Given $3$ positive real numbers $x,\,y,\,z$ and $x+ y+ z= 3$. Prove that: $$4\geqslant x^2y+ y^2z+ z^2x$$ This problem was homogenized so I set $x+ y+ z= 3$ to cancel stuff. Now I'm stuck. I have noticed a funny equality condition: If $(\,x,\,y,\,z\,)=(\, 2,\,1,\,0\,)$ then we have equality we don't have equality if they are all equal. To me$,$ that implied a weighted AM-GM but I couldn't make that work. Thanks in advance for any help!
Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$.  How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$    In other words, if $a,b,c$ are non-negative real numbers, then how to prove inequality $$ 27(ab^2+bc^2+ca^2)\le 4(a+b+c)^3.\tag {**} $$ $\color{gray}{\mbox{(Without using "universal" Lagrange multipliers method).}}$  Thanks!
Why does the reputation need to be publicly displayed? Possible Duplicate:          The idea of getting greater privileges with greater reputation is good, but what is the benefit of publicly displaying that reputation? Since SO claims to be about good questions and answers, that purpose is better served if the reputation is only visible to the account owner (and, of course, the site admins). This would better ensure that content gets pushed up solely by its merit, and not based on the author. (See )   Edit (copied from comment to an answer below): Once you have gathered high reputation, you can also gather more reputation with ordinary answers, especially from newbies (newbies on the topic, not on the site) who usually choose to accept or upvote an answer on the assumption that "since a high-rep user has posted it, it is probably right". Can you guarantee that doesn't happen often? Intuition on the Kullback-Leibler (KL) Divergence On the one hand, I'm aware that  in the US do not have a valid government ID. On the other hand, traveling within the US requires you to show a piece of ID wherever you go - even when taking a Greyhound.   So let's say you're an American who has a large number of cash in their pocket but no ID whatsoever. Would you be able to get from New York to San Francisco or from Seattle to Miami? If so, what are your options? Even though I do have plenty of identification documents, I find it enjoyable to be able to travel anonymously. show that a distance function is continuous Let $X$ be a metric space with metric $d$. Define $d: X \times X \to \mathbb{R}$, show that $d$ is continuous.  I would like to show that the function is continuous the topology way (since it is a course on topology).   Let $(a,b)$ be a basic open set in $\mathbb{R}$, then $d^{-1}(a,b) = \{(x,y): a&lt;d(x,y)&lt;b\}$. Define $A= \bigcup_{x \in X} B_{b} (x)$, where $B_b (x) = \{y \in X| d(x,y)&lt;b$}. Clearly $A$ is open. Now define $C = \{(x,y):d(x,y)&gt;a \}.$ Essentially I hope that $A \cap C$ is open. But I am stuck in showing that $C$ is open.  Any help please? :) Why isn't Higgs coupling considered a fifth fundamental force? When I first learned about the four fundamental forces of nature, I assumed that they were just the only four kind of interactions there were. But after learning a little field theory, there are many other kinds of couplings, even in the standard model. So why isn't the Yukawa Higgs coupling considered a fifth fundamental force? To be a fundamental force, does there need to be a gauge boson mediating it? There is an alignment issue in the stackexchange.com search page .  The bottom bar is missing in the search input box.  Sample URL:   Screen shot for reference:    Firefox version: 54.0b8 (32-bit) OS Version: Windows 10 Pro  Also happens with Chrome latest version. Killing thread after some specified time limit in Java There seems to be something wrong with the sorting of Stack Exchange sites by name:    elementary OS and Русский язык appear in the wrong place at the end of the list. How do I unit test code that includes a callout? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  How do I unit test code that includes a SOAP or REST-based callout, or indirectly calls code that makes a callout? I'm getting an error saying that callouts are not supported. Legendre symbol: Showing that $\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0$ I have a question about Legendre symbol.  Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies: $$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$  I know that $\displaystyle\sum_{m=0}^{p-1} \left(\frac{m}{p}\right)=0$, but how do I connect this with the previous formula?  Any help is appreciated. E.g.  I know that I can browse the chat when I'm not logged on. But what if I am logged on and want to spare the inconvenience of logging off/multiple browsers? When can a noun be used attributively? Nouns can modify nouns: cat food, coffee cup, gold ring, laser surgery, flood insurance. It seems to me there are even cases where a noun sounds better than the corresponding adjective: sociology papers sounds a bit better than sociological papers.  Some noun-noun combinations don&rsquo;t work. You wouldn't say America gangsters or measurement spoons or greed crimes. If you spilled water on your keyboard you probably wouldn't call it a keyboard spill or even a water spill.  Are there rules that tell which combinations are acceptable? Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$ Let $f$ be analytic on an open set $U$, let $a \in U$, and $f'(a)\neq 0$.  Show that  \begin{align*} \frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz \end{align*} where $C$ is a small circle centered at a.  I thought that this is very similar with Cauchy's integral formula. But, I can't apply the Cauchy's integral formula. How to apply that?   Any help is appreciated..   Thank you! How do I  prevent .xsession-errors from eating disk space? What does the Asterisk * mean in Objective-C? A simple regex search and replacement in php for minifying/compressing javascript? Get files in a directory sorted by last modified? Visualforce Date formatting I just spent the last 10 minutes trying to put a new set of grips on to my mountain bike handlebars..  The old ones came off fairly agreeably and I've got the new ones 80-90% of the way on, however, at this point both my hands are stinging from the aggressive twisting and pushing - I have tried gloves, but I feel it's best left till my hands don't hurt anymore.  Is there a tried and tested way of doing this? Or is there a knack to this that I am missing? We know that $221 = 17*13$. So we can check if the system has roots to both of those equations separately, which it does:  $x^{5} \equiv 2$ mod $13$ has the solution $6 + 13n$ and $x^{5} \equiv 2$ mod $17$ has the solution $15 + 17n$.   I got these numbers from wolfram, I have no idea how to solve this problem WITHOUT a calculator. And even after finding these numbers. How would one obtain a solution modulo $221$? I was thinking the Chinese Remainder Theorem but I am under the assumption that CRT only applies to problems with powers of $x$ which are $1$.  Thanks.