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Prove $(\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x})^3+12\ge 13(x^3+y^3+z^3)$ for positives $xyz = 1$

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#algebra-precalculus #inequality #substitution #symmetric-polynomials #uvw
52
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Prove inequality involving elementary symmetric polynomial

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#inequality #roots #symmetric-polynomials
42
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Solve system solutions

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#trigonometry #systems-of-equations #quadratics #symmetric-polynomials
68
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What happens when we raise a polynomial with a power how many roots do we get?

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#algebra-precalculus #polynomials #quadratics #quadratic-forms #symmetric-polynomials
36
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What are elementary symmetric polynomials?

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#polynomials #graphing-functions #symmetric-polynomials
215
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Problematic inequality & hint

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#inequality #symmetric-polynomials #muirhead-inequality
63
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Solution for $x$ and $y$.

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#polynomials #systems-of-equations #symmetric-polynomials
75
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Invariant to inversions 2

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#multivariable-calculus #inequality #symmetric-polynomials #invariant-theory
472
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If $a,b,c$ are distinct positive real numbers and $a^2+b^2+c^2=1$, then prove $ab+bc + ac$ will be less than 1.

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#systems-of-equations #quadratics #symmetric-polynomials
49
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invariant to inversions

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#multivariable-calculus #inequality #symmetric-polynomials #invariant-theory
51
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Solution to a system of equations

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#systems-of-equations #cubics #symmetric-polynomials
127
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Solve system of equations $ y + 3x = 4x^3$ and $x + 3y = 4y^3$

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#systems-of-equations #cubics #symmetric-polynomials
86
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For $ab+bc+ca=3$, when does $\prod_\text{cyc}(a^8+1)(a^4+1)=\prod_\text{cyc}(a^2+1)^2$?

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#inequality #products #symmetric-polynomials #cauchy-schwarz-inequality
97
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Prove that in any triangle

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#inequality #triangles #factoring #symmetric-polynomials #geometric-inequalities
68
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$ 27 + ( 2 + \frac{a^{2}}{bc} ) (2 + \frac{b^{2}}{ac}) (2 + \frac{c^{2}}{ab}) \ge 6 (a+b+c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $

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#inequality #a.m.-g.m.-inequality #cauchy-schwarz-inequality #symmetric-polynomials

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