# Tagged Questions

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### Find the smallest $a>1$ such that $\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$ for all $x \leq y$

Can anyone please help me with the following question: Find the smallest $a>1$ such that $$\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$$ for all $x \leq y$ My attempt: I think we should rearrange ...
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### minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
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### Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
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### Linear Inequalities - Allocation Problem

The problem at hand can be summarized as follows: we have to allocate a ressource to $n$ production units. The allocation to production unit $i$ is $x_i$. Each of the production unit will produce at ...
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### Convex Minimization Problem with double sum

Given fixed natural number $n$ and two real numbers $A$ and $B$. I'd like to find $c_{12},\dots c_{(n-1)n}$, i.e., ${n\choose2}$ real numbers, such that $\sum_{1\le i<j\le n}^nc_{ij}=1$ which ...
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### Bounds of the solution space

I have a continuous function $f(x)=a x^2+b x+c$ that is defined on $]0,X[$. I know that the function values are bounded, $Fu <f(x) <Fl$ for all values of $x$. I want to find the bounds on the ...
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### Calculus-based proof that $x_1^{p_1}\cdots x_n^{p_n}\le p_1x_1+\dots+p_nx_n$ when $\sum p_i=1$

Let $$g(x_1...x_n)=x_1^{p_1}\cdot...x_n^{p_n}$$ $$u(x_1...x_n)=p_1x_1+...p_nx_n$$ Where $\sum p_i = 1$. I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've ...
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### Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
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### How do you prove ${n \choose k}$ is maximum when k is $\lceil \frac n2 \rceil$ or $\lfloor \frac n2\rfloor$?

How do you prove ${n \choose k}$ is maximum when k is $\lceil \frac{n}{2} \rceil$ or $\lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
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### Maximum of linear combination

I have an range like this: $$x + 2y \leq 40$$ $$4x + 3y \leq 120$$ $$x \geq 0, y \geq 0$$ I made an plot using wolfram alpha. Now I have a linear combination $$4x+5y$$ and I want to find the maximum ...
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### Solve the system of inequalities. Optimization problem.

I have a set of linear inequalities as follow: ...
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### Prove that $(x-1)(y-1)(z-1)\geq 8$.

Let $x, y, z$ be positive integers, such that $\frac{1}{x}+ \frac{1}{y}+ \frac{1}{z} \leq 1$. Find $\inf_{x, y, z} (x-1)(y-1)(z-1)$. After a few trials, I'd say the answer is $8$, but I can't ...
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### Maximum and minimum of weighted sum

For $w_i\ge 0$ and some constants $\alpha_i , i=1,...,n$, what is the maximum and minimum of $\sum_{i=1}^{n}\alpha_i w_i$ subjected to $\sum_{i=1}^{n}w_i=1$? Intuitively, I put all weight on the ...
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### Minimum value of the function $\sqrt{(1+1/m)(1+1/n)}$

If $m, n$ are positive real variables whose sum is a constant $k$, then what is the minimum value of $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
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### Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$

$a;b;c\in \mathbb{R}^+$ such that $a+b+c=6$. Find the minimum of : $P=(1+\frac{1}{a^3})(1+\frac{1}{b^3})(1+\frac{1}{c^3})$ Thanks :) I have no ideas about this problem ! :(
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### Relation between softmax and max

For two vectors $X$ and $Y$ in $\mathbf{R}^n$, does the inequality below hold? $\left| \text{softmax} X - \text{softmax} Y \right| \leq \text{max} | X - Y |$ Softmax is the same as log-sum-exp: ...
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### Find : $\min P=2x^2-xy-y^2$?

$x;y\in \mathbb{R}$ such that : $x^2+2xy+3y^2=4$. Find : $\min P=2x^2-xy-y^2$ ? Thanks :) P/s : I have no ideas about this problem ! :(
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### Proving $~\sum_{cyclic}\left(\frac{1}{y^{2}+z^{2}}+\frac{1}{1-yz}\right)\geq 9$

$a$,$b$,$c$ are non-negative real numbers such that $~x^{2}+y^{2}+z^{2}=1$ show that $~\displaystyle\sum_{cyclic}\left(\dfrac{1}{y^{2}+z^{2}}+\dfrac{1}{1-yz}\right)\geq 9$
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### Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
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### Range Of Quartic Polynomial Of Two Variables

$a$,$b$ are real numbers such that $~3\leq a^{2}+ab+b^{2}\leq 6$. I would like to find the range of $~a^{4}+b^{4}$. Is it possible to find it with (well-known) AM-GM, CS, etc...?
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### find conditions on input data such that a linear system has (no) feasible points

As a result of the apllication of Farkas' lemma I obtained the following problem: Let $m,n,q \in \mathbb{N}$, $b \in \mathbb{N}^m, l \in \mathbb{N}^m$ with $l_i \mid q$ for all $i=1,\ldots,m$. ...
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### Minimum value of: $x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: $$x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$$ I put it in the form $x^6y +x^6z+y^6x+y^6z+z^6x +z^6y$. I tried AM-GM but ...
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### Maximum of the sum of cube

(1) $-2\leq a_{i} \leq 2$ $~(i=1,2,3,4,5)$ (2) $\displaystyle\sum_{cyclic}a_{i}=0$ then, find the maximum value of $\displaystyle\sum_{cyclic}a_{i}^{3}$ also, can it be generalized as for ...
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### $\inf$ and $\sup$ of a set.

Let $n\geq3$ be an arbitrarily fixed integer. Take all the possible finite sequences $(a_{1},...,a_{n})$ of positive numbers. Find the supremum and the infimum of the set of numbers ...
### How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$
let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...