2
votes
2answers
56 views

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 ...
1
vote
2answers
50 views

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 ...
2
votes
3answers
73 views

Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$

What is the maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$ here $a,b \in \mathbb{Z^+}$? What I have gotten so far: From the above, $\frac{a+b}{ab} = ...
0
votes
0answers
8 views

Linear complementarity preconditioner for projected Gauss-Seidel

Given an LCP in the form: $Ax + b \ge 0$ $x \ge 0$ $x^T(Ax +b) = 0$ Where $A$ is a symmetric positive definite matrix. If we have a pre-conditioning matrix $P$, so that the problem is transformed ...
0
votes
0answers
27 views

Minimum of summed sequence

Define M non-negative sequences, \begin{equation} a_{m,1}\geq a_{m,2}\geq,...,\geq a_{m,K}\quad \text{for}\ m=1,..,M \end{equation} and cyclic shifted versions $a^{\zeta_m}_{m,k}$ with shift value ...
0
votes
2answers
39 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
2
votes
1answer
84 views

Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
1
vote
2answers
91 views

Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
0
votes
0answers
22 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
0
votes
0answers
40 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
5
votes
4answers
143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
3
votes
2answers
83 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
3
votes
2answers
148 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
2
votes
1answer
45 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
1
vote
3answers
39 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
11
votes
5answers
294 views

How find this maximum of the $\sin^2{\theta_{1}}+\sin^2{\theta_{2}}+\cdots+\sin^2{\theta_{n}}$

Question: let $\theta_{1},\theta_{2},\cdots,\theta_{n}\ge 0$,and such $$\theta_{1}+\theta_{2}+\theta_{3}+\cdots+\theta_{n}=\pi$$ find the $P$ the maximum of value $P(n)$ ...
12
votes
2answers
172 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
2
votes
1answer
54 views

Can this multidimensional non-linear equation with constraints be minimized analytically?

I wish to find the vector of real numbers, $\mathbf{w}$, that minimizes the function: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
2
votes
3answers
60 views

Find the minimum of $(x(1+y)+y(1+z)+z(1+x))/\sqrt{xyz}$ over positive integers $x,y,z$

Let $x,y,z$ be positive integers.The least value $$\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{\frac 12}}$$ I tried sum using arithmetic-geometric means inequality (seems promising as the denominator is ...
0
votes
3answers
78 views

Minimum value of $\sqrt{(1+1/y)(1+1/z)}$

If $y,z > 0$ and $y + z = c$ where $c$ is a constant, then what's the minimum value of $$\sqrt{\left(1+\frac1y\right)\left(1+\frac1z\right)}$$ I am having a hard time solving this.
5
votes
3answers
114 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
42 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
40 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
32 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
33 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
34 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
1answer
31 views

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 ...
0
votes
0answers
104 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 ...
4
votes
3answers
150 views

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 ...
2
votes
2answers
83 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
75 views

Solve the system of inequalities. Optimization problem.

I have a set of linear inequalities as follow: ...
1
vote
3answers
73 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
52 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
41 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
61 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
60 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
78 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
65 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
46 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
27 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
157 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
50 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
46 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
67 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
133 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
158 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
537 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
436 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 < ...