Find all values of real numbers $a,b,c, d$ satisfying the following condition  $$(b+c+d)^{2010}=3a$$ $$(a+c+d)^{2010}=3b$$ $$(a+b+d)^{2010}=3c$$ $$(a+b+c)^{2010}=3d$$
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations  $(b+c+d)^{2010}=3a$  $(a+c+d)^{2010}=3b$  $(a+b+d)^{2010}=3c$  $(a+b+c)^{2010}=3d$  I tried to find the solutions using hit and trial and by using some logic also I find two solutions which are $(0,0,0,0)$ and $(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3})$  I think these are the only solutions that exist.
The code below compiles, but has different behavior for the char type than for the int types.  In particular     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;   result in 3 instantiations of templates for three types: int8, uint8 and char. What gives?  The same is not true for ints: int and uint32 which result in the same template instantiation, and signed int another.  The reason seems to be that C++ sees char, signed char and unsigned char as three different types. Whereas int is the same as a signed int. Is this right or am I missing something?  #include &lt;iostream&gt;  using namespace std;  typedef   signed char       int8; typedef unsigned char      uint8; typedef   signed short      int16; typedef unsigned short     uint16; typedef   signed int        int32; typedef unsigned int       uint32; typedef   signed long long  int64; typedef unsigned long long uint64;  struct TrueType {}; struct FalseType {};  template &lt;typename T&gt; struct isX {    typedef typename T::ikIsX ikIsX; };   // This  int==int32 is ambiguous //template &lt;&gt;            struct isX&lt;int  &gt;    { typedef FalseType ikIsX; };  // Fails template &lt;&gt;            struct isX&lt;int32  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint32 &gt;  { typedef FalseType ikIsX; };   // Whay isn't this ambiguous? char==int8 template &lt;&gt;            struct isX&lt;char  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;int8  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint8 &gt;  { typedef FalseType ikIsX; };   template &lt;typename T&gt; bool getIsTrue(); template &lt;&gt;           bool getIsTrue&lt;TrueType&gt;() { return true; } template &lt;&gt;           bool getIsTrue&lt;FalseType&gt;() { return false; }  int main(int, char **t ) {    cout &lt;&lt; sizeof(int8) &lt;&lt; endl;  // 1    cout &lt;&lt; sizeof(uint8) &lt;&lt; endl; // 1    cout &lt;&lt; sizeof(char) &lt;&lt; endl;  // 1     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;     cout &lt;&lt; getIsTrue&lt; isX&lt;int32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;int&gt;::ikIsX  &gt;() &lt;&lt; endl;  }   I'm using g++ 4.something Jutifying: irrational + rational = irrational I am not sure how to show this. This is what I tried.   Proposition: If x is rational number and y is an irrational number then x+y is an irrational number.  Logically speaking this is  P=x is rational  Q= Y is irrational  R= x+y is irrational  $P\wedge Q \rightarrow R$  Explanation: What I did  $x=\frac{p}{q}$ P and Q integer  Y cannot be written as the a rational by definition  So therefore  $x+y=\frac{p}{q}+y$  Will not be a rational number as there cannot be a common denominator. I would like to update VirtualBox on my 32 bit Ubuntu 15.04 machine from 4.3.26 to 4.3.28, using a Debian package I manually downloaded from . But simply double-clicking the package opens up the Software Centre which gives an error that installing it would break an existing package (the exact message was:   Breaks existing package 'virtualbox' conflict: virtualbox-4.3 ()   and it does not appear to have an option to ignore this and proceed with the installation anyway. Is there a command-line work-around this problem, perhaps? Traffic (views, visits) is not correctly registered on Site Analytics or Area 51 I saw this question  but it doesn't have a full solution. What is the full solution to the 99 bottles of beer on the wall for Vim adventures?  Context: for the sake of being a good SO member I will give more details but a full solution will answer my question. It seems from reading the other post that my register is incorrect for some reason. I can't get the &lt;NL&gt;. I think that's the issue since pasting won't work... How to check if element is visible after scrolling? I'm loading elements via AJAX. Some of them are only visible if you scroll down the page. Is there any way I can know if an element is now in the visible part of the page? Identifying and solving javax.el.PropertyNotFoundException: Target Unreachable Make program first read from pipe, then from keyboard Prove inequality using Schwarz's Lemma I'm having trouble proving the following:  Let $f$ be analytic on $D(0,R)$ and suppose there is a real constant A for which $\operatorname{Re}f(z) &lt; A$ for all $z \in D(0,R)$. If $f(0) = 0$, show that $|f(z)| \leq \frac{2A|z|}{R-|z|}$.   I was told to first start by showing that $|\frac{f(z)}{2A-f(z)}| \leq 1$ and also use the following result, which I was able to prove:  Let $f$ be analytic on $D(0,R)$ and let $f(0)=0$. If $|f(z)| \leq M$ for all $z \in D(0,R)$, then $|f(z)| \leq M|z|/R$ for all $z\in D(0,R)$.  Thanks in advance! Deriving the conditional distributions of a multivariate normal distribution We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into  $$\boldsymbol\mu = \begin{bmatrix}  \boldsymbol\mu_1 \\  \boldsymbol\mu_2 \end{bmatrix} $$ $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\  {\boldsymbol y}_2 \end{bmatrix}$$  with a similar partition of $\Sigma$ into $$  \begin{bmatrix} \Sigma_{11} &amp; \Sigma_{12}\\ \Sigma_{21} &amp; \Sigma_{22} \end{bmatrix} $$ Then, $({\boldsymbol y}_1|{\boldsymbol y}_2={\boldsymbol a})$, the conditional distribution of the first partition given the second, is  $\mathcal{N}(\overline{\boldsymbol\mu},\overline{\Sigma})$, with mean $$ \overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\Sigma_{22}}^{-1}({\boldsymbol a}-\boldsymbol\mu_2) $$ and covariance matrix $$ \overline{\Sigma}=\Sigma_{11}-\Sigma_{12}{\Sigma_{22}}^{-1}\Sigma_{21}$$  Actually these results are provided in Wikipedia too, but I have no idea how the $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ is derived. These results are crucial, since they are important statistical formula for deriving Kalman filters. Would anyone provide me a derivation steps of deriving $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ ? Thank you very much! Why does pdfTeX hang on this file? Forcing a SimpleXML Object to a string, regardless of context How do I transfer a file to my server using PuTTY? What's an intuitive way to think about the determinant? I've been thinking of animating some dice and while most are easy to make, I'm having some trouble figuring out a way to generate a regular dodecahedron. I could hand build it but it would be potentially uneven. Would using the base of a 5 sided cylinder and then some variation on spin be the best option or is there a better way? The adverb "quite" has the following meanings :     1: wholly, completely ("not quite   finished")      2: to an extreme : positively "quite   sure" —often used as an intensifier   with a "quite a swell guy" "quite a   beauty"      3: to a considerable extent : rather   ("quite near")   My understanding is that #1 and #2 conflict with #3. For example, when you say "She's quite right", do you mean "She's completely right" or "To a considerable extent, she's right"? Notice that in the latter case you're not saying that she is completely right.  Another example is "I'm quite sure". Does it necessarily mean "I'm completely sure"?   Does the word really have conflicting meanings that can possibly cause ambiguity? Or am I missing something? Why does "Object reference not set to an instance of an object" not refer the variable name or column? ext4 partition size / free space discrepancies While creating a 250GiB backup partition for my data, I have noticed lots of discrepancies between reported partition size and free space in Nautilus, gParted, df, tune2fs, etc.  At first I thought it was a GiB / GB confusion. It was not.  Then I thought it could be ext4's reserved blocks. It was not.  Im completely puzzled. Here are some images. Here are the steps:   First, NTFS. 524288000 sectors x 512 bytes/sector = 268435456000 bytes = 268.4 GB = 250 GiB.      Nautilus say "Total Capacity: 250.0 GB" (even though its actually GiB, not GB). Apart from that minor mislabeling, so far, so good   Now, same partition, formated as ext4 with gparted:     First, Last and Total sectors are the same. It IS the same 250GiB partition. Used size is 4.11GiB (reserved blocks maybe?)    Nope. Looks like reserved blocks are 12.7 GiB (~5%. ouch!). But... why Total Capacity is now only 246.1 GiB ???. That difference (sort of) matches the 4.11 GiB reported by gparted. But... if its not from reserved blocks, what is it? And why gparted didnt report that 12.7GiB of used space?  $ df -h /dev/sda5 Filesystem            Size  Used Avail Use% Mounted on /dev/sda5             247G  188M  234G   1% /media/BACKUP   df matches Nautilus in reported free space. But.. only 188M used? Shouldnt it be ~12GB? And total capacity is still wrong. So i ran tune2fs to find some clues. (irrelevant output is ommited)  $ sudo tune2fs -l /dev/sda5 tune2fs 1.41.12 (17-May-2010) Filesystem volume name:   BACKUP Filesystem UUID:          613d592e-47f5-4206-96a7-210090d340ef Filesystem features:      has_journal ext_attr resize_inode dir_index filetype extent flex_bg sparse_super large_file huge_file uninit_bg dir_nlink extra_isize Filesystem flags:         signed_directory_hash  Filesystem state:         clean Filesystem OS type:       Linux Block count:              65536000 Reserved block count:     3276800 Free blocks:              64459851 First block:              0 Block size:               4096   65536000 total blocks * 4096 bytes/block = 268435456000 bytes = 268.4 GB = 250 GiB. It matches gparted.  3276800 reserved blocks = 13421772800 bytes = 13.4 GB = 12.5 GiB. It (again, sort of) matches Nautilus.  64459851 free blocks = 264027549696 bytes = 264.0 GB = 245.9 GiB. Why? Shouldnt it be either 250-12.5 = 237.5 (or 250-(12.5+4.11)=~233) ?  Removing reserved blocks:  $ sudo tune2fs -m 0 /dev/sda5 tune2fs 1.41.12 (17-May-2010) Setting reserved blocks percentage to 0% (0 blocks)  $ sudo tune2fs -l /dev/sda5 tune2fs 1.41.12 (17-May-2010) Filesystem volume name:   BACKUP Filesystem UUID:          613d592e-47f5-4206-96a7-210090d340ef Filesystem features:      has_journal ext_attr resize_inode dir_index filetype extent flex_bg sparse_super large_file huge_file uninit_bg dir_nlink extra_isize Filesystem flags:         signed_directory_hash  Filesystem state:         clean Filesystem OS type:       Linux Block count:              65536000 Reserved block count:     0 Free blocks:              64459851 Block size:               4096   As expected, same block count, 0 reserved blocks, but... same free blocks? Didnt I just freed 12.5 GiB ?  $ df -h /dev/sda5 Filesystem            Size  Used Avail Use% Mounted on /dev/sda5             247G  188M  246G   1% /media/BACKUP     Looks like I did. Avaliable space went up from 233 to 245.9 GiB. gparted didnt care at all, showing exactly same info! (useless to post an identical screenshot)  What a huge mess!  I tried to document it as best as I could... So, please can someone give me any clue on what's going on here?   What are those misterious 4.11 GiB missing from NTFS -> ext4 formatting? Why there are so many discrepancies between gparted, Nautilus, tune2fs, df? What is wrong with my math? (questions in bold scattered this post)   Any help is appreciated. While I can not figure what is going on, I am serilously considering giving up on ext4 in favor of NTFS for everything but my / partition.  Thanks! Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? I am a resident of Turkey and I will go to Austria for 6 days beginning of June and then I will come back to Turkey. After staying 10 days I will go to Italy for 11 days at the end of June. Do I need two single-entry visa, or a multiple-entry visa, and which country do I apply to? (My first destination is Austria but I will stay more in Italy) How do i install drush for a drupal site which is on shared hosting?