Question

Why $\prod_i (1+x_i)$ is not considered a symmetric function, $\prod_i (1+x_i) \notin \Lambda$

score 180 · Answer 1 · 2026-03-25 16:01:28

180

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Why $\prod_i (1+x_i)$ is not considered a symmetric function, $\prod_i (1+x_i) \notin \Lambda$

Published on 25 Mar 2026 - 16:01

score 328 · Answer 2 · 2026-03-25 06:12:16

328

Views

show this inequality with $xy+yz+zx=3$

Published on 25 Mar 2026 - 6:12

score 134 · Answer 3 · 2026-03-25 17:45:30

134

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inner product in the algebra of shifted symmetric function, $\Lambda^*$

Published on 25 Mar 2026 - 17:45

score 115 · Answer 4 · 2019-05-26 05:04:51

115

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Solve for $x, y \in \mathbb R$: $x^2+y^2=2x^2y^2$ and $(x+y)(1+xy)=4x^2y^2$

Published on 26 May 2019 - 5:04

score 76 · Answer 5 · 2019-05-30 18:08:16

76

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Solve $x+y+z=3$; $x^3+y^3+z^3=15$; $x^5+y^5+z^5=83$

Published on 30 May 2019 - 18:08

score 203 · Answer 6 · 2019-06-03 22:56:02

203

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Rationalizing $\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}$

Published on 03 Jun 2019 - 22:56

score 104 · Answer 7 · 2026-03-25 09:29:13

104

Views

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Published on 25 Mar 2026 - 9:29

score 150 · Answer 8 · 2019-06-05 13:14:11

150

Views

Simultaneous Equations $a^2 - b^2 = -16$ and $2ab = 30$

Published on 05 Jun 2019 - 13:14

score 86 · Answer 9 · 2019-06-05 15:38:07

86

Views

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

Published on 05 Jun 2019 - 15:38

score 157 · Answer 10 · 2019-06-07 15:08:55

157

Views

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

Published on 07 Jun 2019 - 15:08

score 104 · Answer 11 · 2019-06-07 17:26:59

Calculate $\frac{a^8 + b^8}{a^8 - b^8} + \frac{a^8 - b^8}{a^8 + b^8}$ based on $\frac{a^2 + b^2}{a^2 - b^2} + \frac{a^2 - b^2}{a^2 + b^2}$.

score 117 · Answer 12 · 2026-03-25 06:13:26

117

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Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

Published on 25 Mar 2026 - 6:13

score 222 · Answer 13 · 2019-06-08 06:07:32

222

Views

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$.

Published on 08 Jun 2019 - 6:07

score 77 · Answer 14 · 2019-06-08 07:46:01

77

Views

Elimination of $x,y,z$ to find relation between $a,b,c$ from some simultaneous equations

Published on 08 Jun 2019 - 7:46

score 66 · Answer 15 · 2019-06-08 09:50:10

66

Views

Prove that $x_1 + x_2 = y_1 + y_2 = z_1 + z_2$.

Published on 08 Jun 2019 - 9:50

List Question

Why $\prod_i (1+x_i)$ is not considered a symmetric function, $\prod_i (1+x_i) \notin \Lambda$

show this inequality with $xy+yz+zx=3$

inner product in the algebra of shifted symmetric function, $\Lambda^*$

Solve for $x, y \in \mathbb R$: $x^2+y^2=2x^2y^2$ and $(x+y)(1+xy)=4x^2y^2$

Solve $x+y+z=3$; $x^3+y^3+z^3=15$; $x^5+y^5+z^5=83$

Rationalizing $\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}$

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Simultaneous Equations $a^2 - b^2 = -16$ and $2ab = 30$

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

Calculate $\frac{a^8 + b^8}{a^8 - b^8} + \frac{a^8 - b^8}{a^8 + b^8}$ based on $\frac{a^2 + b^2}{a^2 - b^2} + \frac{a^2 - b^2}{a^2 + b^2}$.

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

Prove that $(x + y + z)^3 + 9xyz \ge 4(x + y + z)(xy + yz + zx)$ where $x, y, z \ge 0$.

Elimination of $x,y,z$ to find relation between $a,b,c$ from some simultaneous equations

Prove that $x_1 + x_2 = y_1 + y_2 = z_1 + z_2$.

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