Let $x_1,...,x_n &gt; 0$. I'm having troubles proving this formula via induction: $$ (x_1 + \ldots + x_n)\left(\frac1{x_1} + \ldots + \frac1{x_n}\right) \ge n^2 $$  So far, I've managed to rewrite it like this: $$ \sum_{k=1}^n x_k \sum_{k=1}^n \frac{1}{x_k} \ge n^2 $$ Also the base case seems simple enough: $1 + 1/1 \ge 1^2$. However, this is where I got stuck and I can't seem to be able to solve this any further.
Prove by induction that $$(a_{1}+a_{2}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)\geq n^{2}$$  where $n$ is a positive integer and $a_1, a_2,\dots, a_n$ are real positive numbers   Hence, show that  $$\csc^{2}\theta +\sec^{2}\theta +\cot^{2}\theta \geq 9\cos^{2}\theta$$  Please help me.  Thank you!
How do you create a dropdownlist from an enum in ASP.NET MVC? I'm trying to use the Html.DropDownList extension method but can't figure out how to use it with an enumeration.  Let's say I have an enumeration like this:  public enum ItemTypes {     Movie = 1,     Game = 2,     Book = 3 }   How do I go about creating a dropdown with these values using the Html.DropDownList extension method?  Or is my best bet to simply create a for loop and create the Html elements manually? Modiagram will not compile The sequence of triangular numbers is well known. Consider now the triangular numbers modulo some positive natural number $k$. That is, let $T_{n,k}$ be the remainder of dividing $T_n$ by $k$. For the regular sequence of triangular numbers, we can take as convention $T_n = T_{n, \infty}$.  Question: For all values of $k, i &lt; k$, does there exists some $n$ such that $T_{n, k} = i$ Can a magus with a high BAB cast two spells or attack and cast, with their full attack action? Suppose you are a level 10 Magus or something similar. (Something with a decent BAB progression and spellcasting abilities) When you hit +6/+1, can you cast a spell, then attack, vice-versa, or simply case two spells? Everywhere I look I can't get a clear answer. Kernel of injective group homomorphism If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?  I think the $Ker\phi = \{x\in G:\phi(x)=1\}$  because you are looking at an identity then it is all reals $0$ does not work a because if $x=0, e'x=0$  but then i realize it is injective how does that affect the kernel? We have two PHP5 objects and would like to merge the content of one into the second. There are no notion of subclasses between them so the solutions described in the following topic cannot apply.    //We have this: $objectA-&gt;a; $objectA-&gt;b; $objectB-&gt;c; $objectB-&gt;d;  //We want the easiest way to get: $objectC-&gt;a; $objectC-&gt;b; $objectC-&gt;c; $objectC-&gt;d;   Remarks:   These are objects, not classes. The objects contain quite a lot of fields so a foreach would be quite slow. So far we consider transforming objects A and B into arrays then merging them using array_merge() before re-transforming into an object but we can't say we are proud if this. I've got to send a bulk email out to a bunch of people in reply to emails they sent me. I've got all the original emails stored in a single outlook folder. I want to extract all of the reply addresses from the emails in that folder so I can send an email to all of them.  The article  shows you how to export the FROM address of these emails, however a large amount of them came from a web service that sends emails 'on behalf of' the person I'm trying to mail to. So I need the reply-to address, and the export wizard doesn't do that.  Does anyone have a tip on how to do this? This is Outlook 2007 on Exchange 2010. Do Stack Overflow and other Stack Exchange sites have an app I can use? After two days of installation, strangely my Ubuntu machine is restarting instead of shutdown from both desktop GUI and command line.  sudo shutdown -h now sudo halt  # (doesn't shutdown, instead freezes on boot screen Plymouth)   I had force shutdown by powering off the machine.  This is not happening all the times.  I have been using Ubuntu for 6 months and never experienced this kind of problem. I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking How can we change root password? Where is a good place to learn classical optics for high school competitions? I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue. "invalid EFI file path" while booting Windows I just got my new working notebook with Windows 8 pre-installed. After installing Ubuntu 12.10 the grub menu says that there is an "invalid EFI file path" and it won't boot Windows. Ubuntu works fine...  Is there anything I can do about this? Proving that two systems of linear equations are equivalent if they have the same solutions I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve:  Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.  I can't seem to figure out how to prove this without resorting to case work where you account for the cases where one of the coefficients are zero and when both are.  Is there an elegant way to prove in general that when two systems of linear equations have the same solutions, they are equivalent? The converse is obvious enough though.  Definition of equivalence from the text :     Let us say that two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations   in the other system. Create instance of generic type in Java? How to monitor who is pinging me? Is there a way to know who is pinging my PC on my network? Is it possible to monitor all ICMP packets? How can I do that? Kill process after it's been allowed to run for some time I want to limit the time a grep process command is allowed to run or be alive.  For example. I want to perform the following:  grep -qsRw -m1 "parameter" /var   But before running the grep command I want to limit how long the grep process is to live, say no longer than 30 seconds.  How do I do this?  And if it can, how do I return or reset to have no time limit afterwards. Mutual Independence Definition Clarification Making overview with linked detailed maps in QGIS? In Random forest algorithm, Breiman (author) constructs similarity matrix as follows:   Send all learning examples down each tree in the forest If two examples land in the same leaf increment corresponding element in similarity matrix by 1 Normalize the matrix with number of trees   He says:     The proximities between cases n and k form a matrix {prox(n,k)}. From their definition, it is easy to show that this matrix is symmetric, positive definite and bounded above by 1, with the diagonal elements equal to 1. It follows that the values 1-prox(n,k) are squared distances in a Euclidean space of dimension not greater than the number of cases.    In his implementation, he uses sqrt(1-prox), where prox is a similarity matrix, to convert it to distance matrix. I guess it has something to do with the "sqaured distances in a Euclidean space"-quoted above.  Can somebody shine a little light on why it follows that 1-prox are squared distances in a Euclidean space and why he uses squared root to get distance matrix?