Question

Duality gap in polynomial optimization problem Lasserre relaxation

score 150 · Answer 1 · 2026-05-02 12:22:18

150

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Duality gap in polynomial optimization problem Lasserre relaxation

Published on 02 May 2026 - 12:22

score 533 · Answer 2 · 2026-03-25 13:05:20

533

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$\;32\displaystyle\left(\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \sum_{c} \frac{x^2+yz}{y+z}+6$

Published on 25 Mar 2026 - 13:05

score 920 · Answer 3 · 2026-03-25 12:53:39

920

Views

Prove that $(x−2y+z)^2 \geq 4xz−8y$

Published on 25 Mar 2026 - 12:53

score 94 · Answer 4 · 2026-04-06 03:27:35

94

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Prove that the inequality is true

Published on 06 Apr 2026 - 3:27

score 132 · Answer 5 · 2026-03-25 07:47:57

132

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Inequality for $a,c,b$ positive real numbers with $a + b + c = 1$

Published on 25 Mar 2026 - 7:47

score 124 · Answer 6 · 2026-03-25 07:45:53

Prove that $\frac{3a^3+7b^3}{2a+3b}+\frac{3b^3+7c^3}{2b+3c}+\frac{3c^3+7a^3}{2c+3a}\ge 3\left(a^2+b^2+c^2\right)-\left(ab+bc+ca\right)$

score 200 · Answer 7 · 2026-03-25 07:45:43

200

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A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

Published on 25 Mar 2026 - 7:45

score 150 · Answer 8 · 2026-03-25 07:46:04

150

Views

If $ab+bc+ca=3$ for non-negative $a$, $b$, $c$, show that $\sum_{cyc}a^2b^2+\sum_{cyc}\frac{12a^2b^2c^2}{(a+b)^2}\ge 12abc$

Published on 25 Mar 2026 - 7:46

score 138 · Answer 9 · 2026-03-25 07:46:20

138

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For $x$, $y$, $z$ the sides of a triangle, show $\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$

Published on 25 Mar 2026 - 7:46

score 54 · Answer 10 · 2026-03-25 09:31:51

54

Views

Inequality triangle Radon substitutions

Published on 25 Mar 2026 - 9:31

score 216 · Answer 11 · 2026-03-25 07:44:55

216

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Inequality. ${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}} + {{1} \over {\sqrt{b}}} + {{1} \over {\sqrt{c}}}$

Published on 25 Mar 2026 - 7:44

score 130 · Answer 12 · 2026-03-25 06:11:00

130

Views

Inequality with $a+b+c=1$, a,b,c positive numbers

Published on 25 Mar 2026 - 6:11

score 217 · Answer 13 · 2026-03-24 23:46:45

217

Views

Prove $\sum \sqrt{\frac{a^2}{6a^2+5ab+b^2}}\le \frac{\sqrt{3}}{2}$

Published on 24 Mar 2026 - 23:46

score 239 · Answer 14 · 2026-03-25 07:43:47

239

Views

Decomposing bivariate quadratic form into sum of two squares

Published on 25 Mar 2026 - 7:43

score 295 · Answer 15 · 2026-03-25 06:06:19

295

Views

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

Published on 25 Mar 2026 - 6:06

List Question

Duality gap in polynomial optimization problem Lasserre relaxation

$\;32\displaystyle\left(\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \sum_{c} \frac{x^2+yz}{y+z}+6$

Prove that $(x−2y+z)^2 \geq 4xz−8y$

Prove that the inequality is true

Inequality for $a,c,b$ positive real numbers with $a + b + c = 1$

Prove that $\frac{3a^3+7b^3}{2a+3b}+\frac{3b^3+7c^3}{2b+3c}+\frac{3c^3+7a^3}{2c+3a}\ge 3\left(a^2+b^2+c^2\right)-\left(ab+bc+ca\right)$

A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

If $ab+bc+ca=3$ for non-negative $a$, $b$, $c$, show that $\sum_{cyc}a^2b^2+\sum_{cyc}\frac{12a^2b^2c^2}{(a+b)^2}\ge 12abc$

For $x$, $y$, $z$ the sides of a triangle, show $\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$

Inequality triangle Radon substitutions

Inequality. ${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}} + {{1} \over {\sqrt{b}}} + {{1} \over {\sqrt{c}}}$

Inequality with $a+b+c=1$, a,b,c positive numbers

Prove $\sum \sqrt{\frac{a^2}{6a^2+5ab+b^2}}\le \frac{\sqrt{3}}{2}$

Decomposing bivariate quadratic form into sum of two squares

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

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