Proof that the product of 3 sequential numbers is divisible by 3
Among any three consecutive positive integers one is a multiple of 3
Number or regions formed when $n$ points on a circle are joined The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$.  This is a fact that I have read in several essays on the dangers of jumping to conclusions in mathematics.  In your opinion, what's the quickest (and/or nicest) way to prove this formula?  Professor  explains somewhere an especially convincing way to tackle this pearl following an idea which a certain high schooler in one of his talks (lectures?) came up with; yet, I don't remember  where it was that I actually read this...  Let me thank you in advance for your insightful replies. Why does ('0' ? 'a' : 'b') behave different than ('0' == true ? 'a' : 'b') How do I avoid rendering glitches from overlapping faces? For instance, here I have two cubes that are offset only on the x-axis. When I render them (using Blender's internal renderer), the overlapping faces interfere with each other and the result looks awful. A quick and easy solution would be to offset them by just a smidget, but is there another solution that allows faces to overlap and not exhibit this graphical glitch? How can I deal with "Low on Space" messages on my Android? I keep getting the "Low on Space: Application data space is low. To free up" [the rest is cut off] message.  I've moved every app I can over to the SD card, cleared the cache, compacted my K9 mail cache.  Advanced Task Killer (ouch!) says I have 98M Available Memory  File Manager says I have 223M/748M (29%) free.  That's not so bad -- but I'm stuck in the no-updates scenario. The market downloads won't really start, gmail won't update, etc. I've rebooted several times, and nothing changes. Aaaargh.  What can be done, other than uninstalling everything ? House address is not listed on Google Maps When I search for my address on Google Maps it cannot be found which means I can't order takeaway etc.  Is there a way of adding my address to Google Maps or even add a street address?  I want the address to be publicly available so others can search it. How can something finite become infinite? How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe? How to remove the "3D Objects" entry from Windows 10 Explorer? The 2017 Fall Creators Update to Windows 10 added a "3D Objects" folder to the "This PC" view of Windows Explorer:    How can this be removed? Did Sidious know the secret of cheating death? I have this value    How to read that in English? translating from my native language, I would say     Ten base of nine.   but is that really correct? Windows Update not working on Windows 2012 R2 Standard I recently inherited the management of a Windows 2012 server at a remote site.    I checked Windows Update and it has not been updating since March.  When I tell Windows to check for updates, it acts like it is checking, but it seems to say that for hours.   If I attempt to restart the windows update service, it seems to never be able to shut down.  My only remedy seems to be rebooting to get back to the point where I can tell Windows Update to check for new updates.    The last successful check for updates says March 20.  The last time updates were installed says March 17 (failed).  The update history shows that one update failed for March 17, a printer driver update, but the history shows 13 failed updates for Feb 17.  Not sure what else to try. Is there any module or some configuration that I can define the meta tags for the views page display in drupal 7 using  3.0? Excuse me for the confusing title. I was asked to find $gcd(x^2+3x+2,x^2+x)$  What i did is i factorized both polynomials $x^2+x=(x+1)x$  $x^2+3x+2=(x+1)(x+2)$  So i expected the gcd to be $x+1$   But using the euclidean algorithm i found out the gcd to be $2x+2$. Why is factorizing wrong? Is it because $K[X]$ is not factorial ? Would the euclidean algorithm also work if the polynomials are in $\Bbb Z[X]$ ??? Why was the MCU Super Soldier program considered a failure? In Marvel Cinematic Universe, military tried to create super soldiers like Hulk, but they failed. So, the program was shut down.  Where did they fail? Didn't Blonsky get Wolverine-type super healing capability? Why couldn't it make the military stronger? Other than this, wasn't Abomination a success (despite the fact that it wasn't an original work of the military)? He was defeated by Hulk, but enemy forces don't have a Hulk. When it comes to anger, who cares as long as you are focused on the target? Abomination was indeed in full control and he was always after Hulk. Silverlight - how do I get the text of the selected item in a combobox Consider this equation :   $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$  where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$.  Now how to proceed ?   This equation arises out of following problem :   A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v&gt;u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$.  See my solution below :   Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat.  Let $DA = dy, \;AD' = dx$  Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line.  Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property.  $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$  $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $  Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? Does the std::stack in the C++ STL expose any iterators of the underlying container or should I use that container directly? Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ Show that every prime $p&gt;3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$ Is there a need to set Objects to Nothing I always read that it is recommended to set objects to nothing, once I am done with them. But I normally use them only in functions inside forms.  Isn't the reference lost and memory released when the function scope is left, regardless of setting objects to Nothing?   i.e. is it really necessary to do:  Set db = Nothing Set record_set = Nothing Solving $\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$ I have the following problem that I don't know how to solve:  Let $\textbf{r}(t)$ be a curve and let $r(t):= |\textbf{r}(t)|$. Let the curve $\textbf{r}(t)$ be defined by: $$\textbf{a}(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ Where $\textbf{a}(t) = \textbf{r}''(t)$ is the acceleration of the particle and $G,M \in \mathbb{R}$. Suppose that $\textbf{r}(t)$ and $\textbf{r}'(t)$ are not parallel. The objective is to determine the trajectory $\textbf{r}(t)$ in polar coordinates.  So basically we have the following equation: $$\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ The thing is that I don't know how I should proceed from now on. If we'd have just normal scalar function this would be a differential equation and I think I would be able to solve it, but like this I have no idea how to approach the problem. How can I solve this?  Edit: If $$\textbf{r}(t):=(x(t),y(t))$$ then the equation turns into the following system of equations: $$x''=\frac{GM}{(\sqrt{x^2 + y^2})^3}x$$ $$y''=\frac{GM}{(\sqrt{x^2 + y^2})^3}y$$ I don't know if this helps but I think that this turns thing easier perhaps. Why are potential differences equal across two capacitors in parallel, but charge on each capacitor is not? Two or more capacitors are said to be connected in parallel if each one of them is connected across the same two points. In a parallel combination of capacitors potential difference across each capacitor is same but each capacitor will store different charge. Why is this true? Why is the potential difference across each capacitor in parallel the same and why will each capacitor store different charge?  Also, in series, why are Potential difference across each capacitor different, while charge is the same?