Expert Help with High School Geometry

Suppose I have $n$ integers ${\mathrm{\mu <!--\; \mu \; -->}}_i$, $i=\{1,2,...,n\}$. Define $\stackrel{¯<!--\; ¯\; -->}{\mathrm{\mu <!--\; \mu \; -->}}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{\mathrm{\mu <!--\; \mu \; -->}}_i$. It is given that all the ${\mathrm{\mu <!--\; \mu \; -->}}_i$'s are either +1 or −1. How can I show that $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}({\mathrm{\mu <!--\; \mu \; -->}}_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{\mathrm{\mu <!--\; \mu \; -->}}{)}^2$ is maximized only when $\frac{\lfloor <!--\; \lfloor \; -->n\rfloor <!--\; \rfloor \; -->}{2}$ or $\frac{\lceil <!--\; \lceil \; -->n\rceil <!--\; \rceil \; -->}{2}$ are +1 and rest are −1's.

I was thinking about proving this inductively, but I am not sure how.

If $X\sim <!--\; \sim \; -->$ Poisson $(\mathrm{\lambda <!--\; \lambda \; -->})$, prove that $P[X=k]$ is maximized at $k=\mathrm{\lambda <!--\; \lambda \; -->}$

How would I go about proving something like this so that the method would work to prove a maximization point of other random variables like say Binomial is max at $k=(n+1)\star <!--\; \star \; -->p$?

If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle?

Let L :${\displaystyle :P_2\Rightarrow P_3}$ be a linear transformation for which we
know that ${\displaystyle L:\left(1\right)=1,L\left(t\right)=t^2,L\left(t^2\right)=t^3=t.}$
(a) Find ${\displaystyle L(2t^2-5t=3).}$
(b) Find ${\displaystyle L(a{t}^2-bt+c).}$

How to find the coefficient a of a
${\displaystyle y=a{x}^2}$
parabola?

Solve below somewhat symmetric equations:
x, y, z subject to
${\displaystyle x^2+y^2-xy=3}$
${\displaystyle {(x-z)}^2+{(y-z)}^2-(y-z)(x-z)=4}$
${\displaystyle {(x-z)}^2+y^2-y(x-z)=1}$
${\displaystyle x,y,z\in R^{+}}$

I am trying to solve the following question:
$\text{Maximize }f(x_1,x_2,\dots <!--\; \dots \; -->,x_n)=2\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i^{t}A{x}_i+\underset{i=1}{\overset{n-<!--\; -\; -->1}{\sum <!--\; \sum \; -->}}\underset{j>i}{\overset{n}{\sum <!--\; \sum \; -->}}(x_i^{t}A{x}_j+x_j^{t}A{x}_i)$
subject to
$x_i^t{x}_j=\{\begin{array}{cc}1& i=j\\ -<!--\; -\; -->\frac{1}{n}& i\ne <!--\; \ne \; -->j\end{array}$
where xi's are column vectors ($m\times <!--\; \times \; -->1$ matrices) with real entries and A is an $m\times <!--\; \times \; -->m$ ($(n<m)$) real symmetric matrix.

From some source, I know the answer as
$f_{max}=\frac{n+1}{n}\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{\mathrm{\lambda <!--\; \lambda \; -->}}_i^{\downarrow <!--\; \downarrow \; -->},$
${\mathrm{\lambda <!--\; \lambda \; -->}}_i^{\downarrow <!--\; \downarrow \; -->}$ being the eigenvalues of $A$ sorted in non-increasing order (counting multiplicity). But I am unable to prove it. I will appreciate any help (preferably with established matrix inequality, or Lagrange's multiplier method).

An apartment complex on Ferenginar with 250 units currently has 193 occupants. The current rent for a unit is 965 slips of Gold-Pressed Latinum. The owner of the complex knows from experience that he loses one occupant every time he raises the rent by 2.5 slips of Latinum. What should be our recommendation for the optimal rent?

The equation I got was $(193-<!--\; -\; -->x)\cdot <!--\; \cdot \; -->(965+2.5x)$. I tried to maximize it and got $x=-<!--\; -\; -->96.5$. The optimal rent is $965+2.5(-<!--\; -\; -->96.5)=723.75$.

This answer is not correct? Where did I make a mistake Thanks

Let us consider a linear programming problem shown below:
$\begin{array}{r}\underset{x}{min}f(x)-<!--\; -\; -->c\end{array}$
where, $f(x)$ is a linear function of $x$. Now, can be following maximization problem be considered as an equivalent to the above minimization problem?
$\begin{array}{r}\underset{x}{max}\left(\frac{1}{f(x)-<!--\; -\; -->c}\right)\end{array}$

Help with Pythagorean theorem proof for Hilbert Spaces
For given functions f and g in a Hilbert space $L^2(a,b)$, prove if the Pythagorean theorem is true for f and g, then it is also true for cf and cg, where c is a constant. What would $<cf,cg>$ be?
Assume the Pythagorean Theorem holds for functions f and g in a Hilbert space $L^2(a,b)$. Then $||f|{|}^2+||g|{|}^2=||f+g|{|}^2$.
I have the beginning of the proof above however I am having trouble continuing.

It is not possible for a part of any of three conic sections to be an arc of a circle.
It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?

Consider the example of maximizing $x^{2}yz$ under the constraint that $x^2+y^2+z^2=5$.

One way to do this is to use lagrange multipliers, solving the system of equations
$2xyz=2x\mathrm{\lambda <!--\; \lambda \; -->}$
$x^{2}z=2y\mathrm{\lambda <!--\; \lambda \; -->}$
$x^{2}y=2z\mathrm{\lambda <!--\; \lambda \; -->}$
$x^2+y^2+z^2=5$
However, couldn't you just substitute $x^2=5-<!--\; -\; -->y^2-<!--\; -\; -->z^2$ into the expression you want to maximize to get: $yz(-<!--\; -\; -->y^2-<!--\; -\; -->z^2+5)$and then just maximize that by setting the $y$ and $z$ partial derivatives equal to zero?

Then you just have to solve the arguably simpler system of equations:
$3y^{2}z+z^3=5z$
$y^3+3y{z}^2=5y$
where the $z$s and $y$s cancel out nicely on both sides.

Why is maximize by lagrange multipliers necessary when you can always substitute and maximize the resulting function?

When should you choose one over the other?

Let $w,a\in <!--\; \in \; -->R^n$, and $B\in <!--\; \in \; -->R_{++}^{n\times <!--\; \times \; -->n}$ (the set of $n\times <!--\; \times \; -->n$ positive definite matrices).
We know that the following function (which is a specific form of the Rayleigh quotient) has a unique maximum, and a closed-form solution in $w$:
$f(w)=\frac{w^{T}a{a}^{T}w}{w^{T}Bw}$
It's maximum is achieved at $w^{\ast <!--\; \ast \; -->}=B^{-<!--\; -\; -->1}a$ and its value is $f(w^{\ast <!--\; \ast \; -->})=a^T{B}^{-<!--\; -\; -->1}a$. (by using the generalized eigenvalue decomposition)

Now here is my question:
If I have $w,a_1,a_2\in <!--\; \in \; -->R^n$, and $B_1,B_2\in <!--\; \in \; -->R_{++}^{n\times <!--\; \times \; -->n}$, and the following function:
$g(w)=\frac{w^T{a}_1{a}_1^{T}w}{w^T{B}_{1}w}+\frac{w^T{a}_2{a}_2^{T}w}{w^T{B}_{2}w}$
Then, what can we say about the maximum of $g(w)$, can we still solve for $w$ in closed-form?

Suppose I'm given two points: $(x_1,y_1)$ and $(x_2,y_2)$ (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those two points (the arc with shorter length)?

Given matrix-valued function $A:{\mathbb{R}}^d\to <!--\; \to \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$ and (tall) matrix $B\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->m}$, where $n><!--\; >\; -->m$, let matrix-valued function $C:{\mathbb{R}}^d\to <!--\; \to \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->m}$ be defined by
$C(x):=B^{T}A(x)B$
I would like to maximize the determinant of $C(x)$ without directly considering $C(x)$. If I maximize the determinant of $A(x)$, does that maximize the determinant of $C(x)$ as well?

If it were $m=n$, then we would have:
$det(C)=det(B{)}^{2}det(A)$
Hence, maximising $det(A)$ would maximize as a consequence $det(C)$. Does it hold something similar in the case with $n><!--\; >\; -->m$? Or is it possible to define some lower bound for $det(C)$? And if so, how to prove it?

If in the previous general case it doesn't hold, maybe it could help that, in my specific case $C$ and $A$ are two positive definite matrices and $B$ is a sparse matrix of zeros and ones. $A$ is also block diagonal. The only way I came up with is using Cauchy-Binet for the determinant of the product of rectangular matrices but I remained stuck with a summation involving the principal minors of the Cholesky factorization of $A$ times minors in $B$.