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minimum of an expression using inequality

score 39 · Answer 1 · 2023-08-01 16:39:04

39

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minimum of an expression using inequality

Published on 01 Aug 2023 - 16:39

score 355 · Answer 2 · 2023-08-02 01:30:39

355

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Very hard AM - GM inequality question: $\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$

Published on 02 Aug 2023 - 1:30

score 93 · Answer 3 · 2023-08-05 00:40:15

93

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Prove $\frac{1}{\sqrt[3]{xy}}+3\ge 2\sqrt{3}\left(\frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{y+2}}\right).$

Published on 05 Aug 2023 - 0:40

score 121 · Answer 4 · 2023-08-05 16:49:48

121

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Let $\alpha\cot x+\beta\cot y+\gamma\cot z=\psi$ then find the minimum value of $\cot^2x+\cot^2y+\cot^2z$

Published on 05 Aug 2023 - 16:49

score 74 · Answer 5 · 2026-03-25 12:28:17

74

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How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$

Published on 25 Mar 2026 - 12:28

score 29 · Answer 6 · 2023-08-06 04:37:57

29

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Proving $\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1$ for positive $a$, $b$, $c$ with $abc=1$with AM-GM

Published on 06 Aug 2023 - 4:37

score 74 · Answer 7 · 2026-03-28 05:23:17

74

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How to prove $\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$?

Published on 28 Mar 2026 - 5:23

score 96 · Answer 8 · 2026-03-25 17:21:45

96

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Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

Published on 25 Mar 2026 - 17:21

score 101 · Answer 9 · 2023-08-07 10:56:11

101

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Exercise 1.26 from Riley's "Mathematical Methods for Physics and Engineering" concerning proof by induction

Published on 07 Aug 2023 - 10:56

score 119 · Answer 10 · 2026-03-26 02:55:57

119

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Finding $\small{\min\limits_{ab+bc+ca=3}\dfrac{1}{4a+bc}+\dfrac{1}{4b+ca}+\dfrac{1}{4c+ab}.}$

Published on 26 Mar 2026 - 2:55

score 54 · Answer 11 · 2023-08-07 14:18:48

54

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How to prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{16(ab+bc+ca)^2}\ge \frac{1}{a+b+c}.\left(\dfrac{1}{\sqrt{abc}}-3\right)$

Published on 07 Aug 2023 - 14:18

score 49 · Answer 12 · 2023-08-08 02:48:35

49

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Find the maximum $P=\frac{bc}{\sqrt{(1-a)^3(a+1)}}+\frac{ca}{\sqrt{(1-b)^3(b+1)}}+\frac{ab}{\sqrt{(1-c)^3(c+1)}},$ when $a+b+c=1$

Published on 08 Aug 2023 - 2:48

score 63 · Answer 13 · 2023-08-09 05:58:07

63

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If $a,b,c\in R: a+b+c=3,$ find maximum $\min\{a,b,c\}+ \sqrt[3]{abc}-\max\{a,b,c\}..$

Published on 09 Aug 2023 - 5:58

score 102 · Answer 14 · 2023-08-10 03:44:27

102

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How to prove $a+b+c+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}+\sqrt[3]{abc}\ge 7\sqrt[7]{\frac{abc(a+b)(b+c)(c+a)(a+b+c)}{24}}$?

Published on 10 Aug 2023 - 3:44

score 86 · Answer 15 · 2026-03-25 14:36:39

86

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Help to prove $\frac{1}{4a^2-8ac+13c^2}+\frac{1}{4b^2-8bc+13c^2}\ge \frac{2}{(a+b+c)^2}$

Published on 25 Mar 2026 - 14:36

List Question

minimum of an expression using inequality

Very hard AM - GM inequality question: $\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$

Prove $\frac{1}{\sqrt[3]{xy}}+3\ge 2\sqrt{3}\left(\frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{y+2}}\right).$

Let $\alpha\cot x+\beta\cot y+\gamma\cot z=\psi$ then find the minimum value of $\cot^2x+\cot^2y+\cot^2z$

How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$

Proving $\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\leq 1$ for positive $a$, $b$, $c$ with $abc=1$with AM-GM

How to prove $\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$?

Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

Exercise 1.26 from Riley's "Mathematical Methods for Physics and Engineering" concerning proof by induction

Finding $\small{\min\limits_{ab+bc+ca=3}\dfrac{1}{4a+bc}+\dfrac{1}{4b+ca}+\dfrac{1}{4c+ab}.}$

How to prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{16(ab+bc+ca)^2}\ge \frac{1}{a+b+c}.\left(\dfrac{1}{\sqrt{abc}}-3\right)$

Find the maximum $P=\frac{bc}{\sqrt{(1-a)^3(a+1)}}+\frac{ca}{\sqrt{(1-b)^3(b+1)}}+\frac{ab}{\sqrt{(1-c)^3(c+1)}},$ when $a+b+c=1$

If $a,b,c\in R: a+b+c=3,$ find maximum $\min\{a,b,c\}+ \sqrt[3]{abc}-\max\{a,b,c\}..$

How to prove $a+b+c+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}+\sqrt[3]{abc}\ge 7\sqrt[7]{\frac{abc(a+b)(b+c)(c+a)(a+b+c)}{24}}$?

Help to prove $\frac{1}{4a^2-8ac+13c^2}+\frac{1}{4b^2-8bc+13c^2}\ge \frac{2}{(a+b+c)^2}$

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