Question

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

score 9.4k · Answer 1 · 2026-04-15 06:01:55

9.4k

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if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

Published on 15 Apr 2026 - 6:01

score 286 · Answer 2 · 2026-04-17 09:51:46

286

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Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

Published on 17 Apr 2026 - 9:51

score 122 · Answer 3 · 2026-04-15 21:22:07

122

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Let $a,b,c>0$ such $ \frac{b+c}{a}+ \frac{c+a}{b}+ \frac{a+b}{c} = 2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac} \right)$

Published on 15 Apr 2026 - 21:22

score 138 · Answer 4 · 2026-04-15 10:29:44

138

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Proving Cauchy inequality involving four expression

Published on 15 Apr 2026 - 10:29

score 204 · Answer 5 · 2026-04-14 05:03:26

204

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Seeking concise proof: $\frac18(a^2+b^2)(b^2+c^2)(c^2+a^2)\ge\frac1{27}(ab+bc+ca)^3$, where $a$, $b$, $c$ are positive numbers

Published on 14 Apr 2026 - 5:03

score 173 · Answer 6 · 2026-03-25 09:42:32

173

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Generalization of the AM-GM inequality for three variables

Published on 25 Mar 2026 - 9:42

score 217 · Answer 7 · 2026-05-06 08:52:17

217

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Problematic inequality & hint

Published on 06 May 2026 - 8:52

score 143 · Answer 8 · 2026-04-10 21:54:56

143

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How do we prove this inequality?

Published on 10 Apr 2026 - 21:54

score 503 · Answer 9 · 2026-04-14 13:36:18

503

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Inequality in 4 variables Vasc's EV

Published on 14 Apr 2026 - 13:36

score 72 · Answer 10 · 2026-04-16 04:50:53

72

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Prove that $\sum_\text{cyc} \frac{ab}{ab+b^2+ca}\le 1$

Published on 16 Apr 2026 - 4:50

score 99 · Answer 11 · 2019-11-27 00:54:02

99

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Prove that $3 a^4+ 2 a^3 b - 3 a^2 b^2 - 2 a b^3 + 3 b^4\geq 0$.

Published on 27 Nov 2019 - 0:54

score 69 · Answer 12 · 2026-04-17 23:11:56

69

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Prove an inequality : $\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$ without Jensen's inequality

Published on 17 Apr 2026 - 23:11

score 303 · Answer 13 · 2026-04-28 19:06:43

303

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If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$

Published on 28 Apr 2026 - 19:06

score 294 · Answer 14 · 2026-04-20 06:45:38

294

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Proving $\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{c}{c^2+2a^2}\geq 1$ when $a^2+b^2+c^2=a^3+b^3+c^3$

Published on 20 Apr 2026 - 6:45

score 206 · Answer 15 · 2026-03-25 07:45:39

206

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Proof verification for $x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9$ (where $xyz=1$ and $x,y,z\in \mathbb{R}^+$)

Published on 25 Mar 2026 - 7:45

List Question

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

Let $a,b,c>0$ such $ \frac{b+c}{a}+ \frac{c+a}{b}+ \frac{a+b}{c} = 2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac} \right)$

Proving Cauchy inequality involving four expression

Seeking concise proof: $\frac18(a^2+b^2)(b^2+c^2)(c^2+a^2)\ge\frac1{27}(ab+bc+ca)^3$, where $a$, $b$, $c$ are positive numbers

Generalization of the AM-GM inequality for three variables

Problematic inequality & hint

How do we prove this inequality?

Inequality in 4 variables Vasc's EV

Prove that $\sum_\text{cyc} \frac{ab}{ab+b^2+ca}\le 1$

Prove that $3 a^4+ 2 a^3 b - 3 a^2 b^2 - 2 a b^3 + 3 b^4\geq 0$.

Prove an inequality : $\sum_{cyc}\frac{a^3}{abu+b^2v}\geq \frac{a+b+c}{u+v}$ without Jensen's inequality

If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$

Proving $\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{c}{c^2+2a^2}\geq 1$ when $a^2+b^2+c^2=a^3+b^3+c^3$

Proof verification for $x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9$ (where $xyz=1$ and $x,y,z\in \mathbb{R}^+$)

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