                    I have production server that is running Debian 6.0.6 Squeeze  #uname -a  Linux debsrv 2.6.32-5-xen-amd64 #1 SMP Sun Sep 23 13:49:30 UTC 2012 x86_64 GNU/Linux   Every day cron executes backup script as root:  #crontab -e  0 5 * * * /root/sites_backup.sh &gt; /dev/null 2&gt;&amp;1   #nano /root/sites_backup.sh  #!/bin/bash str=`date +%Y-%m-%d-%H-%M-%S` tar pzcf /home/backups/sites/mysite-$str.tar.gz /var/sites/mysite/public_html/www mysqldump -u mysite -pmypass mysite | gzip -9 &gt; /home/backups/sites/mysite-$str.sql.gz cd /home/backups/sites/ sha512sum mysite-$str* &gt; /home/backups/sites/mysite-$str.tar.gz.DIGESTS cd ~   Everything works perfectly, but I notice that Munin's memory graph shows increase of cache and buffers after backup.  Then I just download backup files and delete them. After deletion Munin's memory graph returns cache and buffers to the state that was before backup.  Here's Munin graph:  Externally hosted image was a dead link. Matrix representation of linear transformation Where does the term "fun fact" originate?-- namely, not with the compositional meaning but rather with the idiomatic usage to introduce some sort of unusual, esoteric, absurd or otherwise "controversial" fact? Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed.  I'm having trouble in showing $F+G$ to be closed. I know that $F$ is itself closed and complete, as it is a finite dimensional subspace of a normed space, and that if $F$ were compact that $G+F$ would be closed. I also know that the closed unit ball of any finite dimensional normed space is compact.  I tried two methods. One was to take a convergent sequence $(x_n)_{n \in \mathbb N}$ in $G+F$. Then we can write $x_n = f_n + g_n$ where $f_n$ and $g_n$ are sequences in $F$ and $G$ respectively, and I attempted to find a way to force the individual components $f_n$ and $g_n$ inside the unit ball which would enable me to say that they had convergent subsequences. I couldn't see how to do this, however.   The other thought was to try and show that $F$ is compact, but I don't see a way to do this as I can't imagine it to be simply true without some other conditions on $F$.  Are one of these methods the right way to go? Or should I go another direction? I would appreciate any help I can get, although I would prefer not to be presented with a full proof so that I can do some work for myself. Thanks! Comparing $\pi^e$ and $e^\pi$ without calculating them How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?