List questions from sum-of-squares-method

Published on 19 Sep 2017 - 10:07

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Nessbit's and Triangle inequality

Published on 24 Sep 2017 - 0:37

214

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Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Published on 25 Mar 2026 - 21:51

3.6k

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Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Published on 25 Mar 2026 - 21:54

292

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Sum-of-squares quartic polynomials in three variables

Published on 25 Mar 2026 - 15:49

68

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Convergence of sum of squares relaxations for global polynomial optimization

Published on 25 Mar 2026 - 15:52

146

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A problem :$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f})^3$

Published on 14 Nov 2017 - 13:30

201

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Prove that, if $a,b,c$ are positive real numbers $\frac {a} {2a+b+c}+ \frac {b} {2b+a+c}+\frac {c} {2c+a+b}<\frac{19}{25}$

Published on 15 Nov 2017 - 22:20

75

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notaion in SOS theorem

Published on 19 Dec 2017 - 12:41

1.8k

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Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

Published on 25 Mar 2026 - 17:40

289

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Inequality : $\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$

Published on 30 Dec 2017 - 9:27

155

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prove this inequality $Σ_{cyc}\frac{a^3+abc}{b^2+c^2}\ge a+b+c$

Published on 04 Jan 2018 - 14:33

598

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proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Published on 22 Mar 2026 - 5:37

370

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Showing that $ \left( \frac{a+2b}{a+2c}\right)^3+\left( \frac{b+2c}{b+2a}\right)^3+\left( \frac{c+2a}{c+2b}\right)^3 \geq3 $

Published on 22 Mar 2026 - 4:42

465

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Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$

Published on 22 Mar 2026 - 6:47

List Question

IMO 2006, A4 Problem, Idea behind the proof

Nessbit's and Triangle inequality

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Sum-of-squares quartic polynomials in three variables

Convergence of sum of squares relaxations for global polynomial optimization

A problem :$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f})^3$

Prove that, if $a,b,c$ are positive real numbers $\frac {a} {2a+b+c}+ \frac {b} {2b+a+c}+\frac {c} {2c+a+b}<\frac{19}{25}$

notaion in SOS theorem

Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

Inequality : $\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$

prove this inequality $Σ_{cyc}\frac{a^3+abc}{b^2+c^2}\ge a+b+c$

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Showing that $ \left( \frac{a+2b}{a+2c}\right)^3+\left( \frac{b+2c}{b+2a}\right)^3+\left( \frac{c+2a}{c+2b}\right)^3 \geq3 $

Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$

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