Diffuse $5$ points on a sphere. Prove there is a closed half-sphere that has at least $4$ points on it.
If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.  It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.  Regards
Comment character counter does not trim, while the server does Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? Is there any other size tires that will fit on a 27 x 1 1/4 rim? How do I install HP P1102W printer driver Dual boot Ubuntu with Windows on Acer Aspire I've installed Ubuntu 16.04 LTS along side with Windows 10 (I shrinked my Disk using Windows Disk Management) on my Acer laptop (ES1-531-C13G), but when I restart/reboot my system no menu (Grub screen) appears from where I can select the operating system that I want to boot with, but instead, system takes me into Windows 10 everytime. I've tried several things from internet including :    Ran bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi from windows command prompt.  Reset  Bootorder i.e. first Ubuntu then Windows from Ubuntu terminal (after successful Ubuntu installation) etc, but nothing works for me so far. Please guide me ! "Try and" versus "Try to"? Print lines between (and including) two patterns Lookup Tables Best Practices: DB Tables... or Enumerations If we have to store the available positions at a company (i.e. Manager, Team Lead, ... etc). What are the best practices for storing it? I have two opinions with comments... "sure, welcoming yours"        Storing it as DB table with columns ID and Name, and deal with it using queries and joins.     Storing it as Enum and forget about the DB table.    In my opinion, I will choose the first solution if I have changing items. So that I won't hard code these options as Enum. I may choose the Enum solution, if I have no doubt that data won't change (for example, Gender: Male, Female).  NOTE: I code in English, and the UI Culture may be Arabic. If I will work with the Enum Solution, I will hard code the culture-based strings in the presentation layer, is it okay from the best practices perspective!!!!  I would like to know your opinions and if my thoughts correspond to what is most recommended "Best Practices"?? If the Universe is expanding, how is that galaxies run into each other? As the title says. If everything is moving outward, thrown by the Big Bang, how could two Galaxies end up with velocity vectors that run them into each other? What details should a good cover letter contain to back up a UK Visit Visa application? Cardinal number subtraction We all know that $|\mathbb{N}| = \aleph_0$. Since $|\{-1\} \cup \mathbb{N}| = \aleph_0$ as well, I guess you could say that $\aleph_0 + 1 = \aleph_0$.  You can go on to derive that $\aleph_0 + \aleph_0 = \aleph_0$, and by induction $n \times \aleph_0 = \aleph_0$ (assuming that $n$ is finite, anyway).  Now, here's a thing: What is $\aleph_0 - \aleph_0$?  Well, $|\mathbb{P}| = \aleph_0$, and $|\mathbb{P} \backslash \mathbb{N}| = \aleph_0$, so perhaps $\aleph_0 - \aleph_0 = \aleph_0$?  No, wait. Consider the set $S = \{n \in \mathbb{N}: n &gt; 5\}$. Now we have $|S| = \aleph_0$, and yet $|\mathbb{N} \backslash S| = 5$. So maybe $\aleph_0 - \aleph_0 = 5$?  But that is absurd, since we can redefine $S$ to make the result any finite number we wish. Or we can define $S$ such that the result is countably infinite.  Does this mean that the notion of $\aleph_0 - \aleph_0$ simply has no definite answer? Or am I just being too simplistic here? How can I make content disappear when a user logs in? I am working in Wordpress, and I would like to make certain elements disappear once a user logs in. How can I do this?  I have social icons on my page that I positioned to the center of my menu bar at the top of my website.  those icons are generated by a plugin  that is what i want to disappear How to search by key=>value in a multidimensional array in PHP Is there a reason the British omit the article when they "go to hospital"? Why do British speakers omit the article in constructions like "go to hospital" or "go on holiday"? Pretty much all American speakers would rephrase those as "go to the hospital" and "go on a holiday", I think. Is there any good reason, or forgotten sense behind those words that might explain why the articles are ommitted?  Are there other common constructions other than those two that the British use that drop the article?  EDIT: I just realized per Kosmonaut's comment that Americans do much the same thing with a few nouns, so this isn't all that special.   Do grammaticists designate nouns that can have their article dropped with anything, i.e. do they have anything in common? How do I ensure a piece of code runs only once? I am trying to align some text to left/center/right in the same line. For example, I want to put my phone number on the left, my name in the center, and my email on the right, how do I do that? Grub not showing on startup for Windows 8.1 Ubuntu 13.10 Dual boot Currently, when flagging an off-topic question, the following choices are offered.   Questions about general computing hardware and software are off-topic for Stack Overflow. Questions concerning problems with code you've written must describe the specific problem and include valid code to reproduce it. Questions must demonstrate a minimal understanding of the problem being solved. Questions on professional server, networking, or related infrastructure administration are off-topic for Stack Overflow unless they directly involve programming or programming tools. Questions asking us to recommend a tool, library or favorite off-site resource are off-topic for Stack Overflow as they tend to attract opinionated answers and spam. This question belongs on another site in the Stack Exchange network   However, there are some questions that fit in none of these categories.  For example, career advice questions are off-topic here (and on programmers.SE), questions like  are off-topic on pretty much any SE site.    Some questions might also be relevant on another SE site not listed when the "belongs on another site" option is selected.  How should those questions be flagged?  Why isn't there a "it's off-topic because it is" option?  EDIT: just to be clear, I don't have the "Other" option under Off-Topic. find all $(x,y,z)$ such that $27^{3x^2 + 2y}+27^{3y^2 + 2z}+27^{3z^2 + 2x}=1$ Find all ($x,y,z$) such that $$ 27^{3x^2 + 2y}+27^{3y^2 + 2z}+27^{3z^2 + 2x}=1$$  I am a high school student and would appreciate it if anyone could solve it using high school mathematics. here is what I have tried till now: $$ 3x^2 +2y &lt; 0 $$ (this can also be said for the other 3 equation. it also means that x,y,z&lt;0) $$ -2y&gt;3x^2 $$ $$ 4y^2&gt;9x^2 $$ ( -2y and 3x^2 are both positive) $$ 3y^2&gt;27/4 x^2 $$ $$ -2z&gt;3y^2&gt;27/4 x^4 $$ $$ 4z^2 &gt; 729/16 x^8 $$ $$ 3z^2&gt; 2187/64 x^8 $$ $$ -128/2187&lt;x^7 $$ $$ -2/3 &lt; x $$ $$ -2/3&lt; x,y,z &lt; 0 $$ i tried this but as you can see didn't do much good How to Measure Luck vs Skill in Games?