Questions tagged [a.m.-g.m.-inequality]
For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.
1,314
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if a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$ [duplicate]
problem:
a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$
my attempt:
$LHS=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=(a+b+c) \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{a+b+...
user1069990
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Weighted AM-GM Inequality
I was tried to learn some tools on inequalities, and I learn the "Weighted AM-GM Inequality" in a small book: in this book there is an exercise and a solution, I did my attempt and it was ...
user1069990
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2
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prove that :$(1+\frac{1}{n})^k<1+\frac{k}{n}+\frac{k^2}{2n^2}$.
if $ n,k \in \mathbb{N^*}$ and $(k-1)^2<n$
prove that :$(1+\frac{1}{n})^k<1+\frac{k}{n}+\frac{k^2}{2n^2}$.
my attempt:
$(1+\frac{1}{n})^k+1=(1+\frac{1}{n}).(1+\frac{1}{n})^{k-1}+1\leq \sqrt{(1+\...
user1065479
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Let $f:\mathbb R^+\to\mathbb R$ be a continuous function satisfied $f(a)+f(b)\ge f(2\sqrt{ab})$ for all $a,b>0$ , is $f$ differentiable? [closed]
Let $f:\mathbb R^+\to\mathbb R$ be a continuous function satisfied $f(a)+f(b)\ge f(2\sqrt{ab})$ for all $a,b>0$ , is $f$ differentiable?
Morever, if for all $a_1,a_2,\cdots,a_n>0$ there holds $$\...
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Prove $ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ 1/a + 1/b+ 1/c}$
Given real numbers $c \ge b \ge a>0$, prove that
$$ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$$
*using well-known inequality
Other solution ...
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An Application of Cauchy Schwarz - AM-GM in Discrete Probability Measures
Suppose $p_m \geq 0$ and $\sum_{m \in \mathbf{Z}} p_m =1 .$ That is $p$ is a probability measure on integers. Then how can I show (is it true) that
$$ \sum_{m \in \mathbf{Z}} (p_m + p_{m+1} + p_{m+2})^...
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Finding a limit using AM-GM?
Doing some calculus papers before going university and I found this question:
Find $\lim_{x\to\infty} \left[\frac{1}{3} \left(3^\frac{1}{x} + 8^\frac{1}{x} + 9^\frac{1}{x} \right)\right]^x$
My ...
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Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$
The number of continuous functions $f:\left[0,\frac{3}{2}\right]\rightarrow (0,\infty)$ satisfying the equation$$4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$$
...
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Let $x,y,z\in[0,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$. [duplicate]
Let $x,y,z\in[0,1]$. Then find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.
Now the given answer is $2\sqrt{2}$ but I am not able to obtain the corresponding values of $x,y,z$. ...
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How can i find minimum value of this inequality
$a_1,a_2,...,a_n$ are 8 distinct positive integers. $b_1,b_2,...,b_n$ are another 8 distinct positive integers ($a_i,b_j$ are not necessarily y distinct for $i, j = 1, 2, ...8$).Enter
the smallest ...
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Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$ Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$
Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$
Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$
My attempts: $3m \le (x-y)^2+(y-z)^2+(z-x)^2$ so I tried to prove $(x-y)^2+(y-z)^2+(z-x)^2 \le \...
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A conditional negative definite quadratic form involving $\ln$ function
Let us consider the following property which is a constrained version of $(\star)$ (see Remark below):
$$\begin{align*}\bbox[#EFF,15px,border:2px solid blue]
{\begin{aligned}\text{For any n, for any} \...
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Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$
Let $x_1,x_2,...,x_k$ be real numbers such that $\sum_{i=1}^k x^2_i=1$. Determine the minimum and maximum (if there is) value of $$\prod_{i=1}^k (x_i-x_{i+1})$$ and determine all values of $(x_1,x_2,...
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Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.
I am not sure if the inequality is true. My first attempt was to try AM-GM inequality in clever ways. I also tried Schur's inequality which gives
$$
x^6+y^6+z^6 + 6x^2y^2z^2 \geq (x^2+y^2+z^2)(x^2y^2+...
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Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus
Note: this has been edited to make the question more general.
I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$.
Of course, I could take the first ...
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How can I prove that the perimeter is at most 60?
Problem: Let $\Delta$ be a triangle in the plane. Let $P$ be the perimeter of the triangle and $A$ be the area. Let $a,b,c$ be the length of the sides and suppose they are positive integers. Suppose ...
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Inequality of positive integers [duplicate]
I have to prove the following inequalities for positive integer $n$:
$$n^n\geqslant\left(\frac{n+1}2\right)^{n+1}$$
$$ 2^ {n(n+1)}> (n+1)^{n+1} \left(\frac{n}{1}\right)^{n} \left(\frac{n-1}{2}\...
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find the maximum of $x+\frac{1}{2y}$ under the condition $(2xy-1)^2=(5y+2)(y-2)$
Assume $x,y \in \mathbb{R}^+$ and satisfy $$\left(2xy-1\right)^2 = (5y+2)(y-2)$$, find the maximum of the expression $x+\frac{1}{2y}$.
$\because (5y+2)(y-2) \geq 0$ , $\therefore y\geq 2$ or $y \leq -\...
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To prove the inequality of positive rational numbers
Show that: $$ \left(\frac{a+b}{a+b+c}\right)^{c} \left(\frac{b+c}{a+b+c}\right)^{a} \left(\frac{a+c}{a+b+c}\right)^{b}< \left(\frac{2}{3}\right)^{a+b+c} ,a\ne b\ne c$$
PS: I am supposed to use ...
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Is there a solution for $(\frac{3}{x+y+z})^n+(\frac{3}{x+y+z})^{5-n}<2$?
Is there a solution for
$$\left(\frac{3}{x+y+z}\right)^n+\left(\frac{3}{x+y+z}\right)^{5-n}<2$$
where $n\in\mathbb Z$ and $x,y,z>0, xyz=1$.
This is the part of my attempts for my homework, that ...
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How would I prove the following using the AGM inequality?
Question 17. Let $x,y\in\mathbb R$, $x,y\geq0$. Prove that $$(\sqrt x+\sqrt y)^2\geq2\sqrt{2(x+y)\sqrt{xy}}.$$
I believe I have to use AGM multiple times, but I am not exactly sure how
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Proving $\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$, for positive values with $xyz=1$, without expansion?
Let, $x,y,z>0$ and $xyz=1$, then prove that
$$\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$$
I know that $$x^2+y^2+z^2\geq x+y+z$$ by Cauchy-Schwarz. So, $$\frac{x^...
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How can I prove that, if $x,y,z>0$ and $xyz=1$, then $2(x^2+y^2+z^2)+9\geq 5(x+y+z)$
How can I prove that, if $x,y,z>0$ and $xyz=1$, then
$$2(x^2+y^2+z^2)+9\geq 5(x+y+z)$$
I used the famous inequality
$$x^2+y^2+z^2+3\geq 2(x+y+z)$$
I got $$2(x^2+y^2+z^2)+9\geq 4(x+y+z)+3\geq 5(x+y+...
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To prove the relation between n numbers and their AM - GM [duplicate]
Given $A$ and $G$ to be the arithmetic and geometric mean of n positive real numbers $a_1, a_2,...,a_n$ then for any $k > 0$ show that $$ (k+A)^n \ge\ (k+a_1)...(k+a_n) \ge\ (k+G)^n .$$ I started ...
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To prove the following inequalities of positive rational numbers
I have to prove the following inequalities:
$$ a^ab^bc^c \ge \ (\frac{a+b}{2})^{\frac{a+b}{2}} (\frac{c+b}{2})^{\frac{c+b}{2}} (\frac{a+c}{2})^{\frac{a+c}{2}} $$
$$(a+b)^{c}(c+b)^{a}(a+c)^{b} < \...
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To prove the following inequality of real numbers
If $a_1,...a_n$ are all positive real numbers then prove that $$\left(\frac{a_1 + a_2 + \dots + a_n}{n}\right)^n \ge a_1a_2\left(\frac{a_3 + a_4 + \dots + a_n}{n-2}\right)^{n-2}.$$
I approached the ...
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To prove the given inequality of n numbers
Given $ a_1 + a_2 + ... + a_n = S$ where $a_1,...,a_n$ are positive reals and all $a_i$s are not equal, then show that $$ \prod_{i=1}^n\dfrac{S-a_i}{n-1} > a_1.a_2...a_n$$
I began by considering ...
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Prove the A.M.-G.M. inequality for n terms using induction
You are required to give a detailed proof of the A.M.-G.M. inequality using induction. I have one answer and I am posting it but it is too brutish. I want a more elegant method of proving A.M.-G.M. ...
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Proving the AM-GM Inequality by induction [closed]
I am required to prove the AM-GM inequality using induction but via this route:
(i) Let $a_1$, $a_2$, ..., an be a sequence of positive numbers. Denote their sum by $s$ and their geometric mean by $G$....
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question based on AM/GM [closed]
If $a, b, c$ are positive real numbers and
\begin{align*}
a^{2}(1 + b^{2})+ b^{2}(1 + c^{2}) + c^{2}(1 + a^{2}) = 6abc
\end{align*}
Find $a + b + c$.
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Solve the inequality $ \sum_{cyc}\frac{a-bc}{a+bc} \le \frac32$
Solve the inequality
$ \displaystyle\sum_{cyc}\frac{a-bc}{a+bc} \le \frac32$ given $a + b + c = 1$ and $a, b, c \in \mathbb{R_{>0}}$
So, I wanted to use the known inequality $9(a+b)(b+c)(c+a) \ge ...
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2
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Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$
Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$ for $a, b, c \in \mathbb{R_{>0}}$
I tried by first using AM-HM inequality on $a, b, c$ to get the following result.
$\frac{a+b+c}3 \...
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1
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Prove $\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $
Deduce whether the statement is true or false.
Suppose $n\in \mathbb N \setminus \{0,1\}$. Then, $$\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $$
I would like to ask what inequality I can apply or any ...
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2
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Showing $\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$ for $x,y,z>0$ and $xyz=1$
Let $x,y,z>0$ with $ xyz=1$ then prove that,
$$\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$$
Let $$5≤\sqrt{4(x^2+y^2+z^2)+6}\implies x^2+y^2+z^2≥\frac {25-6}{4}=\frac {19}{4}$$ then the ...
- 1,773
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3
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Prove that $(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$
Let, $x,y,z>0$ such that $xyz=1$, then prove that
$$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$$
My progress:
Using the Cauchy-Schwars inequality I got
$$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(xy+yz+xz)(x+...
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2
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Is this nice-looking inequality actually trivial?
Let, $x,y,z>0$ such that $ xyz=1$, then prove that
$$(xy+yz+xz)(x^2+y^2+z^2+xy+yz+xz)≥2(x+y+z)^2 $$
I tried to use the inequality
$$x^2+y^2+z^2≥xy+yz+xz$$
Then I got,
$$(xy+yz+xz)(x^2+y^2+z^2+xy+...
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2
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Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$
Let, $x,y,z>0$ such that $ xyz=1$, then prove that
$$(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$$
I tried to use the following inequalities:
$$x^2+y^2+z^2≥xy+yz+xz$$
and The Cauchy–...
- 1,773
4
votes
2
answers
110
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Find minimum of the function using AM-GM
Problem: Find the minimum of the function $f(x,y)=x + \frac{8}{y(x-y)}$, where $x>y>0$ using AM-GM.
My attempt:
$$f(x,y)=2\cdot \frac{x+\frac{8}{y(x-y)}}{2} \ge 2 \sqrt{\frac{8x}{y(x-y)}}$$
But ...
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1
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Is the geometric mean bounded above by this value?
It is clear that the geometric mean is bounded above by the arithmetic mean:
$$
\prod_{k=1}^{M} x_k^{\alpha_k} \leq \sum_{k=1}^{M}\alpha_k x_k
$$
Moreover, it is clear that the arithmetic mean is ...
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2
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Alternate proofs of the inequality $\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$.
If $x,y,z\in(0,\infty),$ prove the inequality,
$$\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$$
I have a solution using the substitution $x+y=a,$ $y+z=b$ and $z+x=c$.
$$\frac{x}{2x+y+z}+\frac{...
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Prove $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$.
Prove:
Let $p$ be an integer greater than $1$. Suppose $a,b,c$ be positive real numbers. Then $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$.
By AM-GM, I get $\frac{a+b+c}{3}\geq (abc)^{1/...
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1
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Prove that for $a,b,c \in \mathbb{R}^+$ with $abc = 1$, $\frac{1}{a^3(b + c)} + \frac{1}{b^3(a + c)} + \frac{1}{c^3(a + b)} \ge \frac{3}{2}$ [duplicate]
I would like confirmation that I did this proof correctly. If I did, it would be a milestone in my mathematical journey as it would be my first IMO problem.
By Cauchy, we have that $$\left( \frac{1}{a^...
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3
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What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$?
What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$?
I know the answer is going to be $\frac{6}{5}$ because it will occur when there is equality ...
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0
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Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\frac ab} + \sqrt[5]{\frac bc} + \sqrt[7]{\frac ca} > \frac 52$
Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\dfrac ab} + \sqrt[5]{\dfrac bc} + \sqrt[7]{\dfrac ca} > \dfrac 52$.
My thought process along with the proof: Since we're dealing with ...
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3
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Proof based on inequalities [closed]
I want to find the minimum value of
$$\frac{(5+x)(2+x)}{1+x}.$$
I brought it down to the fact that it depends on the value of
$$(x^2 + x)\sqrt{7x+10}.$$
Also $x\geq -10/7$, but don't know what to do ...
1
vote
0
answers
76
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Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 4abc.$ Find the maximum value of
$$\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}.$$
First, I tried using $\dfrac{4 \sqrt{...
- 748
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1
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Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.
Find the largest real number $\lambda$ such that
$$a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$$ for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$
I tried using AM-GM on like $a^2+\dfrac{b^2}{4}...
- 748
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1
answer
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Hard AM-GM Inequality
Find the sum of all positive integers $n,$ where the inequality
$\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$ holds for all nonnegative real numbers $a,$ $b,$ and $c.$
I tried squaring both sides ...
- 77
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1
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question about AM-GM inequality ab<(a^2+b^2)/2 [closed]
I have a question about the AM-GM inequality if a,b>0 then
ab≤(a+b)^2/4;
ab≤a^2+b^2;
I wonder if these 2 inequality are true for all a,b>0, if not which one is correct?
- 1
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0
answers
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Proof that $\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$ for $x+y+z=3$ [duplicate]
If $x+y+z=3$, where $x$, $y$ and $z$ are sides of a triangle, prove that $$\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$$
We have $0< x,y,z<\frac{3}{2}$ from the triangle ...
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