Tagged Questions

0
votes
0answers
57 views

A minimax problem

Let $\omega$ be a non zero real number, $(\lambda_i)_{i = 1}^n$ be a sequence of $n$ real numbers and $$u_{i}:=1-(1-\lambda_{i})\,\omega.$$ Show that if we pick ...
1
vote
0answers
32 views

How to solve this type of equation with posynomial form?

I have an equation with the following form where the goal is to find $x$: $$ \sum_k c_k x^{\gamma_k} = 1$$ where $c_k, \gamma_k \in \Re^+$ and $\gamma_k > 1$ Alternatively using $y = \log(x)$ I can ...
5
votes
4answers
121 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
0
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1answer
33 views

How to maximize this function of X,Y?

I have 2 input $X$ and $Y$ which are both positive integers. I have to maximize this function Let $A=\min(Y/4,X/2)$ , $B=\min(W/2,Y/2)$, $C=\max(A,B)$, and $D=\max(X-W,Y)$. Then $$ ...
0
votes
3answers
89 views

How to write “the parameter maximizing the maximum of the maximum value of two functions continuous in the domain of maximization”

Say you have $f(x),g(x)$ continuous where they need to be and you want to express the following: Give me the biggest value of $f$ for $x \leq X_f$ , give me the biggest value of $g$ for $x \leq X_g$, ...
0
votes
1answer
79 views

Finding the minimum of $N = \frac{(a+3c)}{(a+2b+c)}+\frac{(7a+6b+3c)}{(a+b+2c)}+\frac{(c-a)}{(2a+b+c)}$ if $a, b, c \in \Bbb R$

Find the minimum of $$N = \frac{(a+3c)}{(a+2b+c)}+\frac{(7a+6b+3c)}{(a+b+2c)}+\frac{(c-a)}{(2a+b+c)}. \qquad (a,b,c \in \Bbb R^+)$$
3
votes
1answer
38 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
0
votes
2answers
43 views

Fit screen resolution given ratio and total number of pixels

Given: width: 1920 height: 1080 total pixels: width * height = 2073600 aspect ratio: 1920 / 1080 ~= 1.8 How do I calculate a new resolution (width and height) ...
3
votes
3answers
122 views

Finding the minimum of $\frac pq + \frac rs$ for distinct integers $p, q, r, s$ from $\{1,2,3,4,5,\ldots,16,17\}$

Here is the question: Four distinct integers $p$, $q$, $r$ and $s$ are chosen from the set $\{1, 2, 3, 4, 5, \ldots, 16, 17\}$. The minimum possible value of $\frac pq + \frac rs$ can be written ...
1
vote
1answer
73 views

Maximum/minimum problem of integers

Let $f$ be the function such that $$f(x,y,z,w)=x+w, \quad x,y,z,w\in{\Bbb Z}$$ where $$ x+y+z+w=400, $$ and $x<y<z<w$. How can I find the maximum of $f$? I think the key point is to use ...
1
vote
1answer
106 views

For integers $a$ and $b \gt 0$, and $n^2$ a sum of two square integers, does this strategy find the largest integer $x | x^2 \lt n^2(a^2 + b^2)$?

Here is some background information on the problem I am trying to solve. I start with the following equation: $n^2(a^2 + b^2) = x^2 + y^2$, where $n, a, b, x, y \in \mathbb Z$, and $a \ge b \gt 0$, ...
0
votes
0answers
68 views

Maximization of a sum subject to constraints on 3 resources

This is a generalization of a subproblem from a past programming competition that I had trouble with. Given input $6$ positive integers: $$r_1, r_2, r_3, x_1, x_2, x_3 \in \mathbb{Z^+}$$ ...
5
votes
2answers
109 views

Minimize sum of smallest and largest among integers on the real line.

Suppose there are 3 non-negative integers $x$, $y$ and $z$ on the real line. We are told that $x + y + z = 300$. Without loss of generality, assume $x$ to be the smallest integer, and $z$ to be the ...
5
votes
3answers
254 views

The minimum value of $(\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)$ if $x+y+z=1$

$x, y, z$ are three distinct positive reals such that $x+y+z=1$, then the minimum possible value of $(\frac{1}{x}-1) (\frac{1}{y}-1) (\frac{1}{z}-1)$ is ? The options are: $1,4,8$ or $16$ ...
29
votes
4answers
771 views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
1
vote
2answers
160 views

How to find the minimum value of $px+qy$ when $xy=r^2$?

The question says: "Find the minimum value of $px+qy$ when $xy=r^2$." No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...
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vote
2answers
307 views

bird traveling to a nest wants to save energy

This is a multiple choice question in one of tests I just wrote and I did not know the answer to it. I was just stuck on this during the test. It is a very weird question, one I find to be impossible. ...
3
votes
3answers
77 views

Prove that $(2-x)^nx^{n-1}$ decreases with $n$ for $0 <x<1$?

How can I show that: $$(2-x)^nx^{n-1}$$ is decreasing with $n$ when $0<x<1$? I think this is generally true, but specifically I am concerned with $n$ as an integer $\geq 2$ and showing that the ...
4
votes
8answers
1k views

Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
3
votes
6answers
274 views

Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
1
vote
3answers
222 views

A “fast” way to ,find the maximum value of $(x^2) \times (y^3)$,if $3x+4y=12$ for $x,y \ge 0$

If $3x+4y=12$ $\forall x,y \ge 0$,the maximum value of $(x^2) \times (y^3)$ is $6 \times (6/5)^5$ $3 \times (6/5)^5$ $ (6/5)^5 $ $7 \times (6/5)^5$ How to approach this problem?I thought of ...
5
votes
5answers
638 views

Local minimum and maximum of the function

Can anyone help me to solve the following question? maximize and minimize the function $(10-x)(10-\sqrt{9^2-x^2})$ over $x\in[0,10]$ This is a high school question, so is there any simple trick help ...
3
votes
2answers
130 views

Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$

What is a good way to calculate max/min of $$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$ where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
4
votes
1answer
326 views

Find the minimum for a trigonometric function

Find the local minimum of the following function: $$\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$$ I am wondering how can I simply this ...
1
vote
1answer
81 views

How can I find $\sup -\frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$ for $x_1 x_2 > 0$?

I need to find a constant $a$ such that for all $x_1 x_2 > 0$: $$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$$ that is to say the supremum of the term on the right hand side. My question is how to ...
-2
votes
1answer
260 views

Polynomial problem

From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf: Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x ...
1
vote
1answer
193 views

Finding the minimum value of a quadratic within a range

Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$. My programmer brain can do it in a branchy, algorithmic way as follows, but is ...
2
votes
0answers
205 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...