Questions tagged [muirhead-inequality]

Inequality proof by using the Muirhead inequality.

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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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-6 votes
1 answer
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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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1 vote
0 answers
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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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2 votes
1 answer
82 views

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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  • 388
2 votes
1 answer
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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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1 vote
1 answer
53 views

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 ...
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7 votes
4 answers
135 views

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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2 votes
1 answer
125 views

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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1 answer
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Question on the Proof of Muirhead's Inquality (cited from AoPS) [closed]

In AoPS' (Art of Problem Solving) proof of Muirhead's inequality, how does the below equality work out? The below equation appears to show two expressions (1 and 2), each under the symmetric sum ...
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1 vote
4 answers
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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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2 votes
1 answer
82 views

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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1 vote
3 answers
276 views

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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2 votes
6 answers
133 views

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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3 votes
3 answers
128 views

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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3 votes
2 answers
197 views

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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  • 179
2 votes
1 answer
91 views

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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  • 516
1 vote
3 answers
96 views

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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3 votes
0 answers
132 views

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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0 votes
2 answers
62 views

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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1 vote
2 answers
78 views

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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-1 votes
3 answers
112 views

Inequality question . [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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3 votes
2 answers
82 views

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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2 votes
1 answer
107 views

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}+\...
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6 votes
3 answers
114 views

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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  • 574
1 vote
1 answer
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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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  • 63
1 vote
1 answer
138 views

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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  • 1,062
0 votes
2 answers
72 views

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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3 votes
2 answers
424 views

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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  • 1,121
0 votes
1 answer
135 views

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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  • 8,358
2 votes
1 answer
86 views

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)...
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1 vote
1 answer
62 views

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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  • 797
3 votes
2 answers
119 views

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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  • 1,114
2 votes
2 answers
282 views

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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2 votes
4 answers
221 views

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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  • 2,707
1 vote
2 answers
56 views

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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0 votes
3 answers
95 views

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 ...
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3 votes
3 answers
64 views

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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4 votes
1 answer
336 views

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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  • 603
2 votes
4 answers
132 views

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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  • 1,288
2 votes
3 answers
210 views

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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  • 197
0 votes
3 answers
142 views

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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4 votes
3 answers
234 views

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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-1 votes
1 answer
62 views

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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1 vote
4 answers
168 views

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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  • 2,707
5 votes
4 answers
194 views

How can we not use Muirhead's Inequality for proving the following inequality?

There was a question in the problem set in my math team training homework: Show that $∀a, b, c ∈ \mathbb{R}_{≥0}$ s.t. $a + b + c = 1, 7(ab + bc + ca) ≤ 2 + 9abc.$ I used Muirhead's inequality to ...
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  • 2,707
1 vote
1 answer
51 views

an interesting inequality with Muirhead

If $x,y,z>0$ I have to prove that $\sum\limits_{cyc}^{} \frac { x(x^3 yz+x^2-x y^3 z-yz) }{(1+x y^2)(1+xyz)} \ge 0$ holds. My approach is that from Muirhead's inequality the inequality is true ...
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  • 732
6 votes
1 answer
166 views

Given three positive numbers $a,b,c$. Prove that $\sum\limits_{cyc}\sqrt{\frac{a+b}{b+1}}\geqq3\sqrt[3]{\frac{4\,abc}{3\,abc+1}}$ .

Ji Chen. Given three positive numbers $a, b, c$. Prove that $$\sum\limits_{cyc}\sqrt{\frac{a+ b}{b+ 1}}\geqq 3\sqrt[3]{\frac{4\,abc}{3\,abc+ 1}}$$ Of course, we've to solve it by $uvw$, before that,...
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5 votes
3 answers
136 views

Prove $x+y+z \ge xy+yz+zx$

Given $x,y,z \ge 0$ and $x+y+z=4-xyz$ Then Prove that $$x+y+z \ge xy+yz+zx$$ My try: Letting $x=1-a$, $y=1-b$ and $z=1-c$ we get $$(1-a)+(1-b)+(1-c)+(1-a)(1-b)(1-c)=4$$ $$-(a+b+c)-(a+b+c)+ab+bc+...
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