Given $x,y,z$ is positive number that $x+y+z=1$. Prove that $$ P = \frac{x}{y^3+2xyz} + \frac{y}{z^3+2xyz} + \frac{z}{x^3+2xyz} \gt \frac{15}{2}$$ My attempt: $$2xyz \leq \frac{2}{27}$$ $$ \Rightarrow P= \frac{x}{y^3+2xyz} + \frac{y}{z^3+2xyz} + \frac{z}{x^3+2xyz} \geq \frac{x}{y^3+\frac{2}{27}} + \frac{y}{z^3+\frac{2}{27}} + \frac{z}{x^3+\frac{2}{27}} $$ $$ \Leftrightarrow P \geq \frac{27}{2} (x+y+z) - \frac{27}{2}\left(\frac{x\cdot y^3}{y^3+\frac{2}{27}} + \frac{y\cdot z^3}{z^3+\frac{2}{27}} + \frac{z\cdot x^3}{x^3+\frac{2}{27}}\right)$$ AM-GM: $$y^3+\frac{2}{27}=y^3+\frac{1}{27}+\frac{1}{27}\geq 3\cdot\sqrt[3]{y^3\cdot\frac{1}{27}\cdot\frac{1}{27}}=\frac{y}{3}$$ Similar we have $$\frac{27}{2}\left(\frac{x\cdot y^3}{y^3+\frac{2}{27}} + \frac{y\cdot z^3}{z^3+\frac{2}{27}} + \frac{z\cdot x^3}{x^3+\frac{2}{27}}\right) \leq \frac{81}{2}\left(xy^2+yz^2+zx^2\right)$$ $$\Leftrightarrow P \geq \frac{27}{2} - \frac{81}{2}\left(xy^2+yz^2+zx^2\right)$$ Now we need prove $$\frac{81}{2}\left(xy^2+yz^2+zx^2\right)\lt \frac{27}{2} -\frac{15}{2} = 6$$ $$\Leftrightarrow xy^2+yz^2+zx^2 &lt; \frac{4}{27}$$ I was stuck here. Help me, thank you so much.
Prove that, for all positive real numbers $x$, $y$ and $z$, $$4(x + y + z)^3 &gt; 27(x^2y + y^2z + z^2x)$$  I've tried expressing it as a sum of squares, but haven't got anywhere.   Hints are also welcome.
What ticket reservation do I need for a Schengen visa? I have a visa appointment this Friday 19 at the Italian Consulate in Philadelphia and I have a question about the flight reservation requirement: According to the Visa requirements "Confirmed flight reservation for a round trip showing complete itinerary."  Does this mean I have to reserve the tickets, pay and print the itinerary?  In this case if my visa gets denied then I will lose the money I paid or can I just show the planned itinerary (written not booked)? in this case the travel agency would not give me the Travel Insurance coverage until I book and pay for the reservation.   I am a US green card holder and applying from North Carolina. Is it possible to prevent DHCP service on SBS 2003 from shutting down due to rogue DHCP server? Show inbox updates in the page title Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. The page titles for the tabs above should be something like the following pair:      2i 2q  Meta Stack Overflow     1i 3q  Super User I'm concerned about an app on my child's phone. 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I tried to change network interfaces name on this new Ubuntu 16.04 LTS version but doesn't have the /etc/udev/rules.d/70-persistent-net.rules.  So, I tried to use /lib/udev/write_net_rules but it not exist.  Why do I need this modification? Because I'm using a tool to Simulate virtual platforms that use flex licensing and the authentication need to be in eth0 interface name.  Any suggestion?  The command ip link returns:  user@laptop:~$ ip link  1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN mode DEFAULT group default qlen 1 link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00 2: enp6s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc pfifo_fast state DOWN mode DEFAULT group default qlen 1000 link/ether b8:2a:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff 3: wlp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc mq state UP mode DORMANT group default qlen 1000 link/ether 5c:e0:xx:yy:xx:yy brd ff:ff:ff:ff:ff:ff  user@laptop:~$ ifconfig  enp6s0    Link encap:Ethernet  HWaddr b8:2a:xx:yy:xx:yy   lo        Link encap:Local Loopback   wlp7s0    Link encap:Ethernet  HWaddr 5c:e0:xx:yy:xx:yy     (Some information were ignored and suppressed)  I already tried some links: How do we know the universe is expanding, and not that its contents are shrinking? 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Screencap for posterity:    Unless this user is  (in which case the full year should be shown, or at least a tooltip), it seems like they should have been unbanned 4 months ago. Let $f\colon(0,\infty)\to\mathbb R$ be differentiable on $(0,\infty)$ and suppose that $\lim\limits_{x\to\infty} f'(x)=L$.         a. Show that for any $h&gt;0$, $\lim\limits_{x\to\infty} \frac{f(x+h)-f(x)}h=L$.      b. Show that $\lim\limits_{x\to\infty} \frac{f(x)} {x} =L$.   this is a differentiation question.  i don't know how to solve this question  Because it is Korean, it is difficult to deal with this cite. So for me, it is not easy to write.  thank you Is there anyway to get tuple operations in Python to work like this:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (4,4,4)   instead of:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (1,2,3,3,2,1)   I know it works like that because the __add__ and __mul__ methods are defined to work like that. So the only way would be to redefine them? How to automatically fetch missing dependencies when installing software from .deb?