List questions from a-m-g-m-inequality

Published on 02 Oct 2017 - 0:43

95

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Normal AM-GM inequality

Published on 02 Oct 2017 - 16:14

26

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Bounding $c_1c_2...c_k$ if $c_1+c_2+...+c_k<=n$ and $c_i\geq 0$.

Published on 03 Oct 2017 - 3:28

78

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What is the minimum value of?

Published on 03 Oct 2017 - 17:42

150

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Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$

Published on 25 Mar 2026 - 21:51

1.3k

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Tighter inequality than Cauchy - Schwarz inequality

Published on 05 Oct 2017 - 12:55

121

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How to prove $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}+ \frac{3\sqrt[3]{abc}}{a+b+c}\geq 4 $ using Middle School methods?

Published on 05 Oct 2017 - 16:33

190

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To prove $abc\leq\frac 18$

Published on 06 Oct 2017 - 12:58

487

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How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$?

Published on 25 Mar 2026 - 13:57

421

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Let $a,b,c$ be positive real numbers then prove the inequality $\frac{abc}{(a+1)(a+b)(b+c)(c+16)}≤\frac1{81}$

Published on 06 Oct 2017 - 16:10

174

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Show that $(a+2b+\frac{2}{a+1})(b+2a+\frac{2}{b+1})\geqslant16$ if $ ab\geqslant1$

Published on 07 Oct 2017 - 7:52

67

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Can someone tell me what should i do next? Should I use the inequality between Arithmetic and Geometric mean?

Published on 26 Mar 2026 - 19:18

475

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Can the inequality $\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} \geq 6$ be proved with differentiation?

Published on 08 Oct 2017 - 10:50

214

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Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Published on 25 Mar 2026 - 21:51

254

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Minimizing $\frac{d}{a^3+4}+\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}$ for nonnegative parameters with $a+b+c+d=4$

Published on 08 Oct 2017 - 19:21

List Question

Another Hard Inequality

Normal AM-GM inequality

Bounding $c_1c_2...c_k$ if $c_1+c_2+...+c_k<=n$ and $c_i\geq 0$.

What is the minimum value of?

Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$

Tighter inequality than Cauchy - Schwarz inequality

How to prove $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}+ \frac{3\sqrt[3]{abc}}{a+b+c}\geq 4 $ using Middle School methods?

To prove $abc\leq\frac 18$

How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$?

Let $a,b,c$ be positive real numbers then prove the inequality $\frac{abc}{(a+1)(a+b)(b+c)(c+16)}≤\frac1{81}$

Show that $(a+2b+\frac{2}{a+1})(b+2a+\frac{2}{b+1})\geqslant16$ if $ ab\geqslant1$

Can someone tell me what should i do next? Should I use the inequality between Arithmetic and Geometric mean?

Can the inequality $\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} \geq 6$ be proved with differentiation?

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Minimizing $\frac{d}{a^3+4}+\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}$ for nonnegative parameters with $a+b+c+d=4$

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