Questions tagged [muirhead-inequality]
Inequality proof by using the Muirhead inequality.
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Proving Muirhead’s inequality by AM-GM
I don't understand the second proof of Muirhead’s inequality in page 10-11
First prove \eqref{constantMuirhead} using AM-GM:
Let $(c_i)_{i=1}^n$ be a sequence of real numbers such that $c_i \neq 0$ ...
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a,b,c>0 and prove $\frac{a}{a+\sqrt{(a+2b)(a+2c)}}+\frac{b}{b+\sqrt{(b+2a)(b+2c)}}+\frac{c}{c+\sqrt{(c+2b)(c+2a)}}\le \frac{3}{4}.$
Let $a,b,c>0$. Prove that $$\frac{a}{a+\sqrt{(a+2b)(a+2c)}}+\frac{b}{b+\sqrt{(b+2a)(b+2c)}}+\frac{c}{c+\sqrt{(c+2b)(c+2a)}}\le \frac{3}{4}.$$
It is from a book. My tries did not lead to anything ...
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Conditions for Muirhead's inequality hold for cyclic sums
I know that Muirhead's inequality apply only for symmetrical sums, but all inequalities with cyclic sums I have seen have the sequence in the greater side majorizing the sequence in the smaller side (...
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Conjecture for the inequality $f\left(x\right)+f\left(y\right)+f\left(z\right)\ge1$ where $f\left(x\right)=\frac{x}{x+\frac{1}{x}+1}$ which seems easy
I come back with an inequality checked with Desmos :
Let $x\in R^*$ then define :
$$f\left(x\right)=\frac{x}{x+\frac{1}{x}+1}$$
Then do we have :
$$f\left(x\right)+f\left(y\right)+f\left(z\right)\ge1$$...
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An inequality with plenty of summations. [closed]
I found a question, but was unable to solve it. The question is:
Let k be a positive integer, and let $x_1, x_2, ..., x_n$ be positive real numbers. Prove that
$$\left(\sum_{i=1}^n \frac{1}{1+x_i}\...
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To prove the following inequalities of positive rational numbers
I have to prove the following inequalities:
$$ 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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$\sum \frac{a}{b+c+d}\le \frac{2\sum a^2}{\sum ab}$ if $\sum a =4$
Let $a, b, c, d$ positive real numbers such that $a+b+c+d=4$. Prove that
$$\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{d+a+b}+\frac{d}{a+b+c}\le
\frac{2(a^2+b^2+c^2+d^2)}{ab+ac+ad+bc+bd+cd}.$$
My idea is ...
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Asking About Other Prove To Use In Solving This Inequality
Let $\ (x,y,z) \ $ be positive real numbers such that $\ \ (x+y)(y+z)(z+x)=8 \ \ $ Prove That
$x^3y^3+y^3z^3+z^3x^3+x^2y^2z^2-4xyz>=0$
Muirhead Sols
First homogenize, by multiplying by $2$ and ...
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Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied
Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied:
$(xy+xz+yz) \left(\dfrac{1}{x+y}+\dfrac{1}{x+z}+\dfrac{1}{y+z}\right)^2 \geq t$.
If there were such an ...
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Prove the inequality $\displaystyle\sum_{s y m} x^{4} y^{2} z \geqslant 2 \sum_{s y m} x^{3} y^{2} z^{2}$
Prove for positive $x,y,z$ the inequality $x^{4} y^{2} z+x^{4} z^{2} y+y^{4} x^{2} z+y^{4} z^{2} x+z^{4} y^{2} x+z^{4} x^{2} y \geqslant 2(x^{3} y^{2} z^{2}+x^{2} y^{3} z^{2}+x^{2} y^{2} z^{3})$.
I ...
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Minimum of $abc$ when $a^3+b^3+c^3\geq a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}.$
Let $a,b,c$ be positive real numbers with $abc=k$ such that the inequality $$a^3+b^3+c^3\geq a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}$$
holds for all $a,b,c$. Find the minimum value of $k$.
I found that $...
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Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression
Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression:
a) $$P=\frac{4a}{b+c-a} + \frac{9b}{c+a-b} + \frac{16c}{a+b-c}$$
b) $$P=\frac{a^3}{2a+bc} + \...
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Problem in normalization inequalities
Let $a,b,c>0$. Prove that: $$\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc} \ge \dfrac{64}{27}(a+b+c)^3$$
First solution:
$\bullet$ Since the inequality is homogeneous, we may normalize $a+b+c=3$, we need to ...
user923177
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Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$
Let $a,b,c>0$:
Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$
My solution:
We have: $\left[\begin{matrix}\frac{1}{x}+\frac{1}{y} \geq \...
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Does Muirhead's Also Work on Cyclic Inequalities?
So I am trying to learn how to solve inequalities, and came across the following in one of the problems I was trying to solve:
$\frac{b^3}{a}$+$\frac{c^3}{b}$+$\frac{d^3}{c}$+$\frac{a^3}{d}\geq ab+bc+...
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Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .
Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .
What I Tried: I have tried by doing $(x+y+z)(x+y+z)=2*2 = 4.$
Also I got $2(xy+yz+xz)+(x^2+...
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For any real positive numbers $a, b, c$, prove that $3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \geq abc(a+b+c)^3$ [duplicate]
My progress is that I applied Hölder’s for this,
$3(a^2b+b^2c+c^2a) \frac{(ab^2+bc^2+ca^2)}{abc} \geq (a+b+c)^3$
whereas $3(a^2b+b^2c+c^2a) \frac{(ab^2+bc^2+ca^2)}{abc} = (1+1+1)(a^2b+b^2c+c^2a)(\...
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Two inequalities with parameters $a,b,c>0$ such that $ca+ab+bc+abc\leq 4$
Let $a,b,c>0$ be such that $bc+ca+ab+abc\leq 4$. Prove the following inequalities:
(a) $8(a^2+b^2+c^2)\geq 3(b+c)(c+a)(a+b)$, and
(b) $\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{a^2b}+\...
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How to prove $\frac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}$?
Give $a,b,c>0$. Prove that: $$\dfrac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}.$$
My direction: (we have the equation if and only if $a=b=c$)
$a^{n+1}+a^nb+a^nc \ge 3a^n\sqrt[3]{abc}$
...
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Proving $\sum_{cyc}\sqrt{a^4+a^2b^2+b^4}\geq \sum_{cyc} a\sqrt{2a^2+bc}$ for non-negative $a$, $b$, $c$
I was trying this question with factorization and other similar methods,
Let $a, b, c \geq 0$. Prove that
$$\begin{array}{c}
\sqrt{a^4+a^2b^2+b^4}+\sqrt{b^4+b^2c^2+c^4}+\sqrt{c^4+c^2a^2+a^4} \\[4pt]
\...
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AM/GM inequalities
I need some help to prove this inequality... I guess one can use Jensen's then AM/GM inequalities.
Let $x_1, x_2, x_3, x_4$ be non- negative real numbers such that
$x_1 x_2 x_3 x_4 =1$.
We want to ...
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Proof of inequality by Muirhead
We have to prove:
$$\frac{\sqrt{pq}}{p+q+2r}+\frac{\sqrt{pr}}{p+r+2q}+\frac{\sqrt{pr}}{p+r+2q}\leq\frac{3}{4}$$
By multiplying it all out we get the following equivalent:
\begin{align*}
4\sum_{cyc}{...
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Inequality 6 deg
For $a,b,c\ge 0$ Prove that $$4(a^2+b^2+c^2)^3\ge 3(a^3+b^3+c^3+3abc)^2$$
My attempt: $$LHS-RHS=12(a-b)^2(b-c)^2(c-a)^2+2(ab+bc+ca)\sum_{sym} a^2(a-b)(a-c)$$
$$+\left(\sum_{sym} a(a-b)(a-c)\right)^2+...
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Prove that if $abc \geq 1$ and $a,b,c > 0$ then $\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$ [duplicate]
Prove that if $abc \geq 1$ and $a,b,c > 0$ then,
$$\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$$
Can anyone help me with this problem?
Tried AM-GM and Cauchy-Schwarz ...
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Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$
For $a,b,c \in (0,1)$ such that $ab+bc+ca=1$ Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$
I tried repleace $1$ by $ab+...
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Inequality involving AM-GM but its wierd [duplicate]
Let a, b, c be positive real numbers. Prove that
$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+ \frac{3\cdot\sqrt[3]{abc}}{a+b+c} \geq 4$
Ohk now i know using AM-GM that
$ \frac{a}{b}+\frac{b}{c}+\frac{c}{...
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Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that $(a + 1)(b + 1)(c + 1) \geq 64$ [duplicate]
Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that
$(a + 1)(b + 1)(c + 1) \geq 64$
Ohk so we are given that $abc=a+b+c$ with that now the inequality becomes $2abc+(a+b+c)+1 \...
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Inequality question.
Let $a,b,c>0$ with $a+b+c=1$. Show that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq 3 + 2\cdot\frac{\left(a^3 + b^3 + c^3\right)}{abc}$$
Ohhhkk. So first off,
\begin{align} a^3 + b^3+ c^3 &...
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Schur inequality
Show that for all positive real numbers $a$, $b$ and $c$ such that $abc=1$, the inequality $a+b+c+2a^4+2b^4+2c^4\ge \dfrac{3}{2}\left(a^2\left(\dfrac{1}{b}+\dfrac{1}{c}\right)+b^2\left(\dfrac{1}{a}+\...
user765855
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Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I ...
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Stuck when transforming and solving this
Given abc=1 ( all positive real numbers). Prove that:
$$\frac ab + \frac bc + \frac ca +3( \frac ba +\frac cb +\frac ac) \ge 2(a +b +c+\frac 1a+ \frac 1b +\frac1c)$$
My attempt:
$$\frac ab + \frac ...
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Proving $\sum\limits_{\rm cyc} 1/(a^2 -b+4) \geq3/4$
Suppose $a,b,c\in\mathbb R^+$ with $a+b+c=3.$ Prove that $$\frac{1}{a^2- b+4}+\frac{1}{b^2-c+4}+\frac{1}{c^2- a+4}\geqslant\frac{3}{4}.$$
I tried various approaches, but nothing seems to work. ...
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Prove $13\sum(a+b)^5 \geq 16 \sum ab(a+b)(4a^2+4b^2+4ab+c^2)$
Given $a,b,c>0$, prove that $$13[(a+b)^5+(b+c)^5+(c+a)^5] \geq 16[ab(a+b)(4a^2+4b^2+4ab+c^2)+bc(b+c)(4b^2+4c^2+4bc+a^2)+ca(c+a)(4c^2+4a^2+4ca+b^2)]$$
I tried subtracting the RHS from the LFS but ...
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Barnard and Child inequality exercise
Prove that,
$$3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)≥abc(a+b+c)^3$$
For positive $a,b,c$
The exercises in this book are making me crazy.
Any help would be appreciated.
My attempts:
I opened the LHS ...
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Another asymmetric inequality $5+\frac{3(a^2+2b^2+c^2)}{(a+b)(b+c)} \geq \frac{9(a+b)(b+c)(c+a)}{(a+b+c)(ab+bc+ca)}$
A while ago I conjectured this inequality and its (little) sister on AOPS. Here is another related inequality in the opposite direction which I strongly suspect is true, although I don't have a proof:
...
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How to prove $a^3+b^3+c^3+d^3+3\left(a+b+c+d\right) \geq 14+2abcd$ when $a^2+b^2+c^2+d^2=4$
This is a problem from AoPS I can't solve:
Let $a,b,c,d\geq0$ with $a^2+b^2+c^2+d^2=4$. How can I prove: $$a^3+b^3+c^3+d^3+3\left(a+b+c+d\right) \geq 14+2abcd$$
My attempt:
I try setting $a=2\cos(x)...
user681002
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For real numbers $x>0, y>0, z>0$ and $x y z=1 .$ prove that $ x^{6}+y^{6}+z^{6} \geq x^{5}+y^{5}+z^{5} $
For real numbers $x>0, y>0, z>0$ and $x y z=1 .$ Prove that $$
x^{6}+y^{6}+z^{6} \geq x^{5}+y^{5}+z^{5} $$
How to show this?
I tried using https://en.wikipedia.org/wiki/Muirhead%...
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Proof verification for $x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9$ (where $xyz=1$ and $x,y,z\in \mathbb{R}^+$)
My teacher has shown me the following problem:
Problem. Let $x,y,z\in \mathbb{R}_+$ with $xyz=1$. Show that:$$x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9.$$
I think I solved the problem using Muirhead's ...
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Proving $\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{c}{c^2+2a^2}\geq 1$ when $a^2+b^2+c^2=a^3+b^3+c^3$
In this answer, @MichaelRozenberg stated the following inequality:
Let $a$, $b$ and $c$ be positive numbers such that $a^3+b^3+c^3=a^2+b^2+c^2.$ Then
$$\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{...
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If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$
The following problem is from CHKMO 2018 Problem 1:
If $ab+bc+ca\ge1$, prove that $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$$
I tried to use Cauchy–Schwarz inequality, by ...
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Prove an inequality : $\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$ without Jensen's inequality
I'm interested in the following problem :
Let $a,b,c>0$ be the variables and $u,v>0$ be constant then we have :
$$\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$$
Rewrrting the ...
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Prove that $3 a^4+ 2 a^3 b - 3 a^2 b^2 - 2 a b^3 + 3 b^4\geq 0$.
Let $a,b\geq 0$. How can I prove $$3 a^4+ 2 a^3 b - 3 a^2 b^2 - 2 a b^3 + 3 b^4\geq 0$$
?
I try using Schur and Muirhead but they didn't work here because Schur is for three variables and Muirhead ...
user708986
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Prove that $\sum_\text{cyc} \frac{ab}{ab+b^2+ca}\le 1$
Hello ladies and gentlemen, here I have another inequality that I am struggling with:
Let $a,b,c>0$ Then
$$\sum_\text{cyc} \frac{ab}{ab+b^2+ca}\le 1.$$
I try to show $$\frac{ab}{ab+b^2+ca}\le\...
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Inequality in 4 variables Vasc's EV
Let $a,b,c,d\geq0$ satisfying $a+b+c+d=4$ . Prove
$$\sqrt{a^3+b^3+c^3+d^3}+2(\sqrt3
-1)abcd\geq\sqrt{3(abc+abd+acd+bcd)}$$
Attempt: $a^3+b^3+c^3+d^3=(a+b+c+d)(a^2+b^2+c^2+d^2-ab-bc-cd-da-ac-bd)+3(...
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How do we prove this inequality?
Suppose $a,b,c > 0$. Prove that
$$\frac{a^2}{b^2} +\frac{b^2}{c^2} + \frac{c^2}{a^2} \geq \frac ab + \frac bc + \frac ca.$$
I've tried multiplying everything by the denominator and then I tried to ...
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Problematic inequality & hint
I would like to ask for hint for proving following inequality:
$$x^3(1+x)+y^3(1+y)+z^3(1+z)\geq \frac{3}{4}(1+x)(1+y)(1+z)$$
for all $x>0$, $y>0$, $z>0$ such that $xyz=1$.
Generally, I tried ...
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Generalization of the AM-GM inequality for three variables
Theorem. Let $a,b,c$ be three non-negative real numbers. Then $$a^6+b^6+c^6\geq 3a^2b^2c^2+\frac12 (a-b)^2 (b-c)^2 (c-a)^2.$$
Remark. This Theorem is a generalization of the AM-GM inequality for ...
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A hard inequality indian olympiad problem [duplicate]
If $x,y,z$ are positive real numbers, prove that:
$\left(x+y+z\right)^2\left(yz+xz+xy\right)^2\le 3\left(y^2 + yz + z^2\right)\left(x^2 + xz + z^2\right)\left(x^2 + xy + y^2\right)$.
I have been ...
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An inequality with four variables
Prove that
$$4[(4;0;0;0)]+12[(2;1;1;0)]\ge 12[(3;1;0;0)]+3[(1;1;1;1)]$$
with $$[(a;b;c;d)]=\frac{1}{4!}\sum_{\sigma\in Sym(4)}x_{\sigma(1)}^{a}\cdot x_{\sigma(2)}^b\cdot x_{\sigma(3)}^c\cdot x_{\...
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How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below? [duplicate]
There was a homework about inequalities (that why I ask a bunch of inequality problems). But I couldn't solve the following:
If $0<a,b,c<1$ and $a+b+c=2$, prove that $8(1-a)(1-b)(1-c)\le abc$
...
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