Let $a$, $b$ and $c$ be real numbers such that $a + b + c = 0$ and define:   $$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}.$$   What is the value of $P$?   This question came in the regional maths olympiad. I tried AM-GM and CS inequality but failed to get a result. Please give me some hint in how to solve this question.
I'm stuck at this algebra problem, it seems to me that's what's provided doesn't even at all.  Provided:  $$a+b+c=0$$  Find the value of:  $$\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$$  Like I'm not sure where to start, and the provided clue doesn't even make sense.  There's no way I can think of to factor this big polynomial into a form like $a\times p+b\times q+c\times r=s$ where $p,q,r,s\in\mathbb{Z}$.  Thanks!
A better serial voting trap %AX = (%AH + %AL)  So why not %EAX = (%SOME_REGISTER + %AX) for some register %SOME_REGISTER? If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any mistakes or places where I may improve. Thanks!  By the definition of continuity at $a$ of $f,g$ we have that $\lim\limits_{x\to a}f(x)=f(a)$ and $\lim\limits_{x\to a}g(x)=g(a)$. Suppose that $f(a)&gt;g(a)$. Then there exists a $\delta&gt;0$ such that $f&gt;g$ for all $x$ satisfying $|x-a|&lt;\delta$. Then for $x$ satisfying $|x-a|&lt;\delta$ we have $h(x)=f(x)$ and $k(x)=g(x)$ and thus $h$ and $k$ are continuous at $a$ because $f$ and $g$ are continuous at $a$. Now if $f(a)&lt;g(a)$ we simply relabel $f=\tilde{g}$ and $g=\tilde{f}$, so that $\tilde{f}(a)&gt;\tilde{g}(a)$ and we apply the previous result to show that again $h$ and $k$ are continuous at $a$.\  Now if $f(a)=g(a)$ then we can distinguish three cases. $f\geq g$ in a small neighborhood about $a$, $f\leq g$  in a small neighborhood about $a$ or $f\geq g$ on one side of $a$ and $f\leq g$ on the other side. In the first case we assume that $f\geq g$ in some small neighborhood about $a$. Then, since $f(a)=g(a)$ we have $h(x) = f(x)$ in this neighborhood and $k(x) = g(x)$ and again we see that $h$ and $k$ are continuous at $a$ due to the continuity of $f$ and $g$ at $a$. Similarly if $f\leq g$ in a neighborhood around $a$ then $h(x)=g(x)$ and $k(x)=f(x)$ in this neighborhood and $h$ and $k$ are continuous at $a$.   Suppose that just to the left of $a$ we have $f\geq g$ and just to the right of $a$ we have $f\leq g$. Then $\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$. Similarly $\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$. So $\lim\limits_{x\to a}h(x)=h(a)$ and $\lim\limits_{x\to a}k(x)=k(a)$ which means that $h$ and $k$ are continuous at $a$. Lastly if $f\leq g$ just to the left of $a$ and $f\geq g$ just to the right of $a$ we can relabel $f=\tilde{g}$ and $g=\tilde{f}$ and apply the last result to show $h$ and $k$ are continuous at $a$.   Again, please point out any mistakes I may have made. Thanks!! Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle What I tried:  Let $A(2,0,4)$, $B(4,1,-1)$, $C(6,7,7)$ then  $$\vec{AB}=(2,1,-5), \vec{AC}=(4,7,3), \vec{BC}=(2,6,8)$$  Then I calculated the angle between vectors:  $$\begin{aligned} \alpha_1 &amp;= \cos^{-1}\left(\frac{(2,1,-5)(4,7,3)}{\sqrt{2^2+1^2+(-5)^2}\sqrt{4^2+7^2+3^2}}\right) \\ &amp;= \cos^{-1}(0)=90° \\ \alpha_2 &amp;= \cos^{-1}\left(\frac{(4,7,3)(2,6,8)}{\sqrt{4^2+7^2+3^2}\sqrt{2^2+6^2+8^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{74}{\sqrt{74}\sqrt{104}}\right)=32.49\\ \alpha_3 &amp;= \cos^{-1}\left(\frac{(2,6,8)(2,1,-5)}{\sqrt{2^2+6^2+8^2}\sqrt{2^2+1^2+(-5)^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{-30}{\sqrt{104}\sqrt{30}}\right)=122.5° \end {aligned}$$  As you can see, these angles don't even form a triangle, what am I doing wrong, any thoughts? Is colour, as represented using primary colours, accurate only to humans? Slightly biological, hopefully physical enough to be answered.  Suppose a magenta hue is represented by a mix of red and blue pigment. This is all very well for a creature with red and blue photoreceptors, but suppose it was seen by a creature which had a magenta-sensitive receptor, but no red or blue. Would the colour appear the same (insomuch as a qualititave concept can appear the same, I suppose I mean 'Would they interpret it as colour of the same wavelength?')?  The crux of my question is, do the particular bands in which photoreceptors are activated affect vision of additive, as opposed to pure, hues?  Finally, completely off-topic, but if there happens to be a biologist around, do animals on the whole have similar photoreceptors, or are they placed largely randomly?  Thanks, Wyatt The client can do several actions per frame, for instance, requesting a movement, shooting etc. Should i send a packet to the server for each action the client performed during the frame, or should i stack them in an unique packet and send it at the end of the frame?  EDIT: i'm using Jmonkeyengine and the provided network API SpiderMonkey. How to turn off magic quotes on shared hosting? I want to turn off PHP's magic quotes.  I don't have access to php.ini.  When I tried to add php_flag magic_quotes_gpc off to my .htaccess file, I get a 500 internal server error. This is what my .htaccess file looks like:  AddType x-mapp-php5 .php php_flag magic_quotes_gpc off   Then I tried to use ini_set('magic_quotes_gpc', 'O'), but that had no effect.  How do I turn magic quotes off? Get total duration of video files in a directory I have a list of .ts files:   out1.ts ... out749.ts   out8159.ts  out8818.ts   How can I get the total duration (running time) of all these files? Derivative of function solved from NDSolve can't be plotted How do I make Units efficiently? I just blew almost all my money on a really cool lookin' ship. How do I regain solvency without spending a lot of time fiddling around in my inventory?  At the moment I've found a sequence of steps that'll turn 50 carbon and 100 Plutonium into ~5000 Units (by creating Suspension Fluid and then turning it into Electron Vapor). Are there better recipes or sequences of recipes I should look for? Or should I just drop the whole thing and go mine a particular mineral? Is any particular punctuation necessary when a quotation is the subject of a sentence? For example, in the sentence below, should there be a comma after export? Or anything else?     "Laughter is America's most important export" is a well-known quote of Walt Disney's. An example of a group such that $G \cong G \times G$ I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group? Where are the default settings saved for the Command prompt defaults? I am trying to perform a multiple regression in R. However, my dependent variable has the following plot:    Here is a scatterplot matrix with all my variables (WAR is the dependent variable):    I know that I need to perform a transformation on this variable (and possibly the independent variables?) but I am not sure of the exact transformation required. Can someone point me in the right direction? I am happy to provide any additional information about the relationship between the independent and dependent variables.  The diagnostic graphics from my regression look as follows:    EDIT  After transforming the dependent and independent variables using Yeo-Johnson transformations, the diagnostic plots look like this:    If I use a GLM with a log-link, the diagnostic graphics are: Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. For any natural number $n\ge 1$, given pairs $(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$ of integer numbers, there exist integer number $c$ and $d$ such that  $$\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$$    My initial approach is  Base Case: $(a_1^2+b_1^2) = a_1^2+b_1^2$ which is true. (Although it is trivial)  Prove the statement is true when $n=2$: We have $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2+(ad+bc)^2$$ (Thanks André Nicolas for pointing it out)   So if $a,b,c,d$ are integers, $ac,bd,ad,bc$ are all integers and integers are closed under addition and subtraction. Hence $(ac-bd),(ad+bc)$ are integers.   Inductive Hypothesis: $\prod_{i=1}^{n}(a_i^2+b_i^2) = c^2+d^2$ is true  Inductive Step: $$\prod_{i=1}^{n+1}(a_i^2+b_i^2) = \prod_{i=1}^{n}(a_i^2+b_i^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2)$$  Where $c$ and $d$ are integers.   But when we apply $n=2$, we have  $(c^2+d^2)\cdot (a_{n+1}^2+b_{n+1}^2) = (e^2 + f^2)$ where $e$ and $f$ are integers.  Hence, by the principle of induction, the statement we needed to prove is true. How to install Python 3.6? Longest line using awk Whenever I try to do a fluid simulation, the fluid domain remains visible.  I have tried:ctrl+a-scale, turn on/off visibility for render/viewport  Here's a video demonstrating my problem:   Here's the blend file in case I have messed up some settings:   Thank you for any help! A classic exponential inequality: $x^y+y^x>1$