Question

How to prove $\frac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}$?

score 142 · Answer 1 · 2026-04-16 03:17:22

142

Views

How to prove $\frac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}$?

Published on 16 Apr 2026 - 3:17

score 340 · Answer 2 · 2026-04-16 07:37:32

340

Views

Two inequalities with parameters $a,b,c>0$ such that $ca+ab+bc+abc\leq 4$

Published on 16 Apr 2026 - 7:37

score 125 · Answer 3 · 2026-04-15 22:44:02

125

Views

For any real positive numbers $a, b, c$, prove that $3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \geq abc(a+b+c)^3$

Published on 15 Apr 2026 - 22:44

score 192 · Answer 4 · 2026-04-16 10:14:00

192

Views

Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .

Published on 16 Apr 2026 - 10:14

score 384 · Answer 5 · 2026-04-12 04:30:19

384

Views

Does Muirhead's Also Work on Cyclic Inequalities?

Published on 12 Apr 2026 - 4:30

score 155 · Answer 6 · 2026-03-25 19:12:07

155

Views

Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$

Published on 25 Mar 2026 - 19:12

score 105 · Answer 7 · 2026-03-25 19:17:44

105

Views

Problem in normalization inequalities

Published on 25 Mar 2026 - 19:17

score 245 · Answer 8 · 2026-03-25 19:11:45

245

Views

Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression

Published on 25 Mar 2026 - 19:11

score 165 · Answer 9 · 2026-03-25 19:11:44

165

Views

Minimum of $abc$ when $a^3+b^3+c^3\geq a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}.$

Published on 25 Mar 2026 - 19:11

score 86 · Answer 10 · 2026-03-25 19:12:01

86

Views

Prove the inequality $\displaystyle\sum_{s y m} x^{4} y^{2} z \geqslant 2 \sum_{s y m} x^{3} y^{2} z^{2}$

Published on 25 Mar 2026 - 19:12

score 70 · Answer 11 · 2026-03-25 19:12:04

70

Views

Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied

Published on 25 Mar 2026 - 19:12

score 100 · Answer 12 · 2026-05-11 05:19:11

100

Views

Asking About Other Prove To Use In Solving This Inequality

Published on 11 May 2026 - 5:19

score 115 · Answer 13 · 2026-03-25 19:11:48

115

Views

$\sum \frac{a}{b+c+d}\le \frac{2\sum a^2}{\sum ab}$ if $\sum a =4$

Published on 25 Mar 2026 - 19:11

score 111 · Answer 14 · 2026-03-25 19:12:08

111

Views

To prove the following inequalities of positive rational numbers

Published on 25 Mar 2026 - 19:12

score 90 · Answer 15 · 2026-03-25 19:10:52

90

Views

An inequality with plenty of summations.

Published on 25 Mar 2026 - 19:10

List Question

How to prove $\frac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}$?

Two inequalities with parameters $a,b,c>0$ such that $ca+ab+bc+abc\leq 4$

For any real positive numbers $a, b, c$, prove that $3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \geq abc(a+b+c)^3$

Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .

Does Muirhead's Also Work on Cyclic Inequalities?

Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$

Problem in normalization inequalities

Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression

Minimum of $abc$ when $a^3+b^3+c^3\geq a\sqrt{b+c}+b\sqrt{c+a}+c\sqrt{a+b}.$

Prove the inequality $\displaystyle\sum_{s y m} x^{4} y^{2} z \geqslant 2 \sum_{s y m} x^{3} y^{2} z^{2}$

Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied

Asking About Other Prove To Use In Solving This Inequality

$\sum \frac{a}{b+c+d}\le \frac{2\sum a^2}{\sum ab}$ if $\sum a =4$

To prove the following inequalities of positive rational numbers

An inequality with plenty of summations.

Trending Questions

Popular # Hahtags

Popular Questions