Find all functions $f:\mathbb{R}_{&gt;0}\rightarrow \mathbb{R}_{&gt;0}$ such that $$ f(xy+f(x))=f(f(x)f(y))+x, \forall x,y\in \mathbb{R}_{&gt;0} $$  If someone can give me a hint for this question I would be very grateful. edit: We can see that $f$ is injective because: $f(x)=f(y)$ then the equation above tell's that $x=y$.
Let $ \mathbb R _ { &gt; 0 } $ be the set of positive real numbers. Find all functions $ f : \mathbb R _ { &gt; 0 } \to \mathbb R _ { &gt; 0 } $ such that $$ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $$ for all positive real numbers $ x $ and $ y $.  What I thought: We could change $ x $ by $ y $, and then subtract. Source: Brazil National Olympiad 2019 #3
Same e-mail user, different address? I was wondering why using foo.bar@gmail.com works but not foo.bar@outlook.com.  In Gmail, foo.bar@gmail.com and foobar@gmail.com (and any combination of periods in the username) is equivalent.  Is this a hidden feature in Gmail? What regression model is the most appropriate to use with count data? I am trying to get a little into statistics, but I am stuck with something.  My data are as follows:  Year   Number_of_genes 1990          1 1991          1 1993          3 1995          4   I now want to build a regression model to be able to predict the number of genes for any given year based on the data. I did it with linear regression until now, but I have done some reading and it does not seem to be the best choice for this kind of data. I have read that Poisson regression might be useful, but I am unsure what to use. So my question is:  Is there a general regression model for this kind of data? If no, what do I have to do to find out which method is the most  appropriate to use ( in terms of what I have to find out about the data)? Why is $\sqrt {12} = 2 \sqrt 3$? I was editing a curve for hair, and pressed W to bring the specials menu and Subdivide it, but instead everything but the curve I had active dissapeared, and my cursor turned into a circle like the one from the circle select mode.  I'm guessing they changed the specials menu and I'm inside some unknown to me function, but I'm not sure how to turn everything back to normal. \RequirePackage{filecontents} \begin{filecontents}{\jobname.bib} @article{Arter95,     Author = {David Arter},     Date-Added = {2013-08-11 17:38:06 +0000},     Date-Modified = {2013-09-01 15:13:22 +0000},     Journal = {Journal of Common Market Studies},     Number = {3},     Pages = {361-287},     Title = {The {EU} {Referendum} in {Finland} on 16 October 1994:              {A} Vote for the {West}, not for {Maastricht}},     Volume = {33},     Year = {1995}}  @article{Detlef95,     Author = {Jahn, Detlef; Storsved,Ann-sofie},     Date-Added = {2013-08-11 17:14:34 +0000},     Date-Modified = {2013-08-11 17:16:45 +0000},     Journal = {West European Politics},     Number = {4},     Pages = {18-37},     Title = {Legitimacy through referendum? The nearly successful domino‚Äêstrategy of the Eu‚Äêreferendums in Austria, Finland, Sweden and Norway},     Volume = {18},     Year = {1995}} \end{filecontents} \documentclass{article} \usepackage[ngerman]{babel} \usepackage{natbib} \bibpunct{(}{)}{;}{a}{}{,} \makeatletter \renewcommand\@biblabel[1]{} \makeatother  \begin{document}  \cite{Arter95}  \bibliography{\jobname} \bibliographystyle{plainnat}  \end{document}   Please help. I want the author's name displayed like this "Arter, D." instead of David Arter Adding values to a C# array Probably a really simple one this - I'm starting out with C# and need to add values to an array, for example:  int[] terms;  for(int runs = 0; runs &lt; 400; runs++) {     terms[] = runs; }   For those who have used PHP, here's what I'm trying to do in C#:  $arr = array(); for ($i = 0; $i &lt; 10; $i++) {     $arr[] = $i; } Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$      Attemption $$ \begin{cases} f(-1) = -a - b - 16\\ f(2) = 8a + 2b - 28 \end{cases} $$ But then when I plug in $$b = 28 - 8a$$. Then I get $$-a -(28- 8a) - 16=0$$$$a = 34/7$$  In the book it says $a = 10$ and $b = -26$. So I am stumped. Please help Functional Analysis. Continuous extensions of linear functionals. I would like to show that a continuous functional on a subspace $W$ of a normed space $(V,\|\cdot \|)$ has a unique continuous extension to $V$ iff $W$ is dense in $V$.  I have proved $(\Leftarrow$). But the converse is currently eluding me! Help would be much appreciated! Calling Python in PHP I have a Python script I recently wrote that I call using the command line with some options. I now want a very thin web interface to call this script locally on my Mac.  I don't want to go through the minor trouble of installing mod_python or mod_wsgi on my Mac, so I was just going to do a system() or popen() from PHP to call the Python script.  Any better ideas? Thanks in advance! Interpretation of interaction term I have a model: $$ \ln({\rm earnings}) = a+b_1{\rm female}+b_2{\rm white}+b_3{\rm female}\times{\rm white} $$ ${\rm female}$ and ${\rm white}$ are dummy variables.  I have interpreted $b_1$ and $b_2$:    $b_1$ = change in female earnings comparing to male given you are non white $b_2$ = change in white earnings comparing to non white given you are male   But I am unable to interpret the coefficient of the interaction term ($b_3$). Please help me with this.  Let me make it more clear what I need out of this regression $$ \ln({\rm earnings}) = 2.618656-.0899657{\rm female}+.382019{\rm white}-.2754126 {\rm female}\times{\rm white} $$ Now i know there is gender pay difference with b1, i also know there is race pay difference with b2. Now with b3 i need to know is their a gender pay gap for whites only. How can i figure that out with regression above and without test. Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions Show that $n$ lines separate the plane into  $\frac{n^2+n+2}{2}$ regions if no two of these lines are parallel and no three pass through a common point.   I know we start with the base case, where, if we call the above equation P(n), P(0), for 0 lines would be 0.  But I really have no idea how to begin the inductive step.  How do we know what k+1 we're supposed to arrive at?  Thanks! What's the typical microclimate in the aircraft passenger cabin? For example:  "Last week, I found out that NASA stands for 'National Aeronautics and Space Administration.'"  or  "Last week, I found out that NASA stood for 'National Aeronautics and Space Administration.'" While talking about ships and countries, is it a good practice to use the feminine form?  For example:  "Her economy" - while referring to a country's economy "Her flag (or deck etc)" - while referring to a ship  Is this practice common? Is it used today? My wife wants a 4th baby, but I don't.  We have been talking about it for the last few days (talking strongly).   I have raised points such as:   We will get less time to spend with our current kids. The age gap between the first and the last will be too wide. It will cost more money. She will be out of work longer. We will need a bigger house. And we don't even have our own house yet (we are still renting). We will have to start again. Sleepless nights, nappies, feeding, new cot, car seat, etc etc.   She says:   She feels like she is missing something. She has always wanted a large family. Things will be the same regardless of having 3, or 4 kids. She agrees that money will be tighter, but wants to push through it.   Our current kids are 9 months, 2 years, and 4 years. We are young married couple (25), and I have a fairly decent job.  What would be the pros and cons of having another child?   How can we rationally talk to the other about having/or not having another child?  And what should we do if we cannot agree? Divisibility by 7 rule, and Congruence Arithmetic Laws I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = (a_ka_{k-1}\ldots a_2a_1a_0)_{10} = \displaystyle{\sum_{j=0}^{k}}a_{k-j}10^{k-j}$. The expression $$ Q_{3}^{\prime}(n) = (a_2a_1a_0)_{10} - (a_5a_4a_3)_{10} + (a_8a_7a_6)_{10} -\ldots  $$ are called alternating sum of the digits of third order of $n$. For example,  $$ Q_{3}^{\prime}(123456789) = 789-456+123=456 $$ Proposition: $7 | n \ \Leftrightarrow \ 7 | Q_{3}^{\prime}(n)$.  proof. ??  Thanks for any help. Maximum temperature laptop internals can handle? Cooling options? So I have a Compaq CQ50-215CA and my sister was using her laptop on her bed and I was just using it quickly to check my email. The surface of the laptop was really hot and I opened Speccy to see that the CPU and GPU temperature was 96 C. A couple months ago, my laptop was on the desk without a cooling pad and I was playing a game. Using GPU-Z back then, I had a reading of 104 C. Even with a cooling pad on a desk, running several processes/games keeps me around 88-90 C.  Is this normal for a laptop and what's the highest temperature the internals can handle? Besides a cooling pad, are there any other more effective ways of cooling a laptop? How to edit / how to format box overlapping with footer when scrolling down $$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$  Does it converge or diverge?   Can we have a rigorous proof that is not probabilistic?   For reference, this question is supposedly a mix of real analysis and calculus. Today i was at a arena. A player from the red team put a Grimer there, which causes my game to crash every time I select the arena (Samsung Galaxy S4 mini). All other pokemons i saw so far did not make any problems  I searched a bit, and found out that this is a known bug, e.G.:   Now my question, is there a way to battle in the arena or even select it without crashing my app? Some option like "hide model in arena",changing some smartphone-settings or another trick? Or do i have to wait till a bugifx from niantec comes or somebody else which has a handy which dont crashes with grimer clears the arena, before selecting the arena?  Because somebody could place him there i guess this is only a bug for specific smartphones, but this could be used to exclude a lot of players from arenafighting by only placing grimers in arenas to let the game crash for them, are there some official notes about this from niantec or how to avoid the crash?