This question has been tantalizing me for a day now:     Given integers $a,b,c,d$ such that $d \neq 0$ (mod $5$) and $m$ an integer for which $am^3 + bm^2 +cm+d \equiv 0$ (mod $5$), prove there exists an integer $n$ for which $dn^3+cn^2+bn+a \equiv 0 $ (mod $5$).   My attempt:    It is easy to see that $\gcd(5,m)=1$. We know from Fermat's little theorem that $5 \mid m^4 -1 \Rightarrow 5 \mid (m+1)(m-1)(m^2+1)$. The proof is trivial that $5$ only divides one of these factors. Hence we have $3$ cases:  Case $1 \rightarrow m \equiv 1$ (mod $5$). If this is true then  it follows that $m^2 \equiv 1$ (mod $5$) and $m^3 \equiv 1$ (mod $5$). We can substitute $m$ into the equation and achieve:  $$a+b+c+d \equiv 0 \pmod5$$ $$\Rightarrow d(1)^3+c(1)^2+b(1)+a \equiv 0 \pmod5$$  Where $1$ satisfies the condition.  Case $2 \rightarrow m \equiv -1$ (mod $5$). If this is true then  it follows that $m^2 \equiv 1$ (mod $5$) and $m^3 \equiv -1$ (mod $5$). We can substitute $m$ into the equation and achieve:  $$a-b+c-d \equiv 0 \pmod5$$ $$\Rightarrow d(-1)^3+c(-1)^2+b(-1)+a \equiv 0 \pmod5$$  Where $-1$ satisfies the condition.  Case $3 \rightarrow m^2 \equiv -1$ (mod $5$). If this is true then  it follows that $m \equiv 2$ (mod $5$) and $m^3 \equiv -2$ (mod $5$). We can substitute $m$ into the equation and achieve:  $$2a+b-2c-d \equiv 0 \pmod5$$  From where I am unable to proceed. Can I get a hint?
Let $F$ be a field and let $a \neq 0$ be a zero of the polynomial $a_0 + a_1x + . . . +a_nx^n$ in $F[x]$.  I want to show that $\frac{1}{a}$ is a zero of the polynomial $a_n + a_{n-1}x + . . . + a_0x^n$  How can I do this?
Reset post score to 0 on migration To be honest, I have no idea how to even start this problem. I'm sorry I don't have any work to show, but I'm just at a blank. Help?  Chapter 2: Problem 27:  University B, once boasted $17$ tenured professors of mathematics. Tradition prescribed that at their weekly luncheon meeting, faithfully attended by all $17$, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forestall any resignations. One fateful year, the department was augmented by a visitor from another university, one Prof. X, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was apprised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. X obtained his revenge at the last luncheon of the year. &quot;I have enjoyed my visit here very much&quot;, he said, &quot;but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department.&quot; What happened the next year?&quot;   Chapter 2: Problem 28:  After figuring out, or looking up, the answer to Problem 27, consider the following: Each member of the department already knew that Prof.X asserted, so how could his saying it change anything? What "exactly" happens when someone turns to the Dark Side? I wonder a generalization of Jensen's inequality: let $\mathbf{X,Y}$ be two positive definite matrices, can we obtain the following Jensen like inequality $$(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq((1-\lambda)\mathbf{X}+\lambda\mathbf{Y})^{-1}$$ for any $\lambda \in (0,1)$, where the notation $\mathbf{A}\succeq \mathbf{B}$ represents matrix $(\mathbf{A-B})$ positive semi-definite. I want to boot Ubuntu into recovery mode but there is no grub menu entry allowing me to do so. I pressed e and now I am able to edit boot options. It looks like that:  setparams 'Ubuntu , with Linux 3.6.0-26-generic'  recordfail gfxmode $linux_gfx_mode insmod gzio insmod part_msdos insmod ext2 set root='(hd0,msdos5)' search --no-floppy --fs-uuid --set=root &lt;uuid&gt; linux /boot/vmlinuz-3.5.0-26-generic root=&lt;uuid&gt; ro ipv6.disable=1 quirt splash $vt_handoff initrd /boot/initrd.img-3.5.0-26-generic   The  documentation says:     Boot to the recovery mode even if it is not listed on the menu:         Remove existing kernel options from the linux line and add single      But this sentence doesn't make any sense, maybe the wiki page is broken somehow. How to modify the boot menu to boot directly into recovery mode?  Note: I don't have access to grub config or anything else, only the boot menu edit page is available. Equivalence Relation problem Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$.  I think I've found this to be reflexive and symmetric, but I'm stuck on transitivity. Can someone check my work so far and assist with testing transitivity?  Reflexive: Let $(x,y)S(x,y)$. Then $xy = yx$. So $S$ is reflexive. Symmetric: Suppose $(a,b)S(c,d)$. Then, $ad = bc$. Therefore, $da = cb$ and $cb = da$. Therefore, $(c,d)S(a,b)$. Thus, $S$ is symmetric.  Transitive: Suppose $(a,b)S(b,c)$. Then $ac = bb$.   This is where I'm stuck. Any ideas? Installing grub2 in efi-mode in usb flash drive I've two different OS, which one is ubuntu. I'm using a PC, NOT a Mac. I use Chimera bootloader that don't find Ubuntu. So, i'd like to install grub 2 in EFI-mode on flash-drive USB. In this way, i can load Ubuntu only if the usb flash-drive is connected.  I followed this guide  replacing "/dev/sda" in  sudo efibootmgr --create --gpt --disk /dev/sda --part 1 --write-signature --label "GRUB2" --loader "\\EFI\\grub\\grub.efi"   with the usb-pen directory. Then, i've copied the grub.cfg file that exists in Ubuntu partition.Result? Grub2 loads, but when i select the menu entries, grub2 doesn't find any command, or disks. How can i fix this problem?  P.S. Here, there are the photo of the error(when i select ubuntu entry in grub2), the usb-drive's contents and the grub.cfg file Problems with pie charts There seems to be in increasing discussion about pie charts.   The main arguments against it seem to be:   Area is perceived with less power than length. Pie charts have very low data-point-to-pixel ratio   However, I think they can be somehow useful when portraying proportions. I agree to use a table in most cases but when you are writing a business report and you've just included hundreds of tables why not having a pie chart?  I'm curious about what the community thinks about this topic. Further references are welcome.  I include a couple of links:         In order to conclude this question I decided to build an example of pie-chart vs waffle-chart. Django: manage.py does not print stack trace for errors Option to be notified when a post I downvoted is edited Prologue:  This is an effort to follow up on  which is marked as a duplicate of  highly voted and status-declined feature request (although they seem to be different, one focusing on the OP and the other on the downvoter).   My New Year's resolution was to downvote more.  So on January 3rd or so, I saw a post that was low quality, and downvoted.  Two days later, I came back to see if it was ready to be deleted or had gotten piled with downvotes, to find it with a significant positive score!  At which point, I noticed that the answer had been significantly improved.  So I retracted my downvote, changed it to an upvote, and immediately resumed my dearth of downvoting again.  I think I have a responsibility to downvote bad posts, despite that oh-so-painful -1 rep penalty.  But if I am also going to have the responsibility to follow up on every downvote I cast, I'm not going to bother. I don't want my downvote to leave a residual penalty on posts that were once low quality, and I am not going to trouble myself to keep checking up on every post I downvoted.  Can a downvoter have an option to be notified when a post they voted down was edited?  I wouldn't make this an automatic thing, because not all downvoters want it, and it isn't applicable to all cases of downvoting (although  that this should be part of the responsibility of a downvoter).  Personally, I would make this a universal toggle on-off option on the user profile page, not a question-by-question option, but I don't have strong feelings about it. Adding multiple CKEditor instances in jquery Custom date format with jQuery validation plugin How can I specify a custom date formate to be validated with the  for jQuery? Solving $a \sin\theta + b \cos\theta = c$ Could someone help me with the steps for solving the below equation $$a \sin\theta + b \cos\theta = c$$ I know that the solution is $$\theta = \tan^{-1} \frac{c}{^+_-\sqrt{a^2 + b^2 - c^2}} - \tan^{-1} \frac{a}{b} $$ I just can't figure out the right steps to arrive at this solution. Does light reflect if incident at exactly the critical angle? A lot of textbooks and exam boards claim that light incident at exactly the critical angle is transmitted along the media boundary (i.e. at right-angles to the normal), but this seems to violate the principle of reversibility in classical physics.  How would a photon or ray travelling in the reverse direction "know" when to enter the higher refracting medium?  It can't know, so I conclude that such light is simply reflected?  Is this correct? How can gravity affect light? How to initialize HashSet values by construction? Blackboard bold variants for Greek letters It is common (or it used to be) to use a blackboard-bold variant of \mu to denote roots of unity in some mathematical papers.  However, despite searching (several times, periodically over several years) I have never been able to locate any package or technique that makes this possible, other than using Metafont or possibly hunting through the font definitions to see if it is something I can create a variant of (along the lines of the suggestions in Section 8 of the Comprehensive LaTeX symbol list).  Thus far, I haven't been sufficiently motivated to learn how to do any of this (although it probably would have resulted in a net time savings), and I feel like I cannot be the only person who would have tried.  Is anyone aware of an existing way to produce this character? How can I check if a program exists from a Bash script? is there a quick way to sort the items of a select element? Or I have to resort to writing javascript?  Please any ideas.  &lt;select size="4" name="lstALL" multiple="multiple" id="lstALL" tabindex="12" style="font-size:XX-Small;height:95%;width:100%;"&gt; &lt;option value="0"&gt; XXX&lt;/option&gt; &lt;option value="1203"&gt;ABC&lt;/option&gt; &lt;option value="1013"&gt;MMM&lt;/option&gt; &lt;/select&gt; Solve Euler Project #9 only mathematically - Pythagorean triplet The "Euler Project" problem 9 () asks to solve:  $a^2$ + $b^2$ = $c^2$ a + b + c = 1000  I find answers solving it with brute-force and programmatically, but is there a way to solve the problem ONLY mathematically? Can someone help, please?  Problem as explained in Project Euler website:  A Pythagorean triplet is a set of three natural numbers, a &lt; b &lt; c, for which,  a^2 + b^2 = c^2 For example, 3^2 + 4^2 = 9 + 16 = 25 = 52.  There exists exactly one Pythagorean triplet for which a + b + c = 1000.  Find the product abc.