Question

Prove $\sqrt[3]{\frac{(a^4+b^4)(a^4+c^4)(b^4+c^4)}{abc}}\ge \sqrt{\frac{(a^3+bc)(b^3+ca)(c^3+ab)}{1+abc}}$

score 127 · Answer 1 · 2026-03-25 14:36:39

127

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Prove $\sqrt[3]{\frac{(a^4+b^4)(a^4+c^4)(b^4+c^4)}{abc}}\ge \sqrt{\frac{(a^3+bc)(b^3+ca)(c^3+ab)}{1+abc}}$

Published on 25 Mar 2026 - 14:36

score 90 · Answer 2 · 2020-03-04 21:19:07

90

Views

Ring of polynomials finite over ring of elementary symmetric polynomial

Published on 04 Mar 2020 - 21:19

score 196 · Answer 3 · 2020-03-07 13:03:08

196

Views

System of equations $x^3+y=y^2\ \& \ y^3+z=z^2\ \& \ z^3+x=x^2$

Published on 07 Mar 2020 - 13:03

score 105 · Answer 4 · 2020-03-07 13:25:28

105

Views

About the equality $\dfrac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt3$ in a triangle. American Mathematical Problems.

Published on 07 Mar 2020 - 13:25

score 123 · Answer 5 · 2026-03-25 19:09:58

123

Views

Given positives $a, b, c$ such that $a + b + c = 3$, prove that $\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{2}$.

Published on 25 Mar 2026 - 19:09

score 1.2k · Answer 6 · 2020-03-14 20:04:39

1.2k

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How to express a symmetric polynomial in terms of elementary symmetric polynomials

Published on 14 Mar 2020 - 20:04

score 587 · Answer 7 · 2026-03-26 19:02:09

587

Views

Can I solve this other than using Newton's sums or Newton's identities?

Published on 26 Mar 2026 - 19:02

score 152 · Answer 8 · 2026-03-25 16:13:54

152

Views

For non-negative reals $a$, $b$, $c$, show that $3(1-a+a^2)(1-b+b^2)(1-c+c^2)\ge(1+abc+a^2b^2c^2)$

Published on 25 Mar 2026 - 16:13

score 56 · Answer 9 · 2026-03-25 12:38:44

56

Views

Prove the following inequality using am gm inequality

Published on 25 Mar 2026 - 12:38

score 197 · Answer 10 · 2026-03-25 12:41:22

197

Views

$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Published on 25 Mar 2026 - 12:41

score 189 · Answer 11 · 2026-03-25 20:42:09

189

Views

Simplify homogeneous polynomial of degree 3 with all coefficients equal 1

Published on 25 Mar 2026 - 20:42

score 172 · Answer 12 · 2026-03-25 12:40:01

172

Views

Proving $a^2 + b^2 + c^2 \geqslant ab + bc + ca$

Published on 25 Mar 2026 - 12:40

score 86 · Answer 13 · 2020-03-20 17:18:04

86

Views

A problem on symmetric polynomials

Published on 20 Mar 2020 - 17:18

score 249 · Answer 14 · 2026-03-25 07:48:36

249

Views

Computing discriminant of $x^4 + px + 1$, need help to find a mistake

Published on 25 Mar 2026 - 7:48

score 211 · Answer 15 · 2020-03-20 20:29:43

211

Views

Symmetric tensor polynomials generated by power sums?

Published on 20 Mar 2020 - 20:29

List Question

Prove $\sqrt[3]{\frac{(a^4+b^4)(a^4+c^4)(b^4+c^4)}{abc}}\ge \sqrt{\frac{(a^3+bc)(b^3+ca)(c^3+ab)}{1+abc}}$

Ring of polynomials finite over ring of elementary symmetric polynomial

System of equations $x^3+y=y^2\ \& \ y^3+z=z^2\ \& \ z^3+x=x^2$

About the equality $\dfrac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt3$ in a triangle. American Mathematical Problems.

Given positives $a, b, c$ such that $a + b + c = 3$, prove that $\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{2}$.

How to express a symmetric polynomial in terms of elementary symmetric polynomials

Can I solve this other than using Newton's sums or Newton's identities?

For non-negative reals $a$, $b$, $c$, show that $3(1-a+a^2)(1-b+b^2)(1-c+c^2)\ge(1+abc+a^2b^2c^2)$

Prove the following inequality using am gm inequality

$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Simplify homogeneous polynomial of degree 3 with all coefficients equal 1

Proving $a^2 + b^2 + c^2 \geqslant ab + bc + ca$

A problem on symmetric polynomials

Computing discriminant of $x^4 + px + 1$, need help to find a mistake

Symmetric tensor polynomials generated by power sums?

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