Let $n\in\mathbb{N}$ and $x_1,x_2,x_3.....,x_n \in\mathbb{Q}$ with $x_i &gt; 0$ and $\prod_{i=1}^{n} x_i = 1$  prove that $\sum_{i=1}^{n}x_i \geq n$ (hint:use induction).  Been stuck on that for hours. It seems as if the terminology of the question is not well defiened because they use "Let $n\in\mathbb{N}$" and not "For each $n\in\mathbb{N}$"   Could I get some insight and help please?
Please give me the proof for the following: Let $a_1,\,a_2,\,\dots\,a_n$ be $n$ positive real numbers whose product is equal to $1$. Prove that $$a_1+\cdots+a_n \geq n$$ and that the equality sign holds only if every $a_k$ is  equal to $1$.
Names of close-voters are not visible in the post notice but in the revision history and timeline I am not a user with close/reopen votes privileges on Travel Stack Exchange, therefore I can't see the names of the close voters :    However, they are visible when going to the revision history:    or the timeline of the post:    This also happens when I am not logged in: I cannot see the banner (and so if the question has been closed), but from the revision history/timeline of the post I can get that information.  Shouldn't names of close-voters be completely hidden from people without close/reopen privileges (even if the revision history is a little hard to find)? Do spellcasters know when a dismissable spell has been broken? Do spell casters know how long their spells are active, and can they tell at a glance?  Specifically lets talk about , as I know its a permanent duration and dismissable. Does the spell caster who put someone under this spell know how long it is active? Could they tell automatically just by seeing the person (without any clues like them walking in a maze without difficulty or reacting to a sound)? So could they tell if someone has broken the spell they cast? SQL query or method to list all sql services running This might be a simple one, but i searched a lot and unable to find one.  Is there a sql query or a way i can run on registered server to find the list of all sql related services running like, ssis , ssas, ssrs, sql, agent, broswer, etc...  Tried googling out and can find sys.dm_server_services dmv but it does not show all the services running Insert Login IP Ranges using apex in salesforce Is it possible to insert to Login IP ranges via apex? Is there a such thing as an exchange program in Harry Potter? In the Harry Potter universe, is there any mention of an exchange student at Hogwarts or one of the Hogwarts students going on exchange at another school?  This happens all the time in the "muggle" school systems I'm sure it would happen in the wizarding schools as well. Open source computer algebra systems for general relativity I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical aspect rather than numerical computations. Looking for book: contact with aliens, five different races, mismatched claws, biological imperative to breed Encouraging users to create an account (and keep it) I have 1:50 wound current transformer and I don't have the datasheet. Could anybody help me to calculate the burden resistor?    Maximum current to be measured: 2A  Max output voltage to ADC: 5V Restricted logins with automatic full permission in databases My scenario: I have some groups with full - and inappropriate - access to all Sql Server instances. I want to change that.  My Plan: Give full and automatic access to all databases (it is customer databases and this groups must be allowed to change any property if necessary).  Restrict permissions on Instance Level, because this groups should not be allowed to do any unexpected change on instance properties.  I don't want to create a user in databases, because our external customer could drop the user ( they have db_owner permission )  Someone have any idea? Tks Do I need a transit visa for a layover in the United Arab Emirates? I've been trying to ignore my burning desire to make some guacamole because every time I go to the supermarket to buy avocados I can't manage to pick out good ones.  They always either become too soft before I have a chance to do anything with them or are hard and not very tasty.  How can I tell when an avocado is perfectly ripe? Trying to find the original story (I read it in Russian translation, original was most likely English unless it was Lem).  The story plot had artificial birds designed to serve as cops. The birds started attacking humans who were percieved as being aggressive towards anything alive. So they created more advanced birds to protect against the first ones. This worked for a while until the second kind started attacking humans as well. It ended with the idea that they would need another generation to attack the second-generation birds.  The story wasn't very long. I read it a while back (mid-1980s, which means the original was written even earlier, probably before 1980).  Despite a somewhat Asimov-like "misbehaving robots" topic, I'm pretty sure it wasn't Asimov. How to run grep with multiple AND patterns? How to run .sh script with keybind Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$? Is there any $2 \times 2$ matrix $A \neq I$ such that $A^3=I$?  In my opinion: no. Thank you very much. Any software for windows Writing custom (intelligent) microsoft windows firewall extension (C#/C++/C) Writing custom (intelligent) firewall for windows Installing nginx etc. on the same computer and use it as proxy with rules.   "You could not solve all Ddos problems with software" is not answer for this question.I want to reduce Ddos vulnarebility of my server.(It will stop low-level Ddos attacks)  Hosted solutions or cloud services are not related with this question.  Closing some of the ports and protocols via Microsot Windows Firewall is not related with this question.  Any idea?  EDIT After Questions  Low level: sample scenerio: 500 zombie clients and each of them sending 100 Custom Get/Post requests per second)   Web Application is on IIS. I want to stop attacks before they reach to IIS.  EDIT After Helpful Comment  How can the DDOS attack achieve its aim?   Exhausting bandwidth ( Not related with my question) Exhausting network hardware of server (NIC) ( Not related with my question) Exhausting server resources (RAM,CPU)  (Yep, I mean this)   If you stop attacks late, then you lost more resources! (Is this false?)   You can stop attacks on Web Application Level (in Asp.net MVc Code or in Handler) You can stop attacks on IIS (Dynamic Ip Restrictions)  You can stop attacks on lower levels (Firewall ..)   EXTRA INFO ABOUT THE LAST ATTACK (7 Hours ago)  When I connected server during the attack IIS Cpu Utilization was %92-%99. And when I try to connect homepage I got this error:  A connection was successfully established with the server, but then an error occurred during the pre-login handshake. (provider: SSL Provider, error: 0 - The wait operation timed out.) Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits How to compute the series: $\sum_{n=0}^{\infty} (-1)^{n-1}\binom{1/2}{n}$ Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance.