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 140 · Answer 1 · 2026-03-25 19:12:05

140

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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 25 Mar 2026 - 19:12

score 84 · Answer 2 · 2026-03-26 22:50:31

84

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How precise is an inequality

Published on 26 Mar 2026 - 22:50

score 64 · Answer 3 · 2020-08-14 14:44:35

64

Views

Prove that an element $(a,b,c)$ belongs to this set such that it satisfies a certain condition

Published on 14 Aug 2020 - 14:44

score 124 · Answer 4 · 2020-08-17 15:48:51

124

Views

Given positive $x,y$ such that $x > y$ and $\sqrt{x} \sqrt{y}(x-y) = x+y $, find minimum $(x+y)$

Published on 17 Aug 2020 - 15:48

score 159 · Answer 5 · 2026-03-27 17:57:35

159

Views

Prove $ac\cos B+ab\cos C-bc\cos A-a^2 \le \frac{c^2}{8\cos^2(90^\circ-C)}$ for $\triangle ABC$

Published on 27 Mar 2026 - 17:57

score 263 · Answer 6 · 2020-08-20 21:27:28

263

Views

Is it true that $\sum_{i=1}^n ( nGx_i^{G} + G^{x_i}) \ge n^2G + G^2n$, for all $x_i>0$, where $G=\prod_{j=1}^nx_j$?

Published on 20 Aug 2020 - 21:27

score 307 · Answer 7 · 2020-08-21 06:07:09

307

Views

AM-GM inequality proof for nonnegative real numbers x, a, b, c

Published on 21 Aug 2020 - 6:07

score 338 · Answer 8 · 2026-03-25 19:10:48

338

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

Published on 25 Mar 2026 - 19:10

score 1.1k · Answer 9 · 2020-08-22 15:25:36

1.1k

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Find maximum value of $(1 + \sin x)(1 + \cos x)$.

Published on 22 Aug 2020 - 15:25

score 99 · Answer 10 · 2020-08-22 22:31:38

99

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Proving $\sum_{cyc}\frac{(a-1)(c+1)}{1+bc+c}\geq 0$ for positive $a$, $b$, $c$ with $abc=1$.

Published on 22 Aug 2020 - 22:31

score 96 · Answer 11 · 2026-03-25 21:49:47

96

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prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

Published on 25 Mar 2026 - 21:49

score 376 · Answer 12 · 2020-08-25 19:43:07

376

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If $a, b, c$ are positive numbers with $abc=1$, then $(a+\frac{1}{b})^2+(b+\frac{1}{c})^2+(c+\frac{1}{a})^2\ge 3(a+b+c+1)$.

Published on 25 Aug 2020 - 19:43

score 144 · Answer 13 · 2020-08-27 11:36:15

144

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On Proving $ a^6 + b^6 + c^ 6 − 3a ^2 b^ 2 c^ 2 + 2(a^ 2 + bc)(b ^2 + ca)(c^ 2 + ab) ≥ 0 $ where $a,b,c$ are real numbers.

Published on 27 Aug 2020 - 11:36

score 240 · Answer 14 · 2020-08-27 18:38:53

240

Views

Show that $\frac 1x \ge 3 - 2\sqrt{x}$ for all positive real numbers $x$.

Published on 27 Aug 2020 - 18:38

score 69 · Answer 15 · 2026-03-25 21:47:33

69

Views

Doubt:regarding proof $\sum_{cyc}a^{2/3}b\le 3$

Published on 25 Mar 2026 - 21:47

List Question

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

How precise is an inequality

Prove that an element $(a,b,c)$ belongs to this set such that it satisfies a certain condition

Given positive $x,y$ such that $x > y$ and $\sqrt{x} \sqrt{y}(x-y) = x+y $, find minimum $(x+y)$

Prove $ac\cos B+ab\cos C-bc\cos A-a^2 \le \frac{c^2}{8\cos^2(90^\circ-C)}$ for $\triangle ABC$

Is it true that $\sum_{i=1}^n ( nGx_i^{G} + G^{x_i}) \ge n^2G + G^2n$, for all $x_i>0$, where $G=\prod_{j=1}^nx_j$?

AM-GM inequality proof for nonnegative real numbers x, a, b, c

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

Find maximum value of $(1 + \sin x)(1 + \cos x)$.

Proving $\sum_{cyc}\frac{(a-1)(c+1)}{1+bc+c}\geq 0$ for positive $a$, $b$, $c$ with $abc=1$.

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

If $a, b, c$ are positive numbers with $abc=1$, then $(a+\frac{1}{b})^2+(b+\frac{1}{c})^2+(c+\frac{1}{a})^2\ge 3(a+b+c+1)$.

On Proving $ a^6 + b^6 + c^ 6 − 3a ^2 b^ 2 c^ 2 + 2(a^ 2 + bc)(b ^2 + ca)(c^ 2 + ab) ≥ 0 $ where $a,b,c$ are real numbers.

Show that $\frac 1x \ge 3 - 2\sqrt{x}$ for all positive real numbers $x$.

Doubt:regarding proof $\sum_{cyc}a^{2/3}b\le 3$

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