Question

How to prove $\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)}$?

score 145 · Answer 1 · 2026-03-25 16:20:53

145

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How to prove $\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)}$?

Published on 25 Mar 2026 - 16:20

score 187 · Answer 2 · 2026-03-25 16:13:45

187

Views

Prove $\sqrt{a+4b}+\sqrt{b+4c}+\sqrt{c+4a}\le 3\sqrt{abc+4}$ when $ a^2+b^2+c^2=3.$

Published on 25 Mar 2026 - 16:13

score 102 · Answer 3 · 2026-03-25 16:20:53

102

Views

How to prove $\frac{1}{\sqrt{3a^2+13b}}+\frac{1}{\sqrt{3b^2+13c}}+\frac{1}{\sqrt{3c^2+13a}} \ge \frac{3}{4}$?

Published on 25 Mar 2026 - 16:20

score 191 · Answer 4 · 2026-03-25 16:20:54

191

Views

Prove that $\sum\limits_{\mathrm{cyc}}\sqrt{\frac{xy}{z}}+6\sqrt{xyz}\ge \sum\limits_{\mathrm{cyc}} \sqrt{3x^3+6xyz}$

Published on 25 Mar 2026 - 16:20

score 132 · Answer 5 · 2026-03-25 16:20:36

132

Views

Find a nice proof for an elegant problem.

Published on 25 Mar 2026 - 16:20

score 89 · Answer 6 · 2026-03-25 16:13:52

89

Views

Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

Published on 25 Mar 2026 - 16:13

score 83 · Answer 7 · 2026-03-25 17:38:59

83

Views

Find proof for sum of cyclic square root inequality where $a+b+c=3.$

Published on 25 Mar 2026 - 17:38

score 89 · Answer 8 · 2026-03-25 16:20:51

89

Views

Determining best constant $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+3k\ge (k+1)\cdot\frac{x+y+z}{\sqrt[3]{xyz}}.$

Published on 25 Mar 2026 - 16:20

score 80 · Answer 9 · 2026-03-25 16:13:49

80

Views

Help to prove $\frac{a}{(b^2+3)^2}+\frac{b}{(c^2+3)^2}+\frac{c}{(a^2+3)^2}\ge\frac{3}{16}$ for $a+b+c=3.$

Published on 25 Mar 2026 - 16:13

score 180 · Answer 10 · 2026-03-25 17:38:59

180

Views

Prove this inequality $a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$

Published on 25 Mar 2026 - 17:38

score 635 · Answer 11 · 2026-03-25 16:20:46

635

Views

cyclic three variable inequality

Published on 25 Mar 2026 - 16:20

score 204 · Answer 12 · 2026-05-03 17:45:23

204

Views

Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$

Published on 03 May 2026 - 17:45

score 2.5k · Answer 13 · 2026-03-25 16:13:49

2.5k

Views

For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)

Published on 25 Mar 2026 - 16:13

score 619 · Answer 14 · 2026-03-25 16:14:02

619

Views

A cyclic inequality $\sum\limits_{cyc}{\sqrt{3x+\frac{1}{y}}}\geqslant 6$

Published on 25 Mar 2026 - 16:14

score 99 · Answer 15 · 2026-03-25 16:20:34

99

Views

Given $a,b,c$. Prove inequality

Published on 25 Mar 2026 - 16:20

List Question

How to prove $\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)}$?

Prove $\sqrt{a+4b}+\sqrt{b+4c}+\sqrt{c+4a}\le 3\sqrt{abc+4}$ when $ a^2+b^2+c^2=3.$

How to prove $\frac{1}{\sqrt{3a^2+13b}}+\frac{1}{\sqrt{3b^2+13c}}+\frac{1}{\sqrt{3c^2+13a}} \ge \frac{3}{4}$?

Prove that $\sum\limits_{\mathrm{cyc}}\sqrt{\frac{xy}{z}}+6\sqrt{xyz}\ge \sum\limits_{\mathrm{cyc}} \sqrt{3x^3+6xyz}$

Find a nice proof for an elegant problem.

Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

Find proof for sum of cyclic square root inequality where $a+b+c=3.$

Determining best constant $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+3k\ge (k+1)\cdot\frac{x+y+z}{\sqrt[3]{xyz}}.$

Help to prove $\frac{a}{(b^2+3)^2}+\frac{b}{(c^2+3)^2}+\frac{c}{(a^2+3)^2}\ge\frac{3}{16}$ for $a+b+c=3.$

Prove this inequality $a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$

cyclic three variable inequality

Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$

For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)

A cyclic inequality $\sum\limits_{cyc}{\sqrt{3x+\frac{1}{y}}}\geqslant 6$

Given $a,b,c$. Prove inequality

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