List questions from rearrangement-inequality

Published on 16 Jan 2024 - 13:30

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Showing $\frac1{ab+4}+\frac1{ac+4}+\frac1{ad+4}+\frac1{bc+4}+\frac1{bd+4}+\frac1{cd+4}\geq\frac65$, for positive $a,b,c,d$ with $ab+bc+cd+da=4$

Published on 06 Mar 2024 - 19:46

83

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Prove that $ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c$

Published on 25 Mar 2024 - 13:30

794

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Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$

Published on 29 Dec 2015 - 18:52

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Find a maximum of: $x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $

Published on 09 Mar 2026 - 19:23

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How to show AM-GM inequality using rearrangement inequality?

Published on 20 Apr 2016 - 12:57

249

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Evaluation Of Maximum Value Without Calculus

Published on 10 Sep 2013 - 9:14

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Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Published on 14 Nov 2013 - 13:35

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Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$

Published on 27 Jan 2014 - 18:34

1.8k

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How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Published on 12 Feb 2014 - 6:04

757

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Proving with AM-GM Inequality

Published on 14 Apr 2017 - 8:04

253

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To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$

Published on 02 May 2017 - 2:33

79

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Proof of an inequality involving 4 algebraic terms

Published on 18 May 2017 - 12:35

183

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Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq\sqrt{x}+\sqrt{y}$

Published on 22 May 2017 - 17:00

179

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Let $a^4+b^4+c^4=3$. Prove that $a^2+b^2+c^2\geq a^2b+b^2c+c^2a$

Published on 16 Jun 2017 - 4:21

List Question

Use of rearrangement inequality

Showing $\frac1{ab+4}+\frac1{ac+4}+\frac1{ad+4}+\frac1{bc+4}+\frac1{bd+4}+\frac1{cd+4}\geq\frac65$, for positive $a,b,c,d$ with $ab+bc+cd+da=4$

Prove that $ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c$

Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$

Find a maximum of: $x^{2016} \cdot y+y^{2016} \cdot z+z^{2016} \cdot x $

How to show AM-GM inequality using rearrangement inequality?

Evaluation Of Maximum Value Without Calculus

Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$

How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Proving with AM-GM Inequality

To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$

Proof of an inequality involving 4 algebraic terms

Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq\sqrt{x}+\sqrt{y}$

Let $a^4+b^4+c^4=3$. Prove that $a^2+b^2+c^2\geq a^2b+b^2c+c^2a$

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