My strategy is proof by contradiction.  So assume the opposite of the proof statement is true.  I could only think of $x^2 - y^2 + y^2 + z^2 = 2xyz$  $=(x+y)(x-y) + y^2 + z^2 $ but I don't think that is a helpful simplification. Please help!   Thanks!
Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$.   My Attempt:  We will calculate for $x,y,z&gt;0$. Then, using the AM-GM Inequality, we have  $$ \begin{cases} x^2+y^2\geq 2xy\\ y^2+z^2\geq 2yz\\ z^2+x^2\geq 2zx\\ \end{cases} $$  So $x^2+y^2+z^2\geq xy+yz+zx$. How can I solve for $(x,y,z)$ after this?
Meta duplicates don't require an answer. Don't suggest there is one when that's not true How to send text to Notepad in C#/Win32? I'm trying to use SendMessage to Notepad, so that I can insert written text without making Notepad the active window.  I have done something like this in the past using SendText, but that required giving Notepad focus.  Now, first I'm retrieving the Windows handle:  Process[] processes = Process.GetProcessesByName("notepad"); Console.WriteLine(processes[0].MainWindowHandle.ToString());   I've confirmed it's the right handle for Notepad, the same shown within Windows Task Manager.  [DllImport("User32.dll", EntryPoint = "SendMessage")] public static extern int SendMessage(int hWnd, int Msg, int wParam, int lParam);   From here, I haven't been able to get SendMessage to work in all my experimentation.  Am I going in the wrong direction? How to clear the interpreter console? Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice:      Show that, for all $n &gt; 1$ $$\frac{1}{n + 1} &lt; \log(1 + \frac1n) &lt; \frac1n.$$    I tried to follow the below steps in order to prove the RHS inequality:   Proving that $f &lt; g$ on $I$ from $a$ to $b$:  Step $1$. Prove that $f' &lt; g'$ on $\operatorname{Int}(I)$.  Step $2$. Show that $f(a) \leq g(a)$ or that $f(a^+) \leq g(a^+)$  Following the above steps, let $f(x) = \log(1 + \frac1x)$ and $g(x) = \frac1x$, for all $x &gt; 1$. One has  $$f'(x) =  -\frac{1}{x^2 + x} \quad \text{and} \quad g'(x) = -\frac{1}{x^2}.$$  We note that, for every $x &gt; 1$, $f'(x) &gt; g'(x)$. Moreover, $f(1^+) = \log(2) &lt; 1 = g(1^+).$   My problem is that I got the wrong inequality sign in Step $1$.    Looking at the solution in my textbook, the author suggests using the MVT but I don't know how to apply it in this case. How to edit Parameters for an Object after it is created? Can a ring of positive characteristic have infinite number of elements? I'm trying to install a dual boot (with Window 10) on my new XPS-15 (with a SSD 256Go).  Everything works well (following  until I go on step "Installation type".  Where I can't select "Something else", and I already have the partition table but completely empty. I can't see any existing partition, or add new one. Problem from Herstein on group theory Resources for Blender Proof of matrix norm property: submultiplicativity I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define matrix norms, and some include it as an additional property. I still haven't been able to work it out for myself or find it anywhere.  Let $A$ and $B$ be $n\times m$ and $m\times l$ matrices respectively, prove that:  $$\begin{align} \|AB\| \le \|A\|\|B\|  \end{align}$$ Is there a limit to the number of web properties one can create in Google Analytics? Is there an easy way to add a hole into an object? If I wanted to add a hole into an object e.g. Create a tube  Is there an easy way to do that?  I haven't seen anything regarding negative space so far in Blender, and it would be really helpful for my project  Thanks in advance! What damage can person/s do if they have your passport number and visa control number? I read  as well as  but it doesn't tell of what damages can be done. I have seen several sites on which people share/show the visa without the numbers and sometimes without name as well.   The only reason which many people perceive is identity theft but am unable to see how is that possible. I have to use public facilities such as printing, scanning etc. where it is easy for the operator to make copies without my consent. Also as shared, both these details are widely circulated when you stay in a hotel or anywhere else (when abroad) where identity information is needed.   Also particular to visas, as I'm given to understand, for short-stay visas (less than 3 months) which most tourist visas operate under, the window of doing anything nefarious activities (such as terrorism) would be extremely small. If I were a terrorist and I needed those bits to make work, then I would probably go for somebody who has a long-term stay as those credentials would look valid and I could jump from place to place under the radar under a fraudulent identity. Also as a terrorist, I would take quite a long view as any such attack takes years of planning, as shown in the investigation of various terrorist attacks in India as well as the U.S. Also, although I don't know (am speculating here) don't most airports might/should be checking on status quo of a passport to ensure issues such as showing 2 entries of a passport without exit (in some database where this information is collected and collated.)   Am I missing something in the logic here ? How do you become friends with someone? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any application for the integral of position with respect to time?  Maybe with respect to something else? Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem) Vertical placing of wide accents in superscript The following code  $\widetilde{aaa} \quad L^{\widetilde{aaa}}$  $\tilde{aaa} \quad L^{\tilde{aaa}}$  $\widehat{aaa} \quad L^{\widehat{aaa}}$  $\hat{aaa} \quad L^{\hat{aaa}}$  $\overline{aaa} \quad L^{\overline{aaa}}$  $\bar{aaa} \quad L^{\bar{aaa}}$   produces the following output:    What is the reason for additional vertical spacing for \widetidle and \widehat in superscript and what can one do about it? How to do a bias-variance analysis on a machine learning modelling process Mental estimate for tangent of an angle (from $0$ to $90$ degrees)