5
votes
3answers
92 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
0
votes
2answers
35 views

Solving Problem by different Method ( non-induction)

I have this problem , which I was able to prove it by induction, but I wonder could be solve by direct method ( for example combinatorial method). I want to find number of solution for $$0 \le ...
1
vote
2answers
34 views

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 ...
0
votes
1answer
19 views

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 ...
0
votes
0answers
19 views

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 ...
0
votes
0answers
30 views

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 ...
0
votes
0answers
50 views

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 ...
1
vote
0answers
33 views

An indirect optimizing problem

we have $z_1=x_1y_1$ , $z_2=x_2y_2$ and $z=xy$. we know $x<x_1+x_2$ and $y<y_1+y_2$. Can we conclude minimizing $z_1$ and $z_2$ will lead us to minimum (or lesser values) of $z$? ...
2
votes
2answers
39 views

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 ...
0
votes
0answers
46 views

Solve the system of inequalities. Optimization problem.

I have a set of linear inequalities as follow: ...
1
vote
3answers
66 views

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 ...
2
votes
1answer
34 views

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 ...
1
vote
1answer
32 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq ...
0
votes
2answers
50 views

Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
3
votes
2answers
57 views

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 $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
1
vote
3answers
69 views

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 ! :(
1
vote
1answer
55 views

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: ...
0
votes
2answers
45 views

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 ! :(
1
vote
0answers
26 views

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 $~\displaystyle\sum_{cyclic}\left(\dfrac{1}{y^{2}+z^{2}}+\dfrac{1}{1-yz}\right)\geq 9$
1
vote
0answers
136 views

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 ...
1
vote
1answer
46 views

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...?
1
vote
0answers
44 views

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 $. ...
4
votes
2answers
61 views

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 ...
2
votes
1answer
127 views

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 ...
2
votes
1answer
144 views

$\inf$ 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 ...
14
votes
4answers
498 views

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 $$A=\dfrac{2}{\sqrt{10}}$$ ...
0
votes
1answer
35 views

Proving an inequality involving multiple constraints

Let $R$ be a discrete set and let $f:{\left[ {0,1} \right]^{\left| R \right|}} \times {\left[ {0,1} \right]^{\left| R \right|}} \to \mathbb{R}$ be defined as $f\left( {{\mathbf{x}},{\mathbf{y}}} ...
5
votes
5answers
414 views

How to find the minimum of $a+b+\sqrt{a^2+b^2}$

let $a,b>0$, and such $$\dfrac{2}{a}+\dfrac{1}{b}=1$$ Find this minimum $$a+b+\sqrt{a^2+b^2}$$ My try: since $$2b+a=ab$$ so ...
0
votes
0answers
24 views

Show that z is only positive when $\min(\frac{x^{0.5}}{y^{0.5}}, \frac{y^{0.5}}{x^{0.5}}) > A$?

where $$z = \frac{x - Ay^{0.5}x^{0.5}}{x + y - 2Ay^{0.5}x^{0.5}}$$ where $-1 < A < 1$. So the two conditions must be: $y > Ay^{0.5}x^{0.5}$ and $x+y > 2Ay^{0.5}x^{0.5}$ OR $y < ...
0
votes
1answer
46 views

Finding Range Of Formula With Constraint

$a$, $b$, $c$, $d$ are positive such that $a^{4}+b^{4}+c^{4}+d^{4}=4$ find the range of $a^{4}+64abcd$ in the case of maximum, it's not easy for me to adjust coefficients in AM-GM.
0
votes
3answers
54 views

help me find the minimum cost

an insurance company employs full and part-time staff, who work 40 and 20 hours per week respectively. Full-time staff are paid $\$800$ per-week and part-time-staff $\$320.$ In addition, it is ...
4
votes
1answer
60 views

To prove the inequality of two positive variables

$m$, $n$ are natural numbers. $a$, $b$ are positive numbers such that $~a^{m}=\dfrac{1}{n+1}$ , $~b^{n}=\dfrac{1}{m+1}$ prove that $~a+b\geq 1~$ without calculus.
4
votes
1answer
101 views

How find this minimum $\sum_{i=1}^{n}a^2_{i}-2\sum_{i=1}^{n}a_{i}a_{i+1},a_{n+1}=a_{1}$

let $a_{1},a_{2},\cdots,a_{n}\ge 0$,and such $a_{1}+a_{2}+\cdots+a_{n}=1$. Find this follow minimum $$I=a^2_{1}+a^2_{2}+\cdots+a^2_{n}-2a_{1}a_{2}-2a_{2}a_{3}-\cdots-2a_{n-1}a_{n}-2a_{n}a_{1}$$ My ...
2
votes
4answers
142 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
14
votes
1answer
291 views

Searching for tighter bounds

I have to solve an equation $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where $$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$ The $z_i$ are all positive and add ...
1
vote
3answers
68 views

Maximum Of Two Variables' Formula

$x$,$y$ are real numbers satisfying $(x-1)^{2}+4y^{2}=4$ find the maximum of $xy$ and justify it without calculus. Does there exist a tricky solution using elementary inequalities (AM-GM or ...
1
vote
2answers
34 views

finding range of function of three variables

Three real numbers $x$, $y$, $z$ satisfy the following conditions. $x^{2}+y^{2}+z^{2}=1~$, $~y+z=1$ Find the range of $~x^{3}+y^{3}+z^{3}~$ without calculus. I solved this problem only with ...
4
votes
1answer
70 views

Evaluation Of Maximum Value Without Calculus

$x$,$y$,$z$ are non-negative numbers. $x+y+z=3$ Find the maximum value of $~$ $x^{2}y+y^{2}z+z^{2}x$ $~$ without calculus.
4
votes
0answers
68 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
4
votes
1answer
135 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
0
votes
0answers
53 views

Von Neumann Entropy Inequality

Suppose $\rho_1$ and $\rho_2$ are density matrices (and thus Hermitian, positive semi-definite matrices) and $\hat{w}$ is the solution of the following optimization problem $$ \hat{w} = ...
6
votes
1answer
266 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
1
vote
0answers
17 views

Inequality of $h(s)=\int_{0}^{\infty}t^{N-1}e^{-t}\prod_{i=1}^M(1+\lambda_{i}ts)^{-1}dt.$

Suppose \begin{align} h(s)=\int_{0}^{\infty}t^{N-1}e^{-t}\prod_{i=1}^M(1+\lambda_{i}ts)^{-1}dt. \end{align} where $M$ and $N$ are positive integers and $N<M$, $\lambda_i$ are distinct positive ...
1
vote
0answers
48 views

Simplistic method for proving inequlaities

I have noticed that many inequalities posed in olympiads or otherwise were solvable using Lagrange Multipliers. However, the method might get tedious in which case, I had noticed that in inequalities ...
3
votes
1answer
106 views

Prove triangle inequality

I want to prove that $d(x,y) = 1- \sum_i {\min(x_i, y_i)}$ where $\sum_i {x_i} = \sum_i {y_i} =1$ and $\forall i: x_i, y_i \geq 0$ satisfies the triangle inequality. The domain of $d$ therefore is ...
2
votes
1answer
166 views

Proof by Contradiction: $100$ Balls & $9$ Boxes

Show, by giving a proof by contradiction, that if $100$ balls are placed in nine boxes, some box contains 12 or more balls. I would like to ask for a hint for this quesiton. Thank you.
1
vote
1answer
133 views

Find the minimum of this expression

This is a problem in my exam and I can't find the solution using elementary inequality knowledge. Can anyone here help me solve this. Thanks $a,b,c $ are positive real numbers which satisfy ...
0
votes
1answer
66 views

Angles inequality in a tetrahedron

Let $ I \in \triangle \text{ABC}$ of tetrahedron $ABCD$. Prove the inequality: $$\angle ADB+\angle BDC+\angle CDA \ge \angle ADI+\angle BDI+\angle CDI$$ Non-Euclidean geometry is very new for me. ...
1
vote
2answers
135 views

Prove $x^2+y^2+z^2 \ge 14$ with constraints

Let $0<x\le y \le z,\ z\ge 3,\ y+z \ge 5,\ x+y+z = 6.$ Prove the inequalities: $I)\ x^2 + y^2 + z^2 \ge 14$ $II)\ \sqrt x + \sqrt y + \sqrt z \le 1 + \sqrt 2 + \sqrt 3$ My teacher said the ...
3
votes
3answers
76 views

Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus

Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b), c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or langrange multiplier): $$74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 ...