Question

How to prove $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.$

score 174 · Answer 1 · 2026-03-26 07:35:35

174

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How to prove $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.$

Published on 26 Mar 2026 - 7:35

score 133 · Answer 2 · 2023-11-09 13:26:57

133

Views

How to prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}$?

Published on 09 Nov 2023 - 13:26

score 61 · Answer 3 · 2023-11-10 23:49:43

61

Views

4-variable inequality

Published on 10 Nov 2023 - 23:49

score 104 · Answer 4 · 2023-11-12 17:03:55

104

Views

Prove an inequality, knowing that $x,y,z$ are strictly positive real numbers

Published on 12 Nov 2023 - 17:03

score 137 · Answer 5 · 2023-11-13 01:55:26

137

Views

Inequality $\frac{a}{\sqrt{b}} + \frac{b}{\sqrt{a}} \ge \sqrt[4]{8}$ on the unit circle

Published on 13 Nov 2023 - 1:55

score 51 · Answer 6 · 2023-11-15 17:14:31

51

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$n\geq 2$ and $\sum_{i=1}^{n}x_i^2=1 \Rightarrow \sum_{i=1}^{n}\frac{|x_i|}{\sqrt{1-x_i^2}}\geq 2$

Published on 15 Nov 2023 - 17:14

score 137 · Answer 7 · 2026-03-26 04:38:07

137

Views

Given that $a,b,c>0$ and $abc=1$, prove that $a+b+c+\frac{3}{ab+bc+ca} \geq 4$

Published on 26 Mar 2026 - 4:38

score 105 · Answer 8 · 2026-03-26 09:34:16

105

Views

prove that $\frac{1}{a(b + c)^{3}} + \frac{1}{b(c + a)^{3}} + \frac{1}{c(a + b)^{3}} \ge \frac{27}{8}$ given $a^{2} + b^{2} + c^{2} = 1$

Published on 26 Mar 2026 - 9:34

score 127 · Answer 9 · 2026-03-26 08:02:31

127

Views

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 26 Mar 2026 - 8:02

score 91 · Answer 10 · 2026-03-26 07:35:30

91

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Prove $\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$

Published on 26 Mar 2026 - 7:35

score 105 · Answer 11 · 2023-11-28 07:05:07

105

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For $x,y,z$ positive real numbers $3x+4y+7z=1$. Find minumum integer value of $1/x+1/y+1/z$

Published on 28 Nov 2023 - 7:05

score 69 · Answer 12 · 2023-11-28 14:44:03

69

Views

the maximum value of $xy/(x+2y)$, given $x^2+4y^2+2xy=1, x>0, y>0$

Published on 28 Nov 2023 - 14:44

score 67 · Answer 13 · 2023-11-29 02:02:49

67

Views

Maximization question using inequalities (Cauchy-Schwarz, AM-GM)

Published on 29 Nov 2023 - 2:02

score 123 · Answer 14 · 2023-11-29 23:54:46

123

Views

Prove $3a+4(b+c)+\sqrt[3]{abc}\ge 2\left( \sqrt{b\left(4b+5a\right) }+ \sqrt{c\left(4c+5a\right) }\right).$

Published on 29 Nov 2023 - 23:54

score 123 · Answer 15 · 2023-11-30 06:25:19

123

Views

How to prove $\sqrt{(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)}\ge 3+\frac{1}{3}\sum_{cyc}\left(\frac{b-c}{b+c}\right)^2.$

Published on 30 Nov 2023 - 6:25

List Question

How to prove $\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\ge \frac{\sqrt{abc+4}+4\sqrt{ab+bc+ca+4}}{2}.$

How to prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}$?

4-variable inequality

Prove an inequality, knowing that $x,y,z$ are strictly positive real numbers

Inequality $\frac{a}{\sqrt{b}} + \frac{b}{\sqrt{a}} \ge \sqrt[4]{8}$ on the unit circle

$n\geq 2$ and $\sum_{i=1}^{n}x_i^2=1 \Rightarrow \sum_{i=1}^{n}\frac{|x_i|}{\sqrt{1-x_i^2}}\geq 2$

Given that $a,b,c>0$ and $abc=1$, prove that $a+b+c+\frac{3}{ab+bc+ca} \geq 4$

prove that $\frac{1}{a(b + c)^{3}} + \frac{1}{b(c + a)^{3}} + \frac{1}{c(a + b)^{3}} \ge \frac{27}{8}$ given $a^{2} + b^{2} + c^{2} = 1$

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 $a+b+c+abc=4.$

Prove $\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$

For $x,y,z$ positive real numbers $3x+4y+7z=1$. Find minumum integer value of $1/x+1/y+1/z$

the maximum value of $xy/(x+2y)$, given $x^2+4y^2+2xy=1, x>0, y>0$

Maximization question using inequalities (Cauchy-Schwarz, AM-GM)

Prove $3a+4(b+c)+\sqrt[3]{abc}\ge 2\left( \sqrt{b\left(4b+5a\right) }+ \sqrt{c\left(4c+5a\right) }\right).$

How to prove $\sqrt{(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)}\ge 3+\frac{1}{3}\sum_{cyc}\left(\frac{b-c}{b+c}\right)^2.$

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