I need to solve this using Lagrange's theorem and simple corollaries. Thanks.
124249
I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?
91833
Please show how to solve? I am stack with lie bracket. Thank you.
18891
The constraints are   $0 \le a_1,a_2....a_n,b_1,b_2....b_n \le 1$. $|a_i-b_i|&lt; \delta$ for all $1 \le i \le n $   How do I go about proving the following $$|\prod_{i=1}^n a_i-\prod_{i=1}^n b_i| &lt; n\delta$$  I tried reducing it to two terms where one term is like $(a-\delta)^n$ so that I can get the $n\delta$ term from binomial,but I am stuck. I would really appreciate just hints.
74792
Using Lagrange multipliers, it can be shown that a triangle with given perimeter has the maximum possible area, if it is equilateral. Is there a simple geometric proof of that fact ?
80609
Let $x = (x_1,\ldots,x_p) \in \mathbb R^p$. I'm solving the constrained optimization problem     $$\begin{align*} \text{min} &amp;\quad \sum_{i=1}^p (y_i-x_i)^2 \\ \text{s.t} &amp;\quad \sum_{i=1}^p y_i - 1 &amp;&amp;=0\\            &amp;\quad\forall  i = \overline{1,p}: -y_i &amp;&amp;\le 0  \end{align*}$$   My attempt:  Let $f(y) = \sum_{i=1}^p (y_i-x_i)^2$, $h(y) = \sum_{i=1}^p y_i - 1$, and $g_i(y) = -y_i$ for all $i = \overline{1,p}$.  We have $f,g_i$ are convex and $h$ is linear. Let $\alpha =(1/p, \cdots, 1/p)$. Then $h(\alpha)=0$ and $g(\alpha) &lt;0$ for all $i = \overline{1,p}$. It follows that Slater's condition is satisfied. By Karush-Kuhn-Tucker conditions, we have $$\begin{aligned} \begin{cases} \forall i = \overline{1,p}:\mu_i &amp;\ge 0 \\ \forall i = \overline{1,p}: g_i(y) &amp;\le 0\\ h(y) &amp;=0 \\ \forall i = \overline{1,p}:\mu_i g_i(y)&amp;=0 \\ \nabla f (y)+ \lambda\nabla h (y)+ \mu_i \nabla g_i (y) &amp;=0 \end{cases} &amp;\iff \begin{cases} \forall i = \overline{1,p}:\mu_i &amp;\ge 0 \\ \forall i = \overline{1,p}:-y_i &amp;\le 0\\ \sum_{i=1}^p y_i - 1&amp;=0 \\ \forall i = \overline{1,p}: -\mu_i y_i &amp;=0 \\ \forall i = \overline{1,p}: 2(y_i - x_i) +\lambda - \mu_i &amp;= 0 \end{cases} \\ \end{aligned}$$  Then I'm stuck at solving the last system of equations.  How can I proceed to solve it? Thank you so much!
47252
I found this problem in a book and while I understand the given solution, I don't quite understand how somebody would arrive at this solution unless they did out the casework and happened to make the right guesses to solve the problem. Any guidance on solving this from scratch would be greatly appreciated!
49663
Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I have a minimization problem of the form $$ \min_{n\in N} \sum_i A_i n_i +\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^2 $$ where $A_i\geq 0$ and $B_{ij}\geq 0$ for all $i,j$. This problem can be written as a convex quadratic integer program.   What are the standard algorithms to solve these problems numerically? Are they guaranteed to find a global solution? How fast are they?
9253
The question states:    I'm not sure what to do here. I used the geometric sum to do $2^{-n}$, but I can't think of a way to sum $\sin(n\theta)$ to get the required RHS.    Further working     Apologies for maybe being slow or missing something obvious here... I don't know how to get it into required form.
5503
I'm currently trying to prove the following:    And this is what I've come up with so far:    And now I'm stuck. Perhaps the next step might be to eliminate for all x?  I'm sorry for using links, but it seems like I'm not allowed to do it otherwise due to needed reputation point.  Help would be appreciated.
104441
I'm interested in the different ways of proving this, any proof is welcome. I understand one way is the cosine sum/difference formula, another is using a right angled triangle. Are there any others? Thanks.
12307
I've tried deriving a proof for this using the handshaking lemma, but that was unsuccessful. Could someone please explain to me how they would do this proof?
37529
I need to find a trapping region for  $u'=-u+vu^2$  $ v'=b-vu^2$  with $b&gt;0$.  I don't know what theory to use or in wich book I can find some examples to find optimal trapping regions.  Thank you very much.
69890
I have asked this question before  The first guy solved it in a way, but  i would like to know is there any efficient way to do so, because i did the same for N = 13 and N = 15  which took long time.  This problem is from a pen and paper test, ZIO-2018 Can i solve it with an efficient algorithm, well i am pretty beginner in algos, so if there is one kindly descibe it how can i do so?  Thank you
71437
Problem: Without solving the equation, show that all nine roots of $(𝑧 − 1)^{10} = 𝑧^{10}$ lie on the line $Re(𝑧) = \frac{1}{2}$. I would like to say I tried something but I am actually clueless. I was thinking maybe I could use De Moivre's theorem but I am not sure how. Any hint would be appreciated.
120749
Please help me to prove this statement. I'm very confused, Thank you! I Have tried prove it using examples but it needs to be proved using mathematical method
20751
I'm not sure how to go about this. Does it involve using MVT?  I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
22092
Q) Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let $F_{1}$ be the set of all pairs of circles $(S_{1}$, $S_{2}$) such that T is tangent to $S_{1}$ at P and tangent to $S_{2}$ at Q, and also such that $S_{1}$ and $S_{2}$ touch each other at a unique point, say, M. Let $E_{1}$ be the set representing the locus of M as the pair ($S_{1}$, $S_{2}$) varies in $F_{1}$. Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point R(1, 1) be $F_{2}$. Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$. Let $C$ be the circle $x^2+y^2+6(2y+7x)=53$. The number of times $C$ intersects $E_{1}$ and $E_{2}$ is (are):