Question

Proving inequality by SOS.

score 164 · Answer 1 · 2026-03-25 17:29:46

164

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Proving inequality by SOS.

Published on 25 Mar 2026 - 17:29

score 106 · Answer 2 · 2026-02-23 04:24:34

106

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Prove $2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \le \left(x^2+y^2+z^2+3xyz\right)^2.$

Published on 23 Feb 2026 - 4:24

score 37 · Answer 3 · 2026-03-25 04:37:45

37

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Any good usable software or software package for decomposition with constraint?

Published on 25 Mar 2026 - 4:37

score 280 · Answer 4 · 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 5 · 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 232 · Answer 6 · 2020-06-04 04:51:45

232

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Prove that $\sum_{\mathrm{cyc}} (40a^6 + 53a^5b) \ge 0$

Published on 04 Jun 2020 - 4:51

score 86 · Answer 7 · 2020-06-04 11:06:03

86

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Prove $P= 7\,{c}^{4}-2\,ab{c}^{2}-2\,ab \left( a+b \right) c+ \left( a+b \right) ^{2} \left( {a}^{2}+{b}^{2} \right) \geqq 0$

Published on 04 Jun 2020 - 11:06

score 108 · Answer 8 · 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 144 · Answer 9 · 2020-06-07 01:12:37

144

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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 07 Jun 2020 - 1:12

score 206 · Answer 10 · 2020-06-09 03:17:48

206

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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 09 Jun 2020 - 3:17

score 136 · Answer 11 · 2020-06-10 01:22:41

136

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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 10 Jun 2020 - 1:22

score 101 · Answer 12 · 2020-06-12 04:56:32

101

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Help in proving this inequality

Published on 12 Jun 2020 - 4:56

score 141 · Answer 13 · 2026-03-25 09:43:09

141

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Prove that if $abc \geq 1$ and $a,b,c > 0$ then $\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$

Published on 25 Mar 2026 - 9:43

score 106 · Answer 14 · 2026-03-25 09:33:50

106

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Inequality 6 deg

Published on 25 Mar 2026 - 9:33

score 257 · Answer 15 · 2020-06-25 19:01:37

257

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Lower bound for the square root of the sum of variables including squares

Published on 25 Jun 2020 - 19:01

List Question

Proving inequality by SOS.

Prove $2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \le \left(x^2+y^2+z^2+3xyz\right)^2.$

Any good usable software or software package for decomposition with constraint?

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 that $\sum_{\mathrm{cyc}} (40a^6 + 53a^5b) \ge 0$

Prove $P= 7\,{c}^{4}-2\,ab{c}^{2}-2\,ab \left( a+b \right) c+ \left( a+b \right) ^{2} \left( {a}^{2}+{b}^{2} \right) \geqq 0$

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

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) $

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

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

Help in proving this inequality

Prove that if $abc \geq 1$ and $a,b,c > 0$ then $\frac{1}{1 + a + b} + \frac{1}{1 + b + c} + \frac{1}{1 + c + a} \leq 1$

Inequality 6 deg

Lower bound for the square root of the sum of variables including squares

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