This was one of the problems in a math contest in India. This is how I tried it: Subtracting the second equation from the first one gives $a^2-b^2=bc-ca$ or $(a-b)(a+b)=c(b-a)$. $a-b=-(b-a)$. Therefore a+b has to be -c. Thus all integers satisfying the condition $a+b=-c$ are such that $a^2=bc+1$ and $b^2=ca+1$. Also if $a-b=b-a,$ it means $a=b.$ Then both $b-a$ and $a-b$ are $0.$ Hence $c$ can thus be any integer. Hence, there are two groups of integers satisfying the required condition $-a=b$ with any $c$ and $a+b=-c.$ Please let me know whether my solution is correct or not.
Find all integers  $a$,$b$ and $c$ that satisfy the following equations  $a^2=bc+1\tag{i}$  $b^2=ac+1\tag{ii}$    I tried solving came out with following results:  $(i)-(ii)$ gives $a+b+c=0$ which implies, $a^3+b^3+c^3=3abc$  Multiplying equation $(i)$ with $a$ and multiplying equation $(ii)$ with $b$ and then subtracting $(i)-(ii)$ we get  $a^2+b^2+ab=1$ which can be written as  $(a+b)^2-ab=1$ , which can be further written as $c^2-ab=1$.
Dominion randomizer deck confusion MTG: Sideboard if maindeck has more than 60 cards How to find the roots of $x^4 +1$ I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ is a positive term as the equation above.  Thanks Remove subscribers from Auto Suppression List How does SO's new auto-login feature work? SO and SU just told me I've been logged in. I'm guessing because I'm already logged into my Google account... but how does SO know that? Why Does the Transaction Log Keep Growing or Run Out of Space? This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$ I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it? How can metagaming retries on poor skill checks be prevented? I'm sure every DM has had this happen: one PC is trying to find something cool on a dead monster's body. The player rolls poorly, and instead of accepting the paltry 2 copper pieces, the PC calls all the party members over to also try looting the corpse. Invariably, a different player rolls well, and everyone walks away satisfied that everything's been taken down to the poor dead monster's last holey sock.   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Does being Hermitian imply that, should there exist eigenfunctions corresponding to this operator, they will form a complete and orthonormal set?  That any wavefunction can be expanded in terms of these eigenfunction?  Just how much is hidden in the word "Hermitian"? Adding a link with a space in front causes the link to not work When you paste a link into the "Insert Hyperlink" box, it adds it automatically as a citation link:  [Example 1][1]    [1]: http://example.com   But when you paste a link starting with a space , it tries to URL encode it, to change the space to %20, and then sticks a http:// on the front to make it a valid link:  [Example 2][2]    [2]: http://%20http://example.com   This, however, means the link doesn't work.    Please could leading whitespace be stripped from the URL? This question is a sequel of sorts to my  about .  In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes.  The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction.  In my previous question, I asked about how particle number density is computed.  In this question, I would like to ask about how the electric potential is computed, given the charge density distribution.  Recall that in CGS (Gaussian) units, the Poisson equation is   $$\nabla^2 \Phi = -4\pi \rho$$  where $\Phi$ is the electric potential and $\rho$ is the charge density.  So the charge density $\rho$ is proportional to the Laplacian of the potential.  Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation.  How can I do this?  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Thanks for your time. I have custom fields in contacts form named checkbox1__c, checkbox11__c, checkbox12__c, checkbox13__c.  When I check checkbox1__c then I want checkbox11__c,  checkbox12__c,  checkbox13__c to be unchecked.  When I unchecked checkbox1__c then I want checkbox11__c, checkbox12__c, checkbox13__c to be checked.  Is there any way to do this by using validation-rules? If not no way to this by validation-rules, then what is the best way to do this? I try to press the F6 key and in the YouTube tutorials those who make the video pressing them all go well, but not me :( The problem   I have tens of thousands of small polygons for which I want to create buffer/doughnuts for, but I don’t get the units right. I want meters and not degrees, and changing to (what I think is) the right CRS my layer doesn’t longer show. The polygons are small farming plots. I work in QGIS 3.4.15.   Let me take you through it  I start by adding a vector layer. 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