Find all functions such that $$\forall x,y \in \mathbb{R} \quad f(x+y)+f(x-y)=2f(x)\cos(y)$$  I got this question from a book. I tried substituting and assuming different values of $x$ and $y$ but ended getting nothing powerful enough to solve it. On taking $y=90$ I got a function in $x$ following cyclicity of a trigonometric function. But it will be valid iff $y$ is $90$. I'm stuck. Any help would be appreciated. Thanks in advance.
How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$    I tried substituting $x=0$ to get $f(y)+f(-y)=2f(0)\cos y$ . Taking $x=y$ gives $f(2x)+f(0)=2f(x)\cos x$. I similar get some more relations like that, but its not really helping me in finding anything useful. I also think the function must either be $\sin$ or $\cos$ looking at the product to sum formulae. Can anyone tell me how should I solve this?
Anime with a boy hiring a creature from a stone, meets a man named Dante and starts a journey to collect crystals If $a$, $b$, $c$ are the roots of $x^3-6x^2+3x+1=0$, find all possible values of $a^2b+b^2c+c^2a$ Let $a$, $b$, $c$ be the roots of $$x^3 - 6x^2 + 3x + 1 = 0$$ Find all possible values of $a^2 b + b^2 c + c^2 a$. Express all the possible values, in commas.  I've already tried to bash out all the roots, Vieta's Formula, and try to manipulate the equation to become easier to work with. However, Vieta's didn't get me anywhere, and I couldn't find a way to make the equation simpler or anything. Any hints to start this problem? Can a Rogue (D&D 3.5) use sneak attack on both the first and second attack? Are there any versions of t-SNE for streaming data? Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$  I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] =r(x+1)^{r-1}=\sum_{n=1}^\infty \binom{r}{n}nx^{n-1}$  However, I am not sure how to "get rid" of the binomial term. The following code gives me an error:  \documentclass[12pt]{article} \usepackage[margin=2cm]{geometry} \usepackage [english]{babel} \usepackage{amsmath, amssymb} \usepackage{enumitem, array} \begin{document} {\begin{minipage}[t]{\textwidth} \begin{enumerate}[label=(\roman*)] {\setlength\itemindent{10pt}\item  \begin{equation*} \begin{array}{&gt;{\centering\arraybackslash}m{1.5cm}&gt;{\centering\arraybackslash}m{1.5cm}&gt;{\centering\arraybackslash}m{1.5cm}&gt;{\centering\arraybackslash}m{1.5cm}&gt;{\centering\arraybackslash}m{1.5cm}}  \hline x &amp; l_{[x]} &amp; d_{[x]} &amp; l_{x+1} &amp; \mathring{e}_{[x]} \\ \hline \end{array} \end{equation*}} \end{enumerate} \end{minipage}} \end{document}   Note: for various reasons, I have to keep minipage there. For some reason, this code gives me an error when it compiles. It gives the following output:     The errors either say there is a } missing or a $ missing. My eyes don't see such a thing! Is there a way to show statistic info? How fast a (relatively) small black hole will consume the Earth? This question appeared quite a time ago and was inspired, of course, by all the fuss around "LHC will destroy the Earth".   Consider a small black hole, that is somehow got inside the Earth. Under "small" I mean small enough to not to destroy Earth instantaneously, but large enough to not to evaporate due to the . I need this because I want the black hole to "consume" the Earth. I think reasonable values for the mass would be $10^{15} - 10^{20}$ kilograms.   Also let us suppose that the black hole is at rest relative to the Earth.  The question is: How can one estimate the speed at which the matter would be consumed by the black hole in this circumstances? The banner ads are in the wrong format for the right-hand ad slot:    These kinds of ads are not useful (because most of it is cropped) and should be excluded from being shown in that spot. Partition of N into infinite number of infinite disjoint sets? Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers? I want to return two objects from a Java method and was wondering what could be a good way of doing so?  The possible ways I can think of are: return a HashMap (since the two Objects are related) or return an ArrayList of Object objects.  To be more precise, the two objects I want to return are (a) List of objects and (b) comma separated names of the same.  I want to return these two Objects from one method because I dont want to iterate through the list of objects to get the comma separated names (which I can do in the same loop in this method).  Somehow, returning a HashMap does not look a very elegant way of doing so. Is this functionality going to be put into a later Java version?  Can someone explain why I can't do this, as in, the technical way Java's switch statement works? Why is the PHP version different in phpinfo() and CLI? I realized that there is a miss matching between the PHP version reported by phpinfo() and php -v (in the CLI).  phpinfo():  5.5.24 php -v: 5.6.9   I am working on a Mac OS X 10.10 (Yosemite) and installed a library (php-version) to try to manage the PHP versions.  It reports me 3 different versions installed on my laptop:    5.4.41   5.5.25 * 5.6.9   I would like to set the same version of PHP in both Apache and CLI.  How can I tell Apache which PHP version use?  I tried to write this line in my httpd.conf:  LoadModule php5_module    /usr/local/opt/php56/libexec/apache2/libphp5.so   Then:  sudo apachectl restart   But all this not solve the problem… In my PHP info I still have: 5.5.24 How to get the type of T from a member of a generic class or method Links in site profiles stripped from network profile I have a link in my profile, advertising a Stack Exchange question of mine (which I believe is acceptable use of the profile page). The link works fine on my individual profile pages. For example,  and . However, on  the links do not show.  The link is present on the GIF and on the text (two separate links, both to the same place). Both work fine on the individual profile pages but both are absent from the network profile page. The GIF and text are present, just without being linked.  Is there a reason for disallowing links from the network profile page, or is this a bug? How to determine if string contains specific substring within the first X characters I want to check whether Value1 below contains "abc" within the first X characters. How would you check this with an if statement?  var Value1 = "ddabcgghh";  if (Value1.Contains("abc")) {     found = true; }   It could be within the first 3, 4 or 5 characters. How do I fix this "E: Type '*' is not known on line * in source list ..." update error? Can't join a community I look for similar questions to this one but my problem is different. I want to join this community:    so I click in     but I got this error:    but then, I found that the community is already in list of communities:    However, if I click on the community, I'm an anonymous member     and even I have the option to join again     but if I click on it, the problem starts again :( what can I do? Prove that   $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that. I have bumped into an issue that doesn't allow me to do anything related to apt-get: install, remove, etc. Whenever I try to do something (e.g. sudo apt-get install firefox -f) it gives me the same error message:  Reading database ... dpkg: unrecoverable fatal error, aborting:  unable to open files list file for package `fontconfig-config': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   I have tried to deal with the package fontconfig-config without success.  I have found that the "Input/output error" is usually linked with physical problems of the hard drive, but I do not think that's the case I am using that hd without any other problem.  I have tried removing the mention to the package in /var/lib/dpkg/status as mentioned .  I have tried deleting all files related to the package in /var/lib/dpkg/info as I found somewhere.  But I still cannot do anything. The funny bit comes when I look for the file that is giving me troubles:   mecho@Ansible-MS-7680:/var/lib/dpkg/info$ ls fontconfig* ls: cannot access fontconfig-config.list: Input/output error fontconfig.list     fontconfig.postinst  fontconfig.preinst  fontconfig.triggers fontconfig.md5sums  fontconfig.postrm    fontconfig.prerm   This is done after I deleted all files ... it looks like fontconfig-config.list still exists but it doesn't show up!  Any idea about how to solve the problem?  I am on kubuntu precise, fontconfig-config_2.8.0-3ubuntu9.1    I do not have smartmontools installed nor any other programme I could use to check the hard disk. The most similar think I seem to have is KDE Partition Manager but it doesn't allow me to check anything (the option appears unavailable). However, I think the problem is not the hd.  How do you suggests me to do the fs check? fcheck is not installed and I cannot think of any other command in bash (as I do not think I have anything else installed that could perform a check).  Thank you.  Edit: I have found the command (silly me), fsck. This is what I got:  /dev/sda1: recovering journal Clearing orphaned inode 4456702 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456700 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456699 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456697 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456694 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456693 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456688 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456481 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456692 (uid=0, gid=0, mode=0100600, size=65536) Clearing orphaned inode 4456689 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 4456709 (uid=1000, gid=1000, mode=0100600, size=714046) Clearing orphaned inode 4456469 (uid=1000, gid=1000, mode=0140775, size=0) Clearing orphaned inode 3672644 (uid=0, gid=0, mode=0100644, size=1595) Clearing orphaned inode 1324021 (uid=0, gid=0, mode=0100640, size=1980) Clearing orphaned inode 1324014 (uid=0, gid=0, mode=0100640, size=126) Clearing orphaned inode 4456499 (uid=1000, gid=1000, mode=0100600, size=0) Clearing orphaned inode 4456483 (uid=1000, gid=1000, mode=0100600, size=65536) Setting free inodes count to 7147142 (was 7147702) Setting free blocks count to 23891195 (was 23781617) /dev/sda1: clean, 151930/7299072 files, 5292549/29183744 blocks