Question

How to prove $\frac{1}{a^3+c^3}+\frac{1}{b^3+c^3}+2\left[a+b-c(1-\sqrt{3})\right]\ge3\left(\frac{1}{\sqrt[3]{a+c}}+\frac{1}{\sqrt[3]{b+c}}\right).$?

score 65 · Answer 1 · 2026-03-26 07:46:32

65

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How to prove $\frac{1}{a^3+c^3}+\frac{1}{b^3+c^3}+2\left[a+b-c(1-\sqrt{3})\right]\ge3\left(\frac{1}{\sqrt[3]{a+c}}+\frac{1}{\sqrt[3]{b+c}}\right).$?

Published on 26 Mar 2026 - 7:46

score 294 · Answer 2 · 2023-08-12 01:19:41

294

Views

Muirhead proof $\implies$ AM-GM proof

Published on 12 Aug 2023 - 1:19

score 108 · Answer 3 · 2023-08-16 16:58:40

108

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Minimum value of $\frac{\sin x}{4} + \frac{1}{\sin x}$? $x \in (0, \pi/2)$

Published on 16 Aug 2023 - 16:58

score 131 · Answer 4 · 2023-08-21 18:00:33

131

Views

Seeking maximum parameter value so that inequality always holds true

Published on 21 Aug 2023 - 18:00

score 60 · Answer 5 · 2026-03-25 06:29:02

60

Views

Can we prove Am-GM Inequality using Karamata’s Inequality?

Published on 25 Mar 2026 - 6:29

score 76 · Answer 6 · 2023-08-23 15:27:58

76

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Logarithmic inequality calculation

Published on 23 Aug 2023 - 15:27

score 73 · Answer 7 · 2023-08-24 10:13:37

73

Views

Proving an inequality using AM/HM or other inequalities

Published on 24 Aug 2023 - 10:13

score 115 · Answer 8 · 2023-09-17 15:19:49

115

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Find the min value of $\frac{xy}{z}+\frac{xz}{y}+\frac{yz}{x}$ when ${x^2} + {y^2} + {z^2} = 1$.

Published on 17 Sep 2023 - 15:19

score 45 · Answer 9 · 2023-09-18 02:31:02

Show that $\sqrt{\frac{a}{b+c}+ \frac{b}{c+a}}+ \sqrt{\frac{b}{c+a}+ \frac{c}{a+b}} + \sqrt{\frac{c}{a+b}+ \frac{a}{b+c}} \ge 3.$

score 26 · Answer 10 · 2026-03-26 07:40:34

26

Views

Homogeneous Inequality problem unsolved

Published on 26 Mar 2026 - 7:40

score 61 · Answer 11 · 2023-09-20 18:59:33

61

Views

Prove $\frac{a+b}{c} + \frac{b+c}{a}+\frac{a+c}{b} + 6 \ge 2\sqrt2(\sqrt\frac{1-a}{a} + \sqrt\frac{1-b}{b} + \sqrt\frac{1-c}{c})$ when $a + b + c = 1$

Published on 20 Sep 2023 - 18:59

score 51 · Answer 12 · 2026-03-26 02:54:26

51

Views

inequality about elementary symmetric polynomials with real variables

Published on 26 Mar 2026 - 2:54

score 119 · Answer 13 · 2026-03-28 06:40:39

119

Views

What is $(3 n+1)+(3 n+2)+\ldots+(5 n)+(5 n+1)=$ equal to?

Published on 28 Mar 2026 - 6:40

score 157 · Answer 14 · 2026-03-26 07:46:33

157

Views

Prove $\frac{a}{\sqrt{5-4bc}} + \frac{b}{\sqrt{5-4ac}} + \frac{c}{\sqrt{5-4ab}} \ge 1$ With the condition $a+b+c=3$

Published on 26 Mar 2026 - 7:46

score 121 · Answer 15 · 2023-09-27 12:01:42

121

Views

Prove $\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\geq \sqrt{6x+6y+6z}$

Published on 27 Sep 2023 - 12:01

List Question

How to prove $\frac{1}{a^3+c^3}+\frac{1}{b^3+c^3}+2\left[a+b-c(1-\sqrt{3})\right]\ge3\left(\frac{1}{\sqrt[3]{a+c}}+\frac{1}{\sqrt[3]{b+c}}\right).$?

Muirhead proof $\implies$ AM-GM proof

Minimum value of $\frac{\sin x}{4} + \frac{1}{\sin x}$? $x \in (0, \pi/2)$

Seeking maximum parameter value so that inequality always holds true

Can we prove Am-GM Inequality using Karamata’s Inequality?

Logarithmic inequality calculation

Proving an inequality using AM/HM or other inequalities

Find the min value of $\frac{xy}{z}+\frac{xz}{y}+\frac{yz}{x}$ when ${x^2} + {y^2} + {z^2} = 1$.

Show that $\sqrt{\frac{a}{b+c}+ \frac{b}{c+a}}+ \sqrt{\frac{b}{c+a}+ \frac{c}{a+b}} + \sqrt{\frac{c}{a+b}+ \frac{a}{b+c}} \ge 3.$

Homogeneous Inequality problem unsolved

Prove $\frac{a+b}{c} + \frac{b+c}{a}+\frac{a+c}{b} + 6 \ge 2\sqrt2(\sqrt\frac{1-a}{a} + \sqrt\frac{1-b}{b} + \sqrt\frac{1-c}{c})$ when $a + b + c = 1$

inequality about elementary symmetric polynomials with real variables

What is $(3 n+1)+(3 n+2)+\ldots+(5 n)+(5 n+1)=$ equal to?

Prove $\frac{a}{\sqrt{5-4bc}} + \frac{b}{\sqrt{5-4ac}} + \frac{c}{\sqrt{5-4ab}} \ge 1$ With the condition $a+b+c=3$

Prove $\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\geq \sqrt{6x+6y+6z}$

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