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.
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Another inequality related to (weighted) AM-GM ...
Let $w_i \in [0,1], \sum_i w_i = 1$ (i.e. weights); $\beta_i \in (0,1)$; and $N \ge 1$.
We can show that:
$$
[A]\qquad1 - \prod_i {(1 - \beta_i)}^{N\,w_i}
\quad\ge\quad
1 - \sum_i w_i {(1 - \beta_i)}^...
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Inequality proof with AM-GM
I'm new to this community. Do you think my solution is correct?
The problem is:
$$a^2+b^2+c^2+ab+bc+ca\geq6, a+b+c=3$$
My solution:
$$a^2+b^2+c^2+2ab+2bc+2ca\geq3(ab+bc+ca)\rightarrow$$
$$3\geq ab+bc+...
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Minimum value of f(θ)= a²sec²θ +b² cosec²θ using AM- GM inequality
If we take and function
$$f(θ)= \dfrac{a^2}{\cos^2\theta} + \dfrac{b^2}{\sin^2\theta}$$
And wish to find minimum value of function using AM -GM inequality
i.e. if we have two no. p & q
Then $$...
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Is there any simple way of proving triangle inequality for $d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$ [duplicate]
$d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$
Following are the approaches I took, but can't think further -
$d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}
= \dfrac{...
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Prove or disprove that the inequality is true if x, y, z are positives and $xyz=8^3$.
Prove or disprove that the inequality $$ \dfrac{1}{\sqrt{x+1}}+\dfrac{1}{\sqrt{y+1}}+\dfrac{1}{\sqrt{z+1}} \geq 1$$ is true if x, y, z are positives and $$xyz=8^3.$$
First, let us rewrite the ...
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Proving $4(x^4+y^4+z^4)\geq \sum_{cyc}xy(x+y)^2\geq 4xyz(x+y+z)$, for $x,y,z\geq0$
Trying to get used to olympiad inequalities. Tried AM-GM and did not succed. Please explain. (I'm an eight grader)
Prove that for $x,y,z\geq 0$, the following inequality holds true:
$$4(x^4+y^4+z^4)\...
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Three-Variable Inequality
Prove that if $a$, $b$, and $c$ are positive real numbers, then
$$\sqrt{a^2 + ab + b^2} + \sqrt{a^2 + ac + c^2} + \sqrt{b^2 + bc + c^2} \ge \sqrt{3} (\sqrt{ab} + \sqrt{ac} + \sqrt{bc}).$$
When does ...
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Find the maximum value of $ab+bc+cd+de+ef+fa,$ given that $a+b+c+d+e+f=1$
After looking at this post, I framed the following question.
If $a,b,c,d,e,f$ are positive real numbers such that $a+b+c+d+e+f=1$, then find the minimum and maximum value of $ab+bc+cd+de+ef+fa$
My ...
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Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$.
Prove or disprove that the inequality $$ \dfrac{x^2}{\left(x^2+1\right)^2}+\dfrac{y^2}{\left(y^2+1\right)^2}+\dfrac{z^2}{\left(z^2+1\right)^2}+\dfrac{u^2}{\left(u^2+1\right)^2} \leq \dfrac{16}{25}$$ ...
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If $xy+yz+zx=3$ and $x,y,z\geq0$, prove that: $\frac{1}{1+3x-p}+\frac{1}{1+3y-p}+\frac{1}{1+3z-p}\leq\frac{3}{1+2p}$
If $xy+yz+zx=3$ and $x,y,z\geq0$, prove that:
$$\sum_{cyc}\frac{1}{1+3x-p}\leq\frac{3}{1+2p}$$
where $p=xyz$.
*some people are not familiar with the $\sum_{cyc}$ notation, alternative would be
$$\frac{...
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JBMO-$2014$ Inequality question [duplicate]
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$ {\left(a+\frac{1}{b}\right)^2}+{\left(b+\frac{1}{c}\right)^2} +{\left(c+\frac{1}{a}\right)^2}≥3(a+b+c+1)$$
My solution:
By Jensen'...
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AM-GM & Minimization Proof [duplicate]
I want to prove that for all $x, y > 0$, $$\cfrac{x+y}{2} \geq \sqrt{xy}$$
Particularly, I want to show that the minimum of $(x+y)/2$ is exactly $\sqrt{xy}$.
This is my attempt:
$\textbf{Proof}$ (...
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If $x_1 ,\dots , x_n$ are real numbers then $(x_1 \dots x_n )^{\frac{1}{n}} \leq \frac{x_1 + x_2 + \dots x_n}{n}$ [closed]
The algebraic mean is biger than or equal to geometric mean. It is easy to prove the case $n=2$. I tried to use induction but I guess it doesn't work. Can anybody give a proof?
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For $x,y,z∈ℝ^{+}$,without using Hölder's inequality prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$.
For $x,y,z∈ℝ^{+}$, prove that $\frac{x}{\sqrt{x^2+8yz}}+\frac{y}{\sqrt{y^2+8xz}}+\frac{z}{\sqrt{z^2+8xy}}\geq1$.
In this question solution used Hölder's inequality, but I am looking a solution ...
user1135351
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Proof or references for strengthened AM-GM
This other question includes the following strengthened version of the arithmetic-mean geometric-mean inequality.
\begin{equation}
\label{1}\tag{1}
\dfrac{a+b}{2} - \sqrt{ab} \geq \dfrac{1}{16 \max \...
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$\left\{(x_1,x_2)\mid \left|f(x_1,x_2)-f(0,0)\right|<\frac{1}{2}\right\}=\left\{(0,0)\right\}\cup\left\{(x_1,x_2)\mid x_1\neq\pm x_2\right\}$
I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka.
Problem 18 on p.194
Let $f$ be a function from $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$ such ...
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Why does the AM-GM inequality not show $25 \csc^2(\theta) +16 \sin^2(\theta)$ has a minimum of $41$ as the graph indicates?
Let's say we have to find range of $f(\theta) = 25 \csc^2(\theta) +16 \sin^2(\theta)$
If I use $AM \ge GM$
Then $f(\theta) \ge 40$
Which tells minimum value of $f(\theta)$ will be $40$
But I checked ...
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Prove or disprove the inequality if $a,b,c>0$, $a \geq b+c$.
Prove or disprove the inequality
$$a^2b+a^2c+b^2a+b^2c+c^2a+c^2b \geq 7abc$$ if $$a,b,c>0, a \geq b+c.$$
I thought to use this evaluation:
$$a^2b+b^2c+c^2a \geq 3abc.$$
So we have:
$$a^2b+a^2c+b^2a+...
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Prove or disprove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1$.
Prove or disprove that the inequality
$$\dfrac{1}{\sqrt{1+x}}+\dfrac{1}{\sqrt{1+y}}+\dfrac{1}{\sqrt{1+z}} \geq 1$$
is valid if $x,y,z$ are positive numbers and $$xyz=1.$$
My solution is:
Let $$x=\...
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Show that $x^{\frac1x}<1.5$ for $x \in \mathbb R$
Well, I've proven that :
$$x^{\frac1x}<1.5$$ for $x \in \mathbb R^+$, or more specifically I've shown that the maximum value of the expression is $1.44...$ for $x = e$. But I used calculus (finding ...
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Proof of a tighter inequality than Cauchy-Schwarz inequality
A few days ago, I found this post which discuss about some tighter versions of Cauchy-Schwarz (or some might prefer the name AM-GM) inequality. User @Michael Rozenberg proposed a very interesting ...
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Prove that the inequality is valid if $a,b,c,d$ are positive numbers. [duplicate]
Is given that $a,b,c,d$ are positive numbers. We have to prove that
$$\dfrac{a}{b+c}+\dfrac{b}{c+d}+\dfrac{c}{a+d}+\dfrac{d}{a+b} \geq 2.$$
What have I done?
I used the substitution $$a=\dfrac{1}{x},b=...
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To prove $1^1\cdot2^2\cdot 3^3...\cdot n^n<(\frac{2n+1}{3})^{\frac{n(n+1)}{2}} $
So we have to prove the following for $n\in N $ $$1^1\cdot 2^2\cdot 3^3...\cdot n^n<\left(\frac{2n+1}{3}\right)^{\frac{n(n+1)}{2}} $$
So I used concept of weighted means (arithmetic and geometric) ...
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Does the AM-GM inequality not hold if there is a variable on the GM side?
To find the minimum value of $x+\dfrac{1}{x}$, $(x>0)$ we can use the AM-GM inequality to say that $$\dfrac{x+\dfrac{1}{x}}{2}\geq \sqrt{x\cdot \dfrac{1}{x}}$$ or $$x+\dfrac{1}{x}\geq 2$$
The ...
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Prove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1.$
Is given that $x,y,z$ are positive numbers and $xyz=1$, prove that
$$\dfrac{\dfrac{1}{x}}{\sqrt{z^2+1}}+\dfrac{\dfrac{1}{y}}{\sqrt{x^2+1}}+\dfrac{\dfrac{1}{z}} {\sqrt{y^2+1}}>\sqrt{2}.$$
What have ...
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Help w/$(\frac{a}{b})^4+(\frac{b}{c})^4+(\frac{c}{d})^4+(\frac{d}{e})^4+(\frac{e}{a})^4\ge\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{e}{d}+\frac{a}{e}$ [closed]
How exactly do I solve this problem? (Source: 1984 British Math Olympiad #3 part II)
\begin{equation*}
\bigl(\frac{a}{b}\bigr)^4 + \bigl(\frac{b}{c}\bigr)^4 + \bigl(\frac{c}{d}\bigr)^4 + \bigl(\frac{d}...
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Sequence of geometric mean subtracted by arithmetic mean
Let $a_1,a_2,a_3,\dots$ be a sequence of positive numbers. Define $$G_n=\sqrt[n]{a_1a_2\dots a_n}~\text{and}~A_n=\frac{a_1+\dots+a_n}{n}.$$ We are supposed to use the result $$u^av^b\leq au+bv \tag{$*$...
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Inequality related to AM-GM?
Let $a$ and $b$ be two positive numbers such that $a+b=1$. I am supposed to show that $u^av^b\leq au+bv$ for all positive $u$ and $v$.
It is known that $\ln x \leq x$ for all positive $x$, so I ...
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prove that inequality holds for all reals if and only if $|k| \leq 2n$
Prove that $$x_1^2+x_2^2+\ldots+x_{2n}^2+k \cdot x_1 \cdot x_2 \cdot \ldots \cdot x_{2n}\geq 0$$
for all reals $x_1, x_2, \ldots, x_{2n}$ if and only if $|k| \leq 2n$, where $n$ and $k$ integers, $n &...
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Let $a,b,c$ be positive real numbers such that $\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$ Prove that $abc \le \frac{1}{8}$.
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
So I tried this problem and came up with a solution, but I'm afraid ...
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Cyclic inequality. $\sum_{cyc} \frac{x+2y}{\sqrt{z(x+2y+3z)}}\geq \frac{3\sqrt{6}}{2}$
Let $x,y,z>0$. Show that
$$\sum_{cyc} \frac{x+2y}{\sqrt{z(x+2y+3z)}}\geq \frac{3\sqrt{6}}{2}$$
Equality case is for $x=y=z$.
A hint I have is that I have to amplify with something convenient, and ...
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Maximum value of a variable in system of two equations
Consider the system of equations
$$ 3a + 2b + c+ d =14 $$
$$ a^{2} + b^{2} + c^{2} + d^{2} =14.$$
Is there any way to find maximum value of $d$ using AM-GM inequality. I am not even able to think, how ...
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AM - GM inequality with Bernoulli's inequality
I am stuck on this proof: The AM-GM
Inequality
is Equivalent
to the Bernoulli
Inequality.
I get the first part with the Bernoulli Inequality, but how did he get from $\frac{A_n}{A_{n-1}}\ge \frac{x_n}{...
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For which $(n,m)$ holds $x^ny^m + x^my^n \le 2$, where $x+y = 2$?
For which $(n,m)$ holds $x^ny^m + x^my^n \le 2$, where $x+y = 2$ and $x,y \ge 0$?
It is $n,m \ge 0$ where $n$ and $m$ do not have to be integers.
The conjectured answer is: for $|n-m | \le \...
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Is this proof using AM-GM valid?
I was reading about the AM-GM inequality at brilliant.org.
In one of the examples, it proves that:
$(a^2+1)(b^2+1)\ge 4ab , ab\in\mathbb{R^+}$
But I thought applying the AM-GM inequality to a term ...
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Prove inequality for positive real numbers $x,y,z$
For positive, real numbers $x,y,z$ prove that
$$\frac{2(x+y+z)}{3}+\frac{3xyz}{xy+yz+zx}\ge3\sqrt[3]{xyz}$$
It's pretty obvious that the solution will have something to do with the $A.M-G.M$ ...
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Let $x,y,z\in\mathbb{R}^{+}$ such that $x+y+z = 1.$ T/F: $\frac{x^2 y^2}{(x+y)^2}+\frac{y^2 z^2}{(y+z)^2}+\frac{z^2 x^2}{(z+x)^2}\geq\frac{9}{4}xyz$
In Springer's "Inequalities" book, Exercise $2.5$ is the following:
Let $x,y,z\in\mathbb{R}^{+}\ $ such that $\ x+y+z = 1.\ $ Then:
$$ xy + yz + zx \geq 9xyz.\qquad (1)$$
Proof: Applying AM $...
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Please help me on how to prove the equality case of the inequality
goal:
for n=2,3,4....
prove that given x1x2...xn=1,
then x1+x2+x3+...+xn=n iff x1=x2=x3=...=xn=1
I tried to prove by induction. While the "if" direction is obvious, but I am kind of stuck in ...
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$ab+bc+ca \le 4{\sqrt 3}\Delta$ for a triangle with sides $a$, $b$ and $c$ with area $\Delta$ [duplicate]
If $a$, $b$ and $c$ are the sides of a triangle with area $\Delta$, prove that $ab + bc + ca \le 4\sqrt3\Delta$ and prove that the equality holds iff the triangle is equilateral.
I tried to approch ...
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Possible flaw in AM GM inequality
AM GM inequality states that for real positive numbers x,y
$$x + y \geq 2(xy)^{1/2}$$
You would get the least value for x = y
We can also write this as,
$$\frac x3 + \frac x3 + \frac x3 + y \geq 4\...
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Find the minimum of $xy+yz+zx+\frac1x+\frac2y+\frac5z$ for $x, y, z > 0$
Let $x,y,z > 0$. What is the smallest value of the below expression?
\begin{align*}
xy+yz+zx+\frac1x+\frac2y+\frac5z
\end{align*}
I tried with Lagrange multiplier but I have not any constraint.
user1006476
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1
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Show an inequality in three variables
Problem :
Let $a,b,c>0$ then we have :
$$\left[4-\frac{3\left(\sqrt{ac}+\sqrt{bc}+\sqrt{ab}\right)}{a+b+c}\right]^{\frac{3}{2}}abc-\left(\frac{16}{9\sqrt{3}}\right)^{\frac{1}{2}}\left(\frac{a+b+c}{...
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I can't come up with the intended solution to $\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18$
I was going trough some easy algebra problems when I encountered
$$
\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18.
$$
As you can see the problem is easily solvable with AM > GM
I fairly ...
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Proving $30x + 3y^2 + \frac{2z^3}{9} + 36 (\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}) \ge 84$ [closed]
If $x, y, z$ are positive real numbers, prove that $$30x + 3y^2 + \frac{2z^3}{9} + 36 \left(\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}\right) \ge 84.$$
I genuinely have no clue on how to proceed. Is ...
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min of $c$ in such that $\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{a/b}+\sqrt{b/a}}2\right)^c$ for all $a_i\in [a,b]$.
Positive integer $n\geq 2$ is given. All $a_i\in[a,b]$($0<a<b$). What's the minimum value of $c$ such that $$\tag{1}\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{\...
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1
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Prove that: $\sum_{1\le j\le n}{\frac{x_j}{\sum_{1\le i\le n} x_i-x_j}}\ge\frac{n}{n-1}$
$\color{red}{\textbf{Problem:}}$
Let, $x_i>0,1\le i\le n$, then
Prove that:
$$\sum_{1\le j\le n}{\frac{x_j}{\sum_{1\le i\le n} x_i-x_j}}\ge\frac{n}{n-1}$$
$\color{red}{\textbf{Proof:}}$
Using AM-...
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votes
3
answers
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Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$
From R. D. Sharma's Objective Mathematics,
Given that $x + y + z = 1$, ($x,y,z$ are positive real numbers) find the minimum value of $$A = xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$$
My attempt:
By A.M.-G.M....
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1
answer
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Geometric-arithmetic mean inequality applied to eigenvalues
I've applied the arithmetic-geometric mean inequality to the eigenvalues of a positive definite matrix $X$, so $det(X)^{1/n}≤tr(x)/n$.
Now I would like to show when equality holds. I already found out ...
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If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$ $( \frac{1}{a} +\frac{1}{b} +\frac{1}{c}) +a +b +c \geq 4\sqrt{3}$
If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$
Show that:
$$\frac{1}{a} +\frac{1}{b} +\frac{1}{c}+a +b +c \geq 4\sqrt{3}.$$
My attempt:
First , I used Holder's : $$\frac{1}{a}+\...
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3
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$a+b+c=1$, prove $(1+a)(1+b)(1+c) \ge 8(1-a)(1-b)(1-c) $, without AM-GM
$a,b,c \in \mathbb{R}^{+}$, if $a+b+c=1$, prove that
$$ (1+a)(1+b)(1+c) \ge 8(1-a)(1-b)(1-c) $$
but without AM-GM as the only tool.
Collecting data:
Since $a+b+c=1$, we cannot have one of the ...
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