Question

Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

score 280 · Answer 1 · 2026-03-25 14:22:52

280

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Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

Published on 25 Mar 2026 - 14:22

score 98 · Answer 2 · 2026-03-25 12:49:53

98

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Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$

Published on 25 Mar 2026 - 12:49

score 95 · Answer 3 · 2020-06-04 11:18:06

95

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Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$

Published on 04 Jun 2020 - 11:18

score 108 · Answer 4 · 2026-03-25 17:42:25

108

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Let $a, b, c>0$. Prove that $\sum \limits_{cyc}{\frac{a}{b+c}\left(\frac{b}{c+a}+\frac{c}{a+b}\right)}\le \frac{(a+b+c)^2}{2(ab+bc+ca)}$

Published on 25 Mar 2026 - 17:42

score 105 · Answer 5 · 2026-03-25 17:36:45

105

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Let $a,$ $b$ and $c$ are positive numbers.

Published on 25 Mar 2026 - 17:36

score 35 · Answer 6 · 2020-06-06 10:42:34

35

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$\frac1{\sqrt{2 c}}\le\frac{\sum_{cyc}a[(b+c)(b+d)(c+d)]^{3/2}}{(\sum_{cyc} abc)^{3/2}}\le\frac2{\sqrt{c}}$ where $c$ is the second smallest value.

Published on 06 Jun 2020 - 10:42

score 188 · Answer 7 · 2026-03-27 15:58:19

188

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Proving $\frac{a(b+c)}{a^2+bc}+\frac{b(a+c)}{b^2+ac}+\frac{c(b+a)}{c^2+ba}\geqq 1+\frac{16abc}{(a+b)(b+c)(c+a)} $

Published on 27 Mar 2026 - 15:58

score 145 · Answer 8 · 2026-03-25 19:11:44

145

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Proving ${ \left\{\sum \left( ab+{b}^{2}+{c}^{2}+ac \right)\right\} }^{4}\geq 27\,{ \sum} ( ab+{b}^{2}+{c}^{2}+ac ) ^{3} ( c+a) ( a+b) $

Published on 25 Mar 2026 - 19:11

score 73 · Answer 9 · 2026-03-25 16:02:36

73

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Let $a,b,c>0$ then prove this inequality holds

Published on 25 Mar 2026 - 16:02

score 71 · Answer 10 · 2026-03-25 09:43:12

71

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Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Published on 25 Mar 2026 - 9:43

score 207 · Answer 11 · 2026-03-25 19:11:43

207

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Sum of Squares for $a^2+b^2+c^2+d^2+abcd+1\ge ab+bc+cd+da + ac+bd$

Published on 25 Mar 2026 - 19:11

score 137 · Answer 12 · 2026-03-25 19:12:06

137

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Proving $\frac{1}{16} \sum \frac{(b+c)(c+a)}{ab} +\frac{9}{4} \geq 4\sum \frac{ab}{(b+c)(c+a)}$

Published on 25 Mar 2026 - 19:12

score 181 · Answer 13 · 2020-06-10 04:03:49

181

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Can any cyclic polynomial in $a, b, c$ be expressed in terms of $a^2b+b^2c+c^2a$, $a+b+c$, $ab+bc+ca$ and $abc$?

Published on 10 Jun 2020 - 4:03

score 83 · Answer 14 · 2020-06-10 08:14:07

83

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Express the strange symmetric polynomial in terms of elementary symmetric polynomials

Published on 10 Jun 2020 - 8:14

score 102 · Answer 15 · 2026-03-25 19:10:43

102

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Help in proving this inequality

Published on 25 Mar 2026 - 19:10

List Question

Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$

Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$

Let $a, b, c>0$. Prove that $\sum \limits_{cyc}{\frac{a}{b+c}\left(\frac{b}{c+a}+\frac{c}{a+b}\right)}\le \frac{(a+b+c)^2}{2(ab+bc+ca)}$

Let $a,$ $b$ and $c$ are positive numbers.

$\frac1{\sqrt{2 c}}\le\frac{\sum_{cyc}a[(b+c)(b+d)(c+d)]^{3/2}}{(\sum_{cyc} abc)^{3/2}}\le\frac2{\sqrt{c}}$ where $c$ is the second smallest value.

Proving $\frac{a(b+c)}{a^2+bc}+\frac{b(a+c)}{b^2+ac}+\frac{c(b+a)}{c^2+ba}\geqq 1+\frac{16abc}{(a+b)(b+c)(c+a)} $

Proving ${ \left\{\sum \left( ab+{b}^{2}+{c}^{2}+ac \right)\right\} }^{4}\geq 27\,{ \sum} ( ab+{b}^{2}+{c}^{2}+ac ) ^{3} ( c+a) ( a+b) $

Let $a,b,c>0$ then prove this inequality holds

Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Sum of Squares for $a^2+b^2+c^2+d^2+abcd+1\ge ab+bc+cd+da + ac+bd$

Proving $\frac{1}{16} \sum \frac{(b+c)(c+a)}{ab} +\frac{9}{4} \geq 4\sum \frac{ab}{(b+c)(c+a)}$

Can any cyclic polynomial in $a, b, c$ be expressed in terms of $a^2b+b^2c+c^2a$, $a+b+c$, $ab+bc+ca$ and $abc$?

Express the strange symmetric polynomial in terms of elementary symmetric polynomials

Help in proving this inequality

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