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Ah, geometry, my favorite :> The answer seems to be correct, although using trigonometry for such a problem is kinda like picking a sledgehammer to crack a nut Notice that the line connecting the centres of the circles (the double radius) is parallel to the hypotenuse, so if you project these points on the other two legs, you'll get a smaller triangle which has the same shape as the original one (i.e. they're similar, and hence their sides are in proportion). From that you should be able to construct an equation for `r` in terms of the sides.

I'd rather say that the error is in the silly notation that confuses the operation with its operands. But even the great Leonhard Euler fell into this trap, which shows that even the greatest minds can err if the notation they use is deceiving. The actual error is right there in the middle, because this is an identity only for real numbers.

Reminds me when my father asked me once to calculate a compound interest on his investment, and when I said "Hmm… the 12th power of 21/20 is…", he yelled: "12th POWER?! ARE YOU MAD?! THIS WOULD BE AN ASTRONOMICALLY LARGE NUMBER!" "Yeah, but it's a 12th power of number just a wee bit larger than 1" – I replied He still didn't get it, so I left him in the hands of other "experts" :J

I'll try to answer in such a way that you know that I know, but without spoiling the fun to others 1. That's a rather trivial one 2. That's a prime example of what a certain Eratosthenes would ponder upon while carrying a sieve 3. I think that it's perfectly possible to square that circle 4. This question might be quite divisive, depending on whether one likes a proper answer or improper I'd go with improper though. OK, but to add some more fun, let me add another question: 5. Which box will contain the biggest number of marbles and why?

More like "Die Hard 3" challenge

Well then I must be a cyborg, because the numbers often reach 500 My absolute record so far is 575 tabs from last year until I started a new session But I use an extension for tab management, so I can flush them into a file as a list of URLs and reload the entire session later. Why that many? Because I often do research, and I open every single page that I can find on a particular subject, and then I go through all of them, closing the chaff and leaving only the good ones. Then I start reading them. And I often have to jump between them to connect all the dots that are scattered here and there. I also do programming, so I often need to have a lot of documentation pages opened and navigate through them while coding. Organization is the key I often reorder the tabs, group them together by subjects so that related stuff is next to each other, etc. And after ordering them, I save the session and dump all the URLs to a file, so that I could organize them further there, with headings, tags & stuff. Although I must admit that, even with browser extensions and my own organization skills, web browsers these days aren't very well suited for such workflow as mine, and there's still a lot of room for improvement One of them being memory usage, obviously. (Even though I compress memory and call the garbage collector pretty often, my 12 GB of RAM is still filled to the brim most of the time Welp, at least I make a good use of it, i guess… )

Anyone who thinks Chrome is so cool, think again. Wanna hear about the day I uninstalled Chrome forever? :q Here it goes… My parents called me one day, because their laptop with Windows 7 was constantly grinding the hard disk and everything was working horribly slow. They thought that it might be some sort of a virus. But they had an antivirus installed and it didn't detect anything. (I actually suspected that it might be the antivirus slowing everything down, so I uninstalled it to check that, but it didn't solve the problem the slightest.) So I launched Filemon (a File Monitor, a program that monitors every access to the hard drive and logs it to a file) and, to my surprise, the log window has been immediately flooded with thousands of disk accesses, all from one single executable file: software_reporter_tool.exe :q It seemed to be greedily going through every directory and every file on the disk, hence the grinding. Delighted that I found the "virus", I did a web search to see what they write about this executable, what it is and where it comes from, an… that was even a bigger surprise, because it turned out to be an integral part of Google Chrome! And it's not a bug, it's a feature! According to them, this program is supposed to go through your every directory, scan all your files, and report them to Google for whatever reason (sic!) Yeeeeeahhhh… isn't this what every malware does, by definition? It was impossible to delete this file, because it was protected by the system. Even if I deleted it "the hacker way" (with some low-level system tools), it soon reappeared again with the next regular update of Google Chrome! They detect that this file is missing, and forcefully reinstall it! If you do a Google search, you will find threads about this malware on Google tech support forums that are being immediately locked by Google, or even removed (those ones you'll rather be unable to find, obviously :q you can only know that they're deleting such threads by starting such a thread yourself and seeing it disappear, or asking other people who tried the same). Here's one example I could still find: https://support.google.com/chrome/a/thread/99323901/the-software-reporter-tool-exe-is-malware-admins-need-control-back-over-this-unwanted-software or another thread from Super User (which is not under Google's control, so they can't remove it there, but apparently there are still people there who claim that this piece of malware is "normal" and "innocuous" and you shouldn't bother your little mind with it ): https://superuser.com/questions/957332/what-is-the-software-reporter-tool-downloaded-by-google-chrome Anyway, once I deleted this file, all the disk grinding stopped and the machine became perfectly usable again. So this was indeed the cause. And knowing that they will reinstall this file back with the next update and the problem will return, I decided to uninstall Google Chrome completely and replace it with Firefox. My parents don't have any problems with this machine anymore since then. That was the day I stopped using Google Chrome altogether, forever, and I advise you to do the same and replace Google Chrome with another browser of your choice (Firefox, Brave, LibreWolf etc.) that doesn't grind your hard disk to go through your every file and rat to Google about it

Forget YouTube and its annoying ads, use it through Invidious instead. It is a YouTube proxy that allows you to watch videos from YouTube and read YouTube comments (as well as comments from other platforms for those videos) without any ads or other annoying distractions through one of multiple independent Invidious instances that you can choose from. And you can even run your own instance locally, since they're all free and open-source. It's like YouTube, only suckless

If you're wondering whether she might do it, then she probably wouldn't. And if you have some doubts about it, then your reader will most likely have similar doubts when reading it. This doesn't necessarily mean that Rainbow Dash couldn't possibly do that – everyone is capable of doing all sorts of things in certain circumstances. So the real question here is why would Rainbow Dash do it, and what circumstances could make her do it. It is your job as a writer to discover her motivation (if you really want to write a story in which she does that, that is). The reader must understand her position well enough in order to still be on her side when she does it, or at least understand what motivated her to that, even if the reader is not very fond of her actions. Motivation of the character is everything in such cases.

Sure thing, but you see, this argument can be twisted the other way around as well (and I've already seen it twisted around that way a couple of times before): One might argue that saying that most MLP fans are nice and friendly people is a generalization as well, and that it "misrepresents" the fandom in some way, because it paints it with more colour than there really is; that saying that, for example, cloppers, or people who enjoy pony gore, or people "from the spectrum" (choose your favourite spectrum here), are "just the minority", is a "misrepresentation of the fandom", or that it "offends" those people by denying their existence or their tastes, because apparently being a fan is just a matter of self-proclaiming oneself to be one, and it doesn't matter what the rest of the fandom thinks about it. Imagine that some really bad person publicly announced that he's a MLP fan. Can a pedo be a fan? Can a serial killer be a fan? Where's the boundary? If there's none, then we have a problem, because this means that nasty people will always be found in the fandom, and their behaviour will be generalized on the entire fandom. If we wanted to stop it from happening, we would have to put boundaries somewhere, and then we're back to square one with that "tolerance" and "inclusiveness" thing :q That's the gist of the paradox right here.

Well, although the whole thing sure is on the border of funny and cringe, the person who made that petition does have a point: there are pretty messed up people in the fandom (as in pretty much any fandom). And the real question here should rather be: what should we do about it? Or can we do anything about it, for that matter? :q One possible solution might be to disclaim such people from the fandom, by setting some conditions one has to meet in order to call oneself a fan. I mean, I'm not saying that we should put the bar very high, it doesn't necessarily have to be something strict, but… oh, I don't know… at least having watched the actual show wouldn't hurt :q I find it rather weird when people call themselves a fan of something and yet they never watched the source material, and all they want is to belong somewhere because they think it's cool. But if we wanted to really clean the air, we would have to require people to follow the moral standards that can be found in the show if they want to call themselves true fans. (Something that I personally call a "Harmony Brony" – someone who lives up to those moral standards and life lessons from the show, and does his/her best to implement them in his/her life.) Then, if someone accused us of the fandom being nasty, we could always say: "But look, we have this here moral code, and those people are not following it, which means that they are not real fans, they're impostors." Unfortunately, there are some problems with this approach, and I'm pretty sure you already thought of them: "Love and tolerate" – anyone remembers that? If you're new in the fandom, I'd like to remind you that this was once our slogan. And that's precisely the thing that would be contradicted when we assumed the approach from the previous paragraph: One could argue that we're not as tolerant and inclusive as we claim, if we don't allow certain people to call themselves fans or refuse them right to do so :q I've already heard this argument multiple times before, every time something nasty happened in the fandom and some people were condemning it – there always was another group who claimed that we have to allow such nasty behaviours in the fandom, because we don't have the right to refuse anyone the right of calling themselves fans; that such people are fans too, and we have to tolerate it, because we're tolerant, aren't we? It's in our motto after all :q The problem is much broader, though, and I even think that it's the predominant problem in our modern society these days. It is a so-called "Tolerance Paradox", which says, that if you want to be really tolerant, you have to also tolerate intolerant people :q Or people who misbehave, or misrepresent your community. The notion of tolerance has been blown up so much these days, and became so much political, that it lose any meaning whatsoever. If you disagree with something, or have a different opinion that a certain group of people, they will call you intolerant. And often it doesn't stop at that – they will also do what they can to stop you from expressing this world view, which… paradoxically, contradicts the very definition of tolerance :J They want you to be tolerant towards them, but they're unwilling to tolerate you if you're not. Sounds familiar? This also reminds me when a couple of years ago, a couple of my friends had enough of censorship in the media, so they started their own medium, an internet radio, whose sole motto was "no censorship whatsoever". They wanted it to be a platform in which everyone could come and say whatever they have to say, and no world views would be excluded. At first, it worked pretty well, and there was a lot of interesting podcasts, interviews, talks etc. But soon after it started to attract all sorts of messed up people with crazy ideas. UFO-catchers, Flat Earthers, lunatics, cult leaders, or people propagating extremism or totalitarianism. All these things started to scare off both listeners who weren't interested in such crazy stuff, as well as guests who didn't want to show up in such company and didn't want to be associated with it. So after a couple of months, crazy woo stuff was the only thing that remained on the platform, because woo people aren't afraid of other woo people and they were the only ones who didn't scare each other. But even among them there was a lot of disagreement about the subjects that should be presented on the platform, so in the end, the platform rot and died with natural death :q So what's the moral of this story? I'm sure you're thinking now: "Well, it means that SOME censorship is required. Isn't it what all the big platforms are doing right now?" Or, if you're more on the libertarian side, you may think: "The fact that such things happen surely doesn't mean we should relinquish our freedoms and censor everything, now does it?" And you've probably guessed that these views seem to be contradictory: We can only have one or another, we can't have both, can we? Well, if you're like me, you would rather try to see past the paradox and find a solution that would satisfy both sides. Because if anything I learned about the paradoxes in my life is this: The only way to escape a paradox is to surpass the limits in which the paradox emerged in the first place. In this case, on one hand, we want to be able to express ourselves freely without anyone telling us what's the right or wrong way of doing it. On the other hand, we don't want other people's views to mess with ours. There are things that are either too disturbing for us, or simply not interesting to us, and we don't want to be forced to deal with them in our lives (that is, we don't really want to tolerate something that we don't accept, and as you know, or should know, tolerance cannot be compelled). And even if we do, we don't want to do that all the time. For example, I don't mind to see some pr0n from time to time (I'm an adult after all), but it doesn't mean that I want to see it all the time. I don't like pony gore, though, and I don't want to have anything to do with it, but I don't mind other people liking it, as long as they keep it to themselves and don't have me watch it, especially unexpectedly. But I can't stop them from doing it, and – to be frank – I should't. So how can we solve this paradox? Well, I'll give you a hint: the solution evolves around what I wrote in another thread some time ago. The problem with the fandom still stands though. And I think it might be harder to solve than the problem with freedom of expression vs. censorship, or the Tolerance Paradox, without introducing some subdivisions in the fandom, or setting some rules / boundaries that I'm pretty sure most people won't be happy with. But creating "safe spaces", forcing tolerance, or increasing censorship, is NOT the way, I can tell you at least that much. And neither is the opposite: just letting everyone do whatever they want, even at the cost of putting the entire fandom in bad light. And what are your thoughts?

There are also ships Or you can visit South American countries too, they're connected with land with North America. Another possibility is to use the fact that the Earth is round, and travel through Canada, then Alaska, and from there, there's a short water gap to Kamchatka in Russia (or perhaps a short trip in a ship to Japan ). Then you can travel through Russia or China, and finally get to Europe without the need of using a plane Once you're in Europe, there are several ways to get to Afrika in case you wanted to visit it too: either a short boat trip through the Strait of Gibraltar, or through Sinai Peninsula around the Mediterranian Sea (which might be a better choice if you're approaching from the East).

Yes, there is: the life itself. Life is a wonderful experience, in which you can learn a lot, visit different places, enjoy wonderful adventures, meet amazing people and spend your time with them. Sure, there are some downsides as well, as well as suffering. But they are experiences too, and they have their place in life as well – without shadows, you wouldn't appreciate the light. Your life also matters for other people, even if you might not realize it. Even the smallest things you do or say, can influence other people and their lives. And that's also part of the meaning of life: that you are part of a bigger whole, and you are here in this world not just for yourself, but for other people as well. If you deconstruct any idea (life included) into pieces, it will always lose all of its meaning, because any of the pieces on its own doesn't have any meaning. The meaning arises from the composition. Our lives mean as much as we all mean to each other. If you want a nice explanation of this idea, I remember seeing a good one in one of the episodes of "Power Corrupts" on DarkMatter2525's channel, the one with Adam in a simulated reality. (But the entire series is pretty interesting and thought-provoking )

@TBD Funny you mention it, because recently I heard pretty much the same amount of bad stuff about "High Guardian Spice", and yet I watched it and enjoyed it a lot, definitely more than some other lauded shows from Netflix or CN that bored me to death after a couple of episodes. Sure, it's loaded with animation errors, poor quality background art, lousy writing, etc. (not to mention the hectolitres of wokeness). But if you disregard all those deficiencies of that show, and if you can read between the lines, there is some more depth into this show, and some more mature subtexts, for those who like overthinking things and theorizing. Things don't always have to be epic, perfectly animated or tightly written to be enjoyable. I mean, many similar things could be said about MLP, and yet here we are

@ExplosionMare in case you wanted to know how the simplest polynomial curve going through your points looks like: just for fun I found it (with the help of Mr. Lagrange) Here it is, along with its formula: You can play with it yourself here: https://www.desmos.com/calculator/ehgi7ngaor Try moving the red points to some other coordinates by clicking and dragging them The curve should then rearrange itself to still fit through all those points. The original curve you started with will still be visible as a dashed line, for comparison. Notice however that in my curve, points B and E aren't local minima, and point C is not a local maximum. Your problem had these requirements, so I can try come up with a curve that satisfies these requirements as well, but something tells me that its formula might end up even more complicated Interestingly, if it wasn't limited to the [-3, 5] domain, there would be one more point at which it crosses the X axis at roughly 6.559 (if you're curious, I can tell you the exact formula for that point's X coordinate too ).

Lack of knowledge is not something to be ashamed of – there is no person on this planet who would know everything; there will always be things that you don't have knowledge of. And the only way to gain knowledge is to ask people who have it and learn from them. It doesn't hurt And learning stuff this way is usually much more fun than in schools. The more damaging is not asking, because then you remain ignorant and it piles up. The most dangerous are those people who don't want to learn even if you give them knowledge on a silver plate, because they think that they already know everything. Because they usually don't, and their stubbornness makes them incapable of learning. That's how people get stupid. And this is a reason to be ashamed. Yup, you pretty much got them right, except one: the (-3 , 5) one, because the problem says that [-3, 5] is the DOMAIN of this function, so it's not supposed to be a point. It is a RANGE (along the X axis) in which this function has values. If you pick any point within this range on the X axis, and move straight up or down, you should be guaranteed to find a point of the graph there, representing the value of that function for that particular argument X. But if you pick some point outside of that range and move straight up or down, you shouldn't be able to find such a point, because outside of that domain, function is supposed to have no values. But other than that, your points seem to be ok. Now you just need to connect them, and @Pentium100 showed you one possible way of doing it. Keep in mind though that these lines don't necessarily have to be straight (in fact, they don't even have to be continuous curves). As long as the problem doesn't specify what kind of function you are supposed to draw, you can pretty much draw any squiggle that goes through those points, provided that it meets all the required conditions. E.g. from x = -3 to x = -2 they require it to decrease, so you can move down (either slow or fast), but no up. If the problem required it to be a polynomial curve of the lowest degree that goes through these points, this question would get much more interesting And in that case we would need to ask Mr. Lagrange for help