I need to prove that $$\cos\dfrac{\pi}{11}+\cos\dfrac{3\pi}{11}+\cos\dfrac{5\pi}{11}+\cos\dfrac{7\pi}{11}+\cos\dfrac{9\pi}{11}=\dfrac{1}{2}$$  How to do it?
Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$ using Euler's formula.  Everything I tried has failed so far.  Here is one thing I tried, but obviously didn't work. $$\Re e \{e^{\frac{\pi}{11}i}(1+e^{\frac{2\pi}{11}i}+e^{\frac{4\pi}{11}i}+e^{\frac{6\pi}{11}i}+e^{\frac{8\pi}{11}i}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{e^{2\pi i}}+\sqrt[11]{e^{4\pi i}}+\sqrt[11]{e^{6\pi i}}+\sqrt[11]{e^{8\pi i}}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}) \}=\frac{1}{2}$$ $$\Re e \{5e^{\frac{\pi}{11}i} \}=\frac{1}{2}$$ $$5\cos(\frac{\pi}{11})=\frac{1}{2}$$ Which isn't true :D  Thanks in advance
101 positive integers placed on a circle A Pigeonhole Principle problem:     101 positive integers are placed on a circle whose sum is 300. Prove   that it is possible to choose from these numbers some consecutive   numbers whose sum is equal to 200.   (I don't know if the word 'consecutive' is appropriate in this case ,I mean that these numbers follow each other on that circle) Apex:OutputText to Currency Format from String Wrapper is there a way to pass a string to the &lt;apex:outputText&gt; so that it can be formatted as a currency or percent value?  Here's what I have inside a column of my visual force table:  &lt;apex:outputText value="{0, number, currecny}"&gt; &lt;apex:param value="{!row.values[1]}" /&gt; &lt;/apex:outputText   this gives me an error at runtime (when the visual force page loads), which I think may be some kind of type mismatch, where &lt;apex:outputText&gt; is expecting a Decimal or number but I am passing a string.  This is the error at runtime:     The value attribute on  is not in a valid format. It must be a positive number, and of type Number, Date, Time, or Choice.   Thanks Clean install Windows 8.1 or Windows 8 from OEM key Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals. I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of $(\mathbb{Q}, \leq)$ and $\epsilon_o = \sup\{\omega, \omega^{\omega}, \omega^{\omega^{\omega}}, ... \}$. Actually,  I'm not finding neither an isomorphism between $K$ and $\omega^{\omega}$.  Thanks in advanced. In Magic: The Gathering, the rules for a player losing in a multiplayer game are quite complex, so I'd like to try to understand them by example. I would be happy to ask each of these situations as a separate question if that is preferred, but they all seem related, and I would love to have one place to look up all of these answers.  Let's say I'm playing with two friends, Bob and Chloe, in a simple free for all. Unfortunately, I lose. When I lose, suppose the following:   I controlled an  which has exiled Bob's , does he get his bear back? I have a  on the stack targeting Chloe, does Chloe take the 2 damage still? Earlier in the turn I cast  to prevent damage to one of Bob's . If it later takes 3 damage, will that damage be prevented? Something I controlled gave Bob a 1/1 soldier token. What happens to the token? Something Chloe controlled gave me a 1/1 soldier token. What happens to the token? I had a  enchanting one of Bob's army of . What happens to the bear? I had also played  on Bob's final . What happens to the last bear? Chloe had  my . What happens to the Clone? Are there any other situations specifically addressed by the rules? How can I cite a range of papers, the output is a range of numbers, instead of a list of numbers? In other words, when I type  ... some dummy text here is due to me \cite{me1, me2, me3, me4} ...   I want the output to be     ... some dummy text here is due to me [3-6] ...   instead of     ... some dummy text here is due to me [3.4.5.6] ...   (I use LaTeX, and am currently using the amsart class and whatever it includes. But I am open to other suggestions.) Was Elrond, in Tolkien's opinion, more inherently powerful than Galadriel? I always thought it was the other way around, but then I read this (Letter 246):     "Of the others only Gandalf might be expected to master him – being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form. In the 'Mirror of Galadriel', 1381, it appears that Galadriel conceived of herself as capable of wielding the Ring and supplanting the Dark Lord. If so, so also were the other guardians of the Three, especially Elrond. But this is another matter. It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power."   This "especially Elrond" confuses me. I don´t think that Elrond is inherently more powerful than Galadriel, who is described as the greatest of the Noldor, except for Fëanor (and possibly even his equal).  A quote from Unfinished Tales says:     "...In this he (Tolkien) emphasized the commanding stature of   Galadriel already in Valinor, the equal if unlike endowments of   Fëanor; and it is said here that so far from joining in Fëanor's   revolt she was in every way opposed to him."   Furthermore,      "A queen she was of the woodland Elves, the wife of Celeborn of   Doriath, yet she herself was of the Noldor and remember the Day before   days in Valinor, and she was the mightiest and fairest (so even   fairer than Arwen I would guess) of all the Elves that remained in   Middle-earth."   (The Silmarillion, of the Rings of Power)  And again...     "Galadriel was the greatest of the Noldor, except Fëanor maybe, though   she was wiser than he, and her wisdom increased with the long years."   (the Shibboleth of Fëanor)  Restating it:     "These two kinsfolk (Fëanor and Galadriel), the greatest of the   Eldar in Valinor, were unfriends for ever." "Who together with the   greatest of all the Eldar, Luthien Tinuviel, daughter of Elu Thingol,   are the chief matter of the legends and histories of the Elves."   (Unfinished Tales)  The latter means that Luthien, Fëanor and Galadriel are the chief matter of the history and legends. It seems to me that Tolkien thought of those three as the creme de la creme of the elven crop.  Reading all that I have serious problems believing that Elrond could be superior to her. As great as Elrond is, descending from Melian, he is no equal of Fëanor, who is said to be the mightiest of all elves, and if Galadriel is his equal, then there is no way that Elrond would be mightier than her.  Maybe I just read this "especially Elrond" wrong; it could as well mean that especially Elrond was deceived into thinking that he could master the ring, because that was the deceit of the ring. And Tolkien said explicitly that only Gandalf could possibly master it.  So, what do you think? How do I find the particular solution of this second order inhomogenous differential equation? (Using undetermined coefficients).  $y''-4y'+5y = 4e^{2x} (\sin x)$  I can find the generel homogenous solutions but I need help for the particular. Can we balance the whitespace above/below headings? In January, some additional whitespace  below h1, h2 and h3 headings,  more whitespace below the headers than above them.  Could we have the whitespace above the headings be more than the whitespace below them?  I find some posts a bit hard to read, like the current rendering on the left:   &nbsp; &nbsp; &nbsp; A Riemann integrable function must have infinitely many points of continuity Regex for quoted string with escaping quotes How do I get the substring " It's big \"problem  " using a regular expression?  s = ' function(){  return " It\'s big \"problem  ";  }'; Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? Determining the characteristic of a field Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions? Let $\varphi: \mathbb{N} \to \mathbb{N}$ be the totient function.   Is there an integer $N &gt; 0$ such that there are infinitely many integers $n &gt; 0$ such that $$\varphi(n) = N?$$ How to delete all files in a current directory starting with a dot? What is the correct vector input for a texture connected to a Cycles lamp so that I can make a projector-like textured light?  I am trying to make an projector-like effect with a textured light in Cycles, currently trying it out with a Spot type of lamp, but I am open to any solution involving other lamp types if they work correctly.  The problem is that whatever vector input I use, the texture is always projected downwards, towards the global $-Z$ axis in scene world space, no matter where I point my spot light. I would like to have the texture always project "forward" towards the the direction the spot light is pointing, in the local $-Z$ axis.    The plan would then be to use this spot at several places in my scene, pointing at different directions, and as it currently stands I would have to have a different node setup for each spot instance, manually compensating for each unique rotation in the scene, which would quickly become unpractical.  I have searched around the internet and I am currently using a solution similar to the one on this .  Is there any better method, perhaps involving some funky Vector Math, or is this not currently supported? Why is "Nature" usually spelt with an initial capital letter in scientific journals? I am mainly referring to life science here, in case this matters.  I am not talking about the obvious cases, like capitalisation in titles, at the beginning of a sentence, etc., but when it appears mid-sentence, where one would normally expect it to be lowercase.  I've heard some explanation that it gives the impression of Nature as a goddess (see comments on this post ) - but this is probably the last thing I would have expected in a scientific article... Find all solutions of the equation $n^m=x^2+py^2$ which satisfy the following properties Prove or disprove that,     There always exists a solution of the equation, $$n^m=x^2+py^2$$ with odd $x$ and $y$ and for all $m\geq k$ for some positive integral $k$. Here $p$ is an odd prime and $n\in 2\mathbb{N}$. Is $k$ dependent on $n$? If so then find a way to calculate the value of $k$.   This is basically one of my conjecture which I am trying to prove for quite sometime. Unfortunately, I have progressed very little in this problem. So far I have been only able to prove that for all $m\geq3$ , $n=2$ and $p=7$ there is always a solution of the equation meeting the constraints.   Any idea how to tackle the problem?    Update  After D. Burde's answer of the original problem (see below) I am now interested in finding all $(n,m,p)$ triplets such that the equation holds. Any ideas regarding this problem? Is $v=\ (r_i)^{-1}\cdot z $, a uniformly random value of a field? We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers.  Let $r_i$ be a value picked uniformly at random from the field such that $r_i&gt;\frac{q}{2}$. Let $z$ be a fixed field element.   Question: Is $v=(\ (r_i)^{-1}\cdot z) \bmod q$, a uniformly random value of the field? I have been stuck for several days on this old Analysis problem (I am doing some study on my own).  I have tried several things (which I'll indicate below), but I cannot seem to figure it out.  Here is how the problem is presented: Problem: &quot;Let $f$ be a continuous real-valued function on $[a,b]$.  Suppose there exists a constant $M \geq 0 $ such that $$|f(x)| \leq M \int_a^x |f(t)| dt$$ for all $x \in [a,b]$.  Show that $f(x)=0$ for all $x \in [a,b]$.&quot; My Thoughts: I have tried using the mean value theorem iteratively, but that seems to always lead me down a dead end road.  I deduced that $f(a)=0$.  If only $f$ were assumed to be differentiable, then maybe I could play with trying to get the derivative to be $0$, but unfortunately it's only continuous.  My other thought was to (somehow) use the condition to show that $\int_a^b |f(x)| dx = 0$.  I also played around a bit with contradiction, but to no avail.  Even if one of these hair-brained thoughts is correct, I am not really sure what to do next. If you have any ideas, suggestions, or solutions, I would really appreciate it if you are willing to share them.  Thank you for your time.