Question

$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz$

score 102 · Answer 1 · 2021-06-19 01:24:47

102

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$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz$

Published on 19 Jun 2021 - 1:24

score 1.5k · Answer 2 · 2021-07-13 16:47:15

1.5k

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Prove that $\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}$.

Published on 13 Jul 2021 - 16:47

score 39 · Answer 3 · 2026-03-25 19:05:38

39

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Are elements in each stage of the Lasserre hierarchy convex?

Published on 25 Mar 2026 - 19:05

score 243 · Answer 4 · 2026-03-26 01:04:48

243

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Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $

Published on 26 Mar 2026 - 1:04

score 100 · Answer 5 · 2021-08-07 10:59:38

100

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show this inequality $\sum_{cyc}\sqrt{a(b+c)}(b^2+c^2-a^2-bc)\ge 0$

Published on 07 Aug 2021 - 10:59

score 152 · Answer 6 · 2021-08-12 12:49:16

152

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Prove $\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3$ when $x^2+y^2+z^2+(x+y+z)^2\leq 4$.

Published on 12 Aug 2021 - 12:49

score 278 · Answer 7 · 2021-08-19 23:36:46

278

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how to prove $\sum_{cyc}(1-a_{1}+a_{1}a_{2})^2\ge\frac{n}{2}$

Published on 19 Aug 2021 - 23:36

score 155 · Answer 8 · 2026-03-25 19:12:07

155

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Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$

Published on 25 Mar 2026 - 19:12

score 109 · Answer 9 · 2021-09-03 14:48:50

109

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How to write a square of a trigonometric polynomial cosine?

Published on 03 Sep 2021 - 14:48

score 128 · Answer 10 · 2021-09-26 15:01:02

128

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Find the minimum value of $a^8+b^8+c^8+2(a-1)(b-1)(c-1)$

Published on 26 Sep 2021 - 15:01

score 162 · Answer 11 · 2026-03-26 21:10:32

162

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How do I prove that $\sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge6(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})$ given $x,y,z > 0$ and $x+y+z=3$?

Published on 26 Mar 2026 - 21:10

score 120 · Answer 12 · 2021-10-07 21:20:25

120

Views

Find another sum of squares for $3^{12}-6^6+2^{12}$

Published on 07 Oct 2021 - 21:20

score 180 · Answer 13 · 2021-10-13 00:31:51

180

Views

Nice problem: Prove that: $ab+bc+ca \ge \sum{\sqrt{a^2+b^2+3}}$

Published on 13 Oct 2021 - 0:31

score 109 · Answer 14 · 2021-10-15 04:06:21

109

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How to prove:$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+a+b+c\ge\sum_{cyc}{\sqrt{2(a^2+b^2)}}$

Published on 15 Oct 2021 - 4:06

score 141 · Answer 15 · 2021-10-16 01:02:28

141

Views

Prove that: $a+b+c+a^2+b^2+c^2\ge\sum_{cyc}{\sqrt{a(a^3+b+c)}}$

Published on 16 Oct 2021 - 1:02

List Question

$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz$

Prove that $\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}$.

Are elements in each stage of the Lasserre hierarchy convex?

Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $

show this inequality $\sum_{cyc}\sqrt{a(b+c)}(b^2+c^2-a^2-bc)\ge 0$

Prove $\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3$ when $x^2+y^2+z^2+(x+y+z)^2\leq 4$.

how to prove $\sum_{cyc}(1-a_{1}+a_{1}a_{2})^2\ge\frac{n}{2}$

Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$

How to write a square of a trigonometric polynomial cosine?

Find the minimum value of $a^8+b^8+c^8+2(a-1)(b-1)(c-1)$

How do I prove that $\sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge6(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})$ given $x,y,z > 0$ and $x+y+z=3$?

Find another sum of squares for $3^{12}-6^6+2^{12}$

Nice problem: Prove that: $ab+bc+ca \ge \sum{\sqrt{a^2+b^2+3}}$

How to prove:$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+a+b+c\ge\sum_{cyc}{\sqrt{2(a^2+b^2)}}$

Prove that: $a+b+c+a^2+b^2+c^2\ge\sum_{cyc}{\sqrt{a(a^3+b+c)}}$

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