Question

Prove $\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$

score 88 · Answer 1 · 2023-11-18 15:16:36

88

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Prove $\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$

Published on 18 Nov 2023 - 15:16

score 72 · Answer 2 · 2023-11-20 00:09:57

72

Views

Find minimal value $P=(a+b)(b+c)(c+a)$ if $ab+bc+ca=\sqrt{a}+\sqrt{b}+\sqrt{c}>0.$

Published on 20 Nov 2023 - 0:09

score 126 · Answer 3 · 2023-11-21 07:21:32

126

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Prove $\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} },$ if $a+b+c+abc=4.$

Published on 21 Nov 2023 - 7:21

score 90 · Answer 4 · 2023-11-23 05:59:53

90

Views

How to prove $\sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\le a+b+c+\frac{1}{2}abc+\sqrt{2}$ for $ab+bc+ca=1.$

Published on 23 Nov 2023 - 5:59

score 267 · Answer 5 · 2026-03-25 18:49:40

267

Views

How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Published on 25 Mar 2026 - 18:49

score 140 · Answer 6 · 2026-03-25 11:16:16

140

Views

Prove $\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{2(ab+bc+ca)+12}$ when $a+b+c=3.$

Published on 25 Mar 2026 - 11:16

score 84 · Answer 7 · 2023-11-25 13:56:48

84

Views

If $ab+bc+ca=3,$ prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$

Published on 25 Nov 2023 - 13:56

score 107 · Answer 8 · 2023-11-26 10:39:35

107

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Prove $\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}$ for $ab+bc+ca=3.$

Published on 26 Nov 2023 - 10:39

score 159 · Answer 9 · 2023-11-27 11:13:43

159

Views

Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$

Published on 27 Nov 2023 - 11:13

score 56 · Answer 10 · 2023-11-28 03:36:06

56

Views

Prove $\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}$ if $a^2+b^2+c^2=3.$

Published on 28 Nov 2023 - 3:36

score 71 · Answer 11 · 2023-11-28 16:04:08

71

Views

Prove $\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}; a+b+c=3.$

Published on 28 Nov 2023 - 16:04

score 81 · Answer 12 · 2023-12-01 06:00:52

81

Views

Prove $\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}$

Published on 01 Dec 2023 - 6:00

score 72 · Answer 13 · 2023-12-01 06:44:01

72

Views

Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $

Published on 01 Dec 2023 - 6:44

score 164 · Answer 14 · 2023-12-01 13:58:57

164

Views

Inequality $\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$

Published on 01 Dec 2023 - 13:58

score 94 · Answer 15 · 2023-12-03 00:54:07

94

Views

How to solve $\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}$?

Published on 03 Dec 2023 - 0:54

List Question

Prove $\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$

Find minimal value $P=(a+b)(b+c)(c+a)$ if $ab+bc+ca=\sqrt{a}+\sqrt{b}+\sqrt{c}>0.$

Prove $\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} },$ if $a+b+c+abc=4.$

How to prove $\sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\le a+b+c+\frac{1}{2}abc+\sqrt{2}$ for $ab+bc+ca=1.$

How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Prove $\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{2(ab+bc+ca)+12}$ when $a+b+c=3.$

If $ab+bc+ca=3,$ prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$

Prove $\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}$ for $ab+bc+ca=3.$

Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$

Prove $\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}$ if $a^2+b^2+c^2=3.$

Prove $\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}; a+b+c=3.$

Prove $\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}$

Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $

Inequality $\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$

How to solve $\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}$?

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