List questions from muirhead-inequality

68

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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} $

Published on 07 Jan 2020 - 13:17

88

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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 $a^2+b^2+c^2+d^2=4$

Published on 15 Jan 2020 - 23:30

149

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Another asymmetric inequality $5+\frac{3(a^2+2b^2+c^2)}{(a+b)(b+c)} \geq \frac{9(a+b)(b+c)(c+a)}{(a+b+c)(ab+bc+ca)}$

Published on 04 Feb 2020 - 16:43

454

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Barnard and Child inequality exercise

Published on 19 Feb 2020 - 4:06

75

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Prove $13\sum(a+b)^5 \geq 16 \sum ab(a+b)(4a^2+4b^2+4ab+c^2)$

Published on 04 Apr 2020 - 4:29

143

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Proving $\sum\limits_{\rm cyc} 1/(a^2 -b+4) \geq3/4$

Published on 10 Apr 2020 - 5:10

73

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Stuck when transforming and solving this

Published on 01 May 2020 - 13:10

133

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Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

Published on 05 May 2020 - 7:36

157

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Schur inequality

Published on 21 Mar 2026 - 18:51

99

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Inequality question.

Published on 21 Mar 2026 - 22:49

129

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Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that $(a + 1)(b + 1)(c + 1) \geq 64$

Published on 01 Jun 2020 - 16:03

87

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Inequality involving AM-GM but its wierd

Published on 03 Jun 2020 - 4:46

70

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Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Published on 07 Jun 2020 - 13:35

140

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Prove that if $abc \geq 1$ and $a,b,c > 0$ then $\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$

Published on 17 Jun 2020 - 15:04

105

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Inequality 6 deg

Published on 24 Mar 2026 - 20:27

List Question

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 prove $a^3+b^3+c^3+d^3+3\left(a+b+c+d\right) \geq 14+2abcd$ when $a^2+b^2+c^2+d^2=4$

Another asymmetric inequality $5+\frac{3(a^2+2b^2+c^2)}{(a+b)(b+c)} \geq \frac{9(a+b)(b+c)(c+a)}{(a+b+c)(ab+bc+ca)}$

Barnard and Child inequality exercise

Prove $13\sum(a+b)^5 \geq 16 \sum ab(a+b)(4a^2+4b^2+4ab+c^2)$

Proving $\sum\limits_{\rm cyc} 1/(a^2 -b+4) \geq3/4$

Stuck when transforming and solving this

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

Schur inequality

Inequality question.

Let $a,b,c>0$ with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that $(a + 1)(b + 1)(c + 1) \geq 64$

Inequality involving AM-GM but its wierd

Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Prove that if $abc \geq 1$ and $a,b,c > 0$ then $\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$

Inequality 6 deg

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