Let $a_1,a_2,a_3,a_4 $ be $4$ real numbers such that $\sum a_i =0 $ and $\sum a_i^2=1$. Then , the minimum value of $(a_1-a_2)^2+(a_2-a_3)^2+(a_3-a_4)^2+(a_4-a_1)^2$ lies in the range :-   A)$ (0,1.5)$ B)$(1.5,2.5)$ C)$(2.5-3.5)$ D) $(3.5-4.5)$  This question was asked in the Mathematics section of the 2017 edition of the KVPY , one of the most prestigious national level examinations conducted in India .  The answer is (B)    Obviously , being an objective exam , the layman’s approach to tackling a question like this would be to assume values (say $-0.5 , -0.5 ,0.5 $ and $0.5$) , but is there a formal way of solving this problem using a standard inequality ? I have tried rearrangement on $\sum a_i^2$ , but this simply gives me a minimum of $0$ , with no case of equality.
If $a_{1},a_{2},a_{3},a_{4}\in \mathbb{R}$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$ and $a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4}=1.$ Then Range of $$E =(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ is  Try: From $$(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ $$=2(a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4})+2(a_{1}a_{2}+a_{2}a_{3}+a_{3}a_{4}+a_{4}a_{1})$$ $$=2-2(a_{1}+a_{3})(a_{2}+a_{4})=2+2(a_{1}+a_{3})^2\geq 2$$ and equality hold when $\displaystyle a_{1}=-a_{3}$ and $a_{2}=-a_{4}$ Could some help me how to find its upper bound, Thanks
Fibonacci Recurrence Relations Solve the recurrence relation $f(n) = f(n − 1) + f(n − 2)$ with initial conditions $f(0)=1$, $f(1)=2.$   So I understand that it grows exponentially so $f(n) = r^n$ for some fixed $r$.  This means substituting this $r^n = r^n-1 + r^n-2$ which gives the characteristic equation of $r ^ 2 - r - 1 = 0$. I'm not 100% sure where to move on from here. Proper usage of "since" and "from" with regard to duration of time Old usernames stick to Community Wiki posts On a whim the other day, I changed my meta name from Lord Torgamus to Popular Demand.  I just noticed that the old name has "stuck" to the author tag of ; everywhere else on meta, my new name displays as expected.  Is this a feature?  Is it a bug?  Mostly I'm just curious, as I can't think of a strong reason in favor of keeping this behavior or a strong reason for changing it.  A question somewhat like this has been raised before as , but the previous poster implied that the issue had something to do with migration, which my situation shows is not the case. Call to a member function prepare() on a non-object PHP Help I would like to be able to write a Java class in one package which can access non-public methods of a class in another package without having to make it a subclass of the other class. Is this possible? Is there a way to mount a disk directly to a specific folder? I am having a hard time with the following real analysis qual problem.  Any help would be awesome.  Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p&lt; + \infty$. Let $T_r(f)(t)=f(t−r)$. Show that $\lim_{r \to 0} \|T_rf−f\|_{L_p} =0$, that is $$ \lim_{r\to 0} \left( \int_{\mathbb R} |f(t-r) - f(t)|^p \mathrm d t\right)^{1/p} =0.$$ How to remove Home Folder icon from desktop in Ubuntu 20.04? When I start Ubuntu I had my &quot;home/user&quot; in my Desktop. test is my User.   How do I make the folder not be on the desktop? count vs length vs size in a collection From using a number of programming languages and libraries I have noticed various terms used for the total number of elements in a collection.  The most common seem to be length, count, and size.  eg.  array.length vector.size() collection.count   Is there any preferred term to be used? Does it depend on what type of collection it is? ie. mutable/immutable  Is there a preference for it being a property instead of a method? do { ... } while (0) — what is it good for? Possible Duplicate:          I've been seeing that expression for over 10 years now. I've been trying to think what it's good for. Since I see it mostly in #defines, I assume it's good for inner scope variable declaration and for using breaks (instead of gotos.)  Is it good for anything else? Do you use it? Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the  definition of continuity.   I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".  Definition of continuity; for $f:A \to \mathbb{R}$     $$\forall \varepsilon &gt;0 \ \exists\delta&gt;0\ \forall x \forall x_0 \in A:|x-x_0|&lt;\delta \Rightarrow |f(x)-f(x_0)|&lt;\varepsilon$$   So from what I understand, the procedure for epsilon-delta proofs are as follows: 1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|&lt;\varepsilon$ 2. Write $x$ in terms of $\varepsilon$ 3. Then plug $x$ into $|x-x_0|&lt;\delta$ 4. Find $\delta$ in terms of $\varepsilon$. Done    At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|&lt;\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?  Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing... How to calculate stays against 90/180 visa rules? Some destinations have a "90/180 rule" which says you can come and go but you must not spend over 90 days out of any 180 days in the area.  concerns this rule.  At least Schengen and Turkey have this rule, probably depending on where the traveller is from and what kind of visa they apply for.  Maybe I'm a dummy but I've always had a lot of trouble trying to add up how many days I've spent in such places when I've entered and left several times over recent months.  Is there an easy way to do this I'm missing or is it difficult for everybody? I'm guessing something based on counting backwards from the end date might be the trick?  Maybe you know of an app or website that can do this for you. One of the answers in a reading exercise in my class today was:   100,000,000,000,000,000,000   ... which was the value of the highest denomination note ever issued. It was a 100,000,000,000,000,000,000 Pengo note, issued by Hungary in 1946. The students in my class wanted to know how they could say that last sentence in words in English. So I told them that I didn't know. They pointed out to me that if they wanted to read this bit of writing to someone they'd want to be able to use the words. Fair point.  Can anyone help?  [In case you're interested, a 100,000,000,000,000,000,000 Pengo note was worth about $0.20!] How to get the file name from a full path using JavaScript? Follow up: instance control in Java without enum Is there a word for when an action has an effect opposite to the one intended? Is there a word or phrase to express the concept of an action having the opposite effect of the expected outcome?  For example, a drug taken to cure headaches that actually causes headaches, or an advertising campaign designed to deter smoking that leads to an increase in smoking. Show that unit circle is not homeomorphic to the real line Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$? Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ that is $a^{2n-1}\mid b^{2n} ; b^{2n}\mid a^{2n+1} , \forall n \in \mathbb Z^+$ , then is it true that $a=b$ ? Why are "baked" and "naked" not pronounced the same? Sort arrays of primitive types in descending order