Question

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

score 165 · Answer 1 · 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 326 · Answer 2 · 2021-09-30 04:58:44

326

Views

Prove that $\sqrt1+\sqrt2+\sqrt3+....+\sqrt n<n\sqrt{\frac{n(n+1)}{2}}$

Published on 30 Sep 2021 - 4:58

score 221 · Answer 3 · 2021-10-08 19:21:34

221

Views

Prove the inequality by mathematical induction

Published on 08 Oct 2021 - 19:21

score 157 · Answer 4 · 2026-03-25 18:49:37

157

Views

In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Published on 25 Mar 2026 - 18:49

score 141 · Answer 5 · 2026-03-27 22:57:30

141

Views

Prove that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$ if $a^2+b^2+c^2+abc=4$.

Published on 27 Mar 2026 - 22:57

score 190 · Answer 6 · 2026-03-26 21:14:04

190

Views

Prove that $\frac{a^4+2a+b(2a^2+b)}{a^2(b^2+2a)}+\frac{b^4+2b+c(2b^2+c)}{b^2(c^2+2b)}+\frac{c^4+2c+a(2c^2+a)}{c^2(a^2+2c)}\ge 6$

Published on 26 Mar 2026 - 21:14

score 85 · Answer 7 · 2021-10-17 01:05:40

85

Views

Prove $\frac{d}{\sqrt{a^2+b^2+c^2}}+\sqrt{3}\ge2\left(\sqrt{\frac{d}{a+b+c}}+\frac{\sqrt{ab+bc+ca}}{a+b+c}\right)$

Published on 17 Oct 2021 - 1:05

score 192 · Answer 8 · 2021-10-19 00:57:12

192

Views

Induction inequality/ n in exponential

Published on 19 Oct 2021 - 0:57

score 144 · Answer 9 · 2021-10-21 15:32:32

144

Views

For positive real numbers $a, b, c$, maximise $(a-3)(b-2)(c-1)$ given $a+b+c=18$.

Published on 21 Oct 2021 - 15:32

score 211 · Answer 10 · 2021-10-25 12:48:49

211

Views

$\sum\limits_{cyc} \sqrt{\frac{a^3}{1+bc}} \geq 2$ for $a, b, c > 0$ which satisfies $abc=1$.

Published on 25 Oct 2021 - 12:48

score 220 · Answer 11 · 2021-11-07 20:36:46

220

Views

Is there a generic reversing of the Arithmetic Mean – Geometric Mean inequality？

Published on 07 Nov 2021 - 20:36

score 144 · Answer 12 · 2026-03-27 02:03:08

144

Views

Prove that: $\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}}+\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq\frac{21}{2}$

Published on 27 Mar 2026 - 2:03

score 250 · Answer 13 · 2021-11-12 14:25:14

250

Views

About an inequality wich have a link with $\sqrt{\dfrac{a^b}{b}}+\sqrt{\dfrac{b^a}{a}}\ge 2$

Published on 12 Nov 2021 - 14:25

score 83 · Answer 14 · 2021-11-12 22:32:29

83

Views

How to prove $(a_1+a_2+a_3+...+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_n}\right)≥n^2$

Published on 12 Nov 2021 - 22:32

score 248 · Answer 15 · 2021-11-15 16:18:58

248

Views

Find the maximum value of $(x+2) \cdot (12-3x)$ , x is a Real number

Published on 15 Nov 2021 - 16:18

List Question

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

Prove that $\sqrt1+\sqrt2+\sqrt3+....+\sqrt n<n\sqrt{\frac{n(n+1)}{2}}$

Prove the inequality by mathematical induction

In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Prove that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$ if $a^2+b^2+c^2+abc=4$.

Prove that $\frac{a^4+2a+b(2a^2+b)}{a^2(b^2+2a)}+\frac{b^4+2b+c(2b^2+c)}{b^2(c^2+2b)}+\frac{c^4+2c+a(2c^2+a)}{c^2(a^2+2c)}\ge 6$

Prove $\frac{d}{\sqrt{a^2+b^2+c^2}}+\sqrt{3}\ge2\left(\sqrt{\frac{d}{a+b+c}}+\frac{\sqrt{ab+bc+ca}}{a+b+c}\right)$

Induction inequality/ n in exponential

For positive real numbers $a, b, c$, maximise $(a-3)(b-2)(c-1)$ given $a+b+c=18$.

$\sum\limits_{cyc} \sqrt{\frac{a^3}{1+bc}} \geq 2$ for $a, b, c > 0$ which satisfies $abc=1$.

Is there a generic reversing of the Arithmetic Mean – Geometric Mean inequality？

Prove that: $\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}}+\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq\frac{21}{2}$

About an inequality wich have a link with $\sqrt{\dfrac{a^b}{b}}+\sqrt{\dfrac{b^a}{a}}\ge 2$

How to prove $(a_1+a_2+a_3+...+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_n}\right)≥n^2$

Find the maximum value of $(x+2) \cdot (12-3x)$ , x is a Real number

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