a, b, c are distinct reals such that $$a + 1/b = b + 1/c = c + 1/a = t $$for some real t. Show that t  = -abc I tried using continued fractions to isolate a,b and c but equations of degree more than 2 are formed.
If $a, b, c$ be distinct reals such that $$a + \frac1b = b + \frac1c = c + \frac1a$$ how do I find the value of $abc$?  The answer says $1$, but I am not sure how to derive it.
In LWC am trying to iterate over a list and show the details from the list, this is working correctly , but as a header want to show the record number in a displayed order, below is desired outcome and simplified sample code.  Desired OutCome:   Employee Record 1: A  Employee Record 2: B  Employee Record 3: C   JavaScript:  Var empNames = ["A" , "B" , "C"];   HTML:  &lt;template for:each={empNames} for:item="employee" for:index="index"&gt; Employee Record{index} {employee} &lt;/template&gt;    Actual outcome  Employee Record 0: A  Employee Record 1: B  Employee Record 2: C   I understand as i am using {index} which indicated the position of element in an array which starts from 0 my actual outcome also starts like Employee Record 0 instead of Employee Record 1, but is there a simple way without writing much logic on JS to handle this in html and to make my headers starts from 1 instead of 0? Can I install a ROM made for a different device? I own a MicroMax A100. I want to install the latest version of a different ROM (Samsung or Nexus).  Is it possible? If so, can you tell me how?  If not, can I install a ROM made for a different device, but from the same manufacturer? For example, could I install a ROM for the MicroMax A110, which is the successor of A100? Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. How to shift EXIF date/time created by time in days, hours, minutes? Recently I asked about the  pertaining to Voldemorts new face. However I was corrected by ...     Well, I think the confusion comes with mixing movie canon with book canon. In Philosopher's Stone, here's what Harry sees when Quirrell unwraps his turban:           Harry would have screamed, but he couldn’t make a sound. Where there should have been a back to Quirrell’s head, there was a face, the most terrible face Harry had ever seen. It was chalk white with glaring red eyes and slits for nostrils, like a snake.          Philosopher's Stone - page 212 - UK - chapter 17, The Man With Two Faces      So it seems at some point between leaving school, and sticking his face to the back of Quirrel's head possessing Quirrel he got a new face some how.   How did he manage that? Is it some dark magic he applies? Is it from possessing snakes in Albania? Prove that $||x|-|y||\le |x-y|$ I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much. Update Garmin Nüvi 255 maps How to Sort Multi-dimensional Array by Value? How do I import a namespace in Razor View Page? Protecting game assets through archiving I'm writing a game in C++ (with OpenGL) and am getting quite far into development. Currently I'm loading the data directly from different directories.  (E.G. I load textures from a /Data/Textures/...png)  This is fine, but not preferable. Ideally I would like to protect these assets without too much of a performance hit. I looked at other games that I own, both Indie and AAA and found that the majority of them archive the data. However, I am unable to access these archives with programs like Winrar Archiver which implies that they have some level of protection.  I'm not particularly knowledgeable in the field of encryption and data storage, so I may be overlooking something obvious but I would like to know how exactly I could achieve something similar.  (I am aware that competent individuals are able to use the GPU to gain access to the data but that's not preventable) What's the difference between adjusting highlights and shadows in the basic panel and in the tone curve in the Develop Module of Lightroom? I wanted to reduce some highlights in a photo and found that there were two possible ways of doing so: one from the basic panel and another from the tone curve. when I reduce the highlights in one pane the value on the other is unchanged. Are both of them the same thing? Which would be better? Abelian groups of order n. Is there a number $n$ such that there are exactly 1 million abelian groups of order $n$?   Can anyone please explain. I would yes because numbers are infinitive, and so any number n can be expressed as a direct product of cyclic groups of order n.  Can anyone please help  me understand.  Thank you. "Stack Overflow" showing up on hamburger menu for all sites to link to /questions Why does it say "Stack Overflow" instead of the site name? Solve the integeral equation (C.S.I.R) Let $\lambda_1, \lambda_2$ be the eigen value and $f_1 , f_2$ be the coressponding eigen  functions for the homogeneous integeral equation  $$ \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) dt = 0 $$  Then   $\lambda_1 = -18 - 6 \sqrt{10} , \lambda_2 = -18 + 6 \sqrt{10}$ $\lambda_1 = -36 - 12 \sqrt{10} , \lambda_2 = -36 + 12 \sqrt{10}$ $\int_0^1f_1(x)f_2(x) dx = 1$ $\int_0^1f_1(x)f_2(x) dx = 0$   I have solved simply and get the values of $\lambda_1 \ \ and \ \ \lambda_2$ belong to (1).  Please tell me about (3) and (4) option.  Thank you. How do I find slimes in Minecraft? In early alpha I remember Minecraft videos showing slimes deep underground. I think you would attack them then they split into several smaller slimes. But I've never found a slime for myself. In what conditions can you find slimes? Are they definitely still in Minecraft? How does the propagation of gravity work for photons? Study path to Bayesian thinking? I am six years into a business role and have a bachelor's in physics and applied math/stats. Sean Carroll's (Caltech physicist)  opened me to the idea that Bayesian statistics is one useful way of thinking about anything - inevitably you hold a prior and you should update your credence as additional information becomes available.   Is there a path to training your intuition to thinking this way? Critically, it would require repeated practice with verifiable answers through either a course, or self study that includes many problems and solutions. I do not believe simply reading will do.  Possible resources, having read every related question on this site I could find:   "Probability Theory" by Jaynes. Pro: analytic; intuitive explanation of bayesian statistics. Con: prerequisites; missing problems/solutions. "Doing Bayesian Data Analysis" by Kruschke. Pro: includes problems &amp; solutions; requires only "algebra and rusty calculus". Con: works in R, which I think provides for less intuitive learning than the analytical (I may be wrong).   If it is a multi-year path I need to take, starting elsewhere, I am happy to do so! Ideally, I would avoid the frequentist methods, as I have no use for them.  My goal is not to be a scientist, but to leverage insight into how reality works to go above and beyond the established thinking in business.  Many thanks for any suggestions! What Hogwarts classes are taught to two houses at once? If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN) Let $(\Omega,\mathcal F,P)$ be a finite measure space.  Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v's  I need to prove that if:  $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation.  If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$.  And I know that on finite measure space Converging in expectations is converging in $\mathcal L^1(\Omega)$ and it's stronger than almost sure convergence.  And I know that from linearity of expectation even if one of the sequence is not in $\mathcal L^1(\Omega)$ then $Y$ is not in $\mathcal L^1(\Omega)$.  How do I continue? I want to quantify the relationship between two variables, A and B, using mutual information. The way to compute it is by binning the observations (see example Python code below). However, what factors determines what number of bins is reasonable? I need the computation to be fast so I cannot simply use a lot of bins to be on the safe side.  from sklearn.metrics import mutual_info_score  def calc_MI(x, y, bins):     c_xy = np.histogram2d(x, y, bins)[0]     mi = mutual_info_score(None, None, contingency=c_xy)     return mi