Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,178
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Generalizing Ramanujan cubic denesting formula to higher powers
We have the following theorems for denesting radicals of degree 2 and 3 :
Denesting theorem for degree 2 :
If $\alpha, \beta$ are the roots of the equation,
\begin{equation}
x^2-ax+b = 0
\end{equation}...
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9
votes
1
answer
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Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions
I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is
\begin{align}
a^{\...
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vote
0
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31
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How to show that $X_m$ is a zero of this polynomial in $R[X_1,…,X_m][X]$?
I posted this question a few days ago but the images didn’t work so nobody knew what I was talking about.
I'm self-studying through Amann/Escher Analysis 1 and I'm stuck on a problem in I.8. Here's ...
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0
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34
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A proof involving Vieta's formulas
If $x,y,z$ are complex numbers satisfying $x+y+z = 1, xy+ xz + yz = 1, x yz = 1,$ then must they be the roots of the polynomial $t^3 - t^2 + t - 1$? Also, can this formula be generalized: if $x_1,\...
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1
vote
0
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20
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Zonal quaternionic polynomials
I've done a R package which computes the Jack polynomials (I started it a couple of years ago). It also computes the zonal polynomials (Jack with $\alpha=2$ up to a normalizing factor), the Schur ...
- 1,317
2
votes
0
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18
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What are the Littlewood-Richardson coefficients for hook-shaped diagrams?
I am trying to compute Littlewood-Richardson coefficients involving hook-shaped diagrams. In particular, if $\lambda$ is a hook-shaped Young diagram, $\rho$ is any diagram, and we consider,
$$
s_\...
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1
vote
1
answer
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proving that $p$ divides $f(1,2,\ldots, p -1)$
Let $p$ be a prime and let $f(x_1,\ldots, x_{p-1})$ be a symmetric polynomial in $p-1$ variables. Suppose $f$ is homogeneous of degree $d$ with $(p - 1) \not | d$. Prove that $p$ divides $f(1,2,\ldots,...
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0
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Numerical methods for solving algebraic equations of symmetric linear forms
Question: Given a set of polynomial algebraic equations for $x_i\in\mathbb{R}^n$ of the form
\begin{equation}\label{eqs}
x_i = f(x_1,...,x_{i-1},x_{i+1},...,x_{d}), \quad \mathrm{for} \quad i=1,...,d,...
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0
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63
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monomial symmetric polynomials
It's necessary to prove
$m_\lambda=\sum\limits_{\nu}^{}s_\nu$
where $\lambda=(\lambda_1,...,\lambda_n)$ is partition of n, $\nu$=$\sigma(\lambda)$, $\sigma\in S_n$, $s_\nu$ is Schur polynomial.
I ...
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2
votes
1
answer
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Dimension of the symmetric\alternating k-tensor over an $n$-dimensional vector space.
I want to solve this question:
Suppose $V$ is a vector of dimension $n$ over a field $F$ of characteristic not equal to 2. Calculate dim $Sym^{k}(V)$(the symmetric k tensor ).
I know that $(Sym^k(V))^*...
- 1,191
1
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1
answer
43
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proving a claim about symmetric polynomials
Define an $S_n$-invariant polynomial $f$ of $\mathbb{Z}[x_1,\cdots, x_n]$ so that $f(x_1,\cdots, x_n) = f(x_{\sigma(1)},\cdots, x_{\sigma(n)})$ for any permutation $\sigma\in S_n$, the set of ...
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$\sigma_1\sigma_2 =\Sigma_{i<j}{~~(z_i^2z_j+z_iz_j^2})+3\Sigma_{i<j<k}{~~z_iz_jz_k}$
Let $P\in\mathbb{R}[X]$ of degree $n$ and $z_1,...,z_n $ the complex roots of $P$. We consider that $\sigma_1(z_1,...,z_n)=\Sigma_{i}z_i$ and $\sigma_2(z_1,...,z_n)=\Sigma_{i<j}z_iz_j$. I read in a ...
- 1,203
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Alternating polynomials under the diagonal action
Consider the (usual) action of the symmetric group $\mathcal{S}_n$ on the polynomials $\mathbb{R}[x_1,\ldots ,x_n]$ given by
$$
\sigma\colon p(x_1,\ldots ,x_n)\mapsto p(x_{\sigma_1},\ldots ,x_{\...
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0
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Implementation of symmetric tensor decomposition algorithm
Context
Any symmetric tensor F of rank $d$ and dimension 2 ($F \in S^d\mathbb{C^2}$ for our purpose) can be associated with a homogeneous polynomial $P(F)\in k[x_0,x_1]_d$ in 2 variables of degree $d$....
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Kind of elementary symmetric product
I have the following polynomial in the variables $A_1, A_2, ..., A_n;B_1, ...,B_n:$
$ f(A_1, A_2, ..., A_n;B_1, ...,B_n) = A_1B_2B_3...B_n + B_1A_2B_3...B_n + ...+B_1...B_{n-1}A_n$
Without the $A_j$ I ...
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How to prove $\operatorname{LM}\left(\prod_{1\le k\le n}\sigma_{n;k}\right)=\prod_{1\le k\le n}\operatorname{LM}(\sigma_{n;k})$?
The "elementary symmetric polynomials" $\sigma_{n;k}$ are:
$$\sigma_{n;k}(x_1,\ldots,x_n)=\sum_{1\le i_1\lt\cdots\lt i_k\le n}x_{i_1}\cdots x_{i_k}$$
where $k=1,\ldots,n$. (...) Every ...
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Efficient way to derive coefficient from (1+t) to t
Suppose I have the coefficient $b_i$ of degree $d$ polynomial defined on variable $1+t$.
Can i get coefficient $c_i$ that defined on variable $t$ in linear time ?
Namely,
$$
\sum_{i=0}^d b_i(1+t)^i = \...
- 1,111
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0
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Relation between two polynomial subrings
Let $\mathbb{Z}[x_1,x_2,\ldots, x_n]$ be ring in $x_1,\ldots, x_n$ with coefficients from $\mathbb{Z}$.
Let $e_1=x_1+x_2+\cdots + x_n$, $e_2=\sum_{i<j} x_ix_j$, $\cdots$, $e_n=x_1x_2\cdots x_n$ : ...
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1
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How to prove this expression is a polynomial
How can we prove that the expression
$$\gamma_{m k}(\lambda)=\sum_{i=1}^{m}\left(\lambda_{m i}+m-1\right)^{k} \prod_{j \neq i}\left(1-\frac{1}{\lambda_{m i}-\lambda_{m j}}\right)$$
is a polynomial in ...
2
votes
0
answers
34
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Fractional exponent elementary symmetric polynomials.
I am wondering if there is any literature on relations between fractional power symmetric polynomials. For a particular example, with the variables $\textbf{x} = (x_1,x_2,\dots x_n),$, can we ...
- 11.7k
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Proof and meanings of Lagrange's theorem for polynomials
I'm trying to understand Lagrange's theorem which I found in Harold Edward's book ''Galois Theory''. I was trying to find this theory in internet but all Lagrange's theorems which I found are not ...
- 479
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Terminology of "algebraically closed rings"
There are various approaches how to generalize the notion of an algebraically closed field to the context of commutative rings. A good survey is R. Raphael, On algebraic closures. I am interested in ...
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1
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Computing certain polynomials related to symmetric functions and $\lambda$-rings in Sage
The definition of a $\lambda$-ring (https://en.wikipedia.org/wiki/%CE%9B-ring) makes use of certain "universal" polynomials $P_n$ and $P_{n,m}$, which basically give you the formulas for ...
- 20.6k
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1
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Prove for general form of function at -x containing derivatives of order n
I have stumbled across multiple casses of functions (explicitly Hermit and Legendre polynomials) for which I wanted to prove the symmetry. While doing so I always ended up with the following equations:...
0
votes
1
answer
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Factoring the sum of polynomial coefficient "division"
Define the coefficient $B = (b_0, \cdots b_n)$ of elementarty polynomial product of $A=(a_0 \cdots, a_n)$. $B = \sum_{i=0}^N b_ix^i = \prod_{i=0}^N(1+a_ix) $.
Use $f(A) = B$ to get it's coefficient.
...
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Invariant polynomial function on Lie algebras.
Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \...
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Sum of powers of variables
Let $n$ be a fixed positive integer, and
$$
\begin{cases}
y_1+y_2+\ldots+y_n=x_1+x_2+\ldots+x_n,\\
y_1^2+y_2^2+\ldots+y_n^2=x_1^2+x_2^2+\ldots+x_n^2,\\
y_1^3+y_2^3+\ldots+y_n^3=x_1^3+x_2^3+\ldots+x_n^...
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If $A$ is a real symmetric matrix with full rank and only simple roots, what can be said about $|B\otimes I+I\otimes B|$, for $B = \sum_k a_k A^k$?
I'm considering a situation in which $A, B > 0$ are real $(p\times p)$ symmetric matrices where $A$ has $p$ distinct eigenvalues $\{\lambda_i\}_{i=1}^p$, and $AB=BA$. I'm interested in the level of ...
0
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0
answers
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How to derive the mentioned relations (in part a-b-c of example 9) from the Weyl denominator formula?
I want to understand Example 9 (a-b-c) of section 1.5 of the book "Symmetric functions and Hall polynomials (https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)" (pages 78-79):
In ...
1
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0
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Solving $\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$ with symmetric polynomials
I need to solve the following with symmetric polynomials, but I do not understand what to do here.
$$\sqrt[3]{\sqrt {x-1} - 2} = 1-\sqrt[3]{9-\sqrt{x-1}}$$
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1
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How to solve this equation with symmetric polynomials? [closed]
Here is what I did
$$x + \sqrt {17 - x^2} + x\times\sqrt{17 - x^2} = 9$$
I can`t undestand how to solve it, any help would be appreciated!
11
votes
1
answer
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What does Heron's formula naturally deform?
Fixing three real numbers $a,b,c>0$ determines a triangle with
side-lengths $a,b,c$ (if admissible). Therefore, the area of a
triangle is a function in $a,b,c$. Due to the geometry of a
triangle, ...
- 1,676
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votes
1
answer
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What is the proper math notation for a particular sum of products?
Suppose we have a sequence of distinct $a_i$'s: $\left\{ a_1, a_2, \ldots, a_n \right\}$. We also have a sequence of not necessarily distinct $b_i$'s: $\left\{ b_1, b_2, \ldots, b_n \right\}$. Each $...
1
vote
0
answers
37
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Efficient way of simplify sum of product of multiple polynomials
Let $A \in \mathbb{R}^{n\times m}, B \in \mathbb{R}^m$.
I'm trying to compute the coefficients of $n$ polynomials $C_i = (c_0^i, c_1^i, \cdots, c_{n-1}^i)$.
where $\displaystyle \sum_{j=0}^{n-1} c_j^i ...
- 1,111
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0
answers
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Determine if a function is monotonic
Let $a, b, c, d \in \mathbb{R}$ and $a, b, c, d\gt 1$.
For any polynomial $A_n = \sum_{i=0}^n a_ix^i$, i'm interested in a quantity $g(A_n) = \frac{1}{n}\langle (a_0, \cdots, a_n), (1/{n \choose 0}, \...
- 1,111
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votes
1
answer
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Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$
Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}$ , $b=\sqrt{3}-\sqrt{5}+\sqrt{7}$, $c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate:
$$\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$$
What I have tried so far is writing the ...
- 359
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votes
1
answer
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Root transformation to make normalized coefficient match
Let $f(x) = \prod_{i=0}^n(1+a_ix), a_i \neq 0$, and $C(f(x)): = (c_0, c_i, \cdots, c_n) $, where $f(x) = \sum_{i=0}^n c_i x^i$.
For a given monomial $g(x) = (1+mx)(1+nx)$, i'm interested in a quantity ...
- 1,111
3
votes
1
answer
73
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Reciprocal binomial coefficient polynomial evaluation
The conventional binomial coefficient can be obtained via
$$
f(x, n) = (1+x)^n = \sum_{i=0}^n { n \choose i} x^i
$$
And the function $f$ can be every efficiently performed on evaluation.
I'm ...
- 1,111
0
votes
0
answers
17
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Is it possible to obtain roots from the weighted sum of other polynomials in their root form?
Given $n$ polynomials $A_1, A_2, .. A_n$with the same degree $M$.
$A_i = \prod_{j=0}^M(1+Q_{ij}x)$.
In their root form $Q$, $Q_{ij} \in \mathbb{R}$. And the function, I'm interested in, $B$ is a ...
- 1,111
0
votes
0
answers
87
views
How does the fundamental theorem of symmetric polynomials imply that this number is rational?
In the Problems from the Book by Titu Andreescu, there is a proof of Example 9 on page 494 with the following:
Example 9. Let $f$ be a monic polynomial with integer coefficients and let $p$ be a ...
- 13.2k
1
vote
0
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38
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General formulation of polynomial invariants.
Reading this work https://arxiv.org/pdf/2012.06452.pdf, I'm wondering if it's true that any fundamental invariant polynomial $\{f_i\}_{i=1}^{N_{inv}}$ can be written as
$$f_i(x) = \sum_{g \in G}\psi(g ...
- 1,702
0
votes
2
answers
43
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Schur polynomial of partition (1,0)
Schur polynomials $s_\lambda$(or Schur functions) for a partition $\lambda$ is given by the bialternant formula of Jacobi, which is a ratio of two Vandermonde determinants:
$$s_\lambda (x_1,x_2,\cdots,...
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votes
0
answers
45
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Galois group of general equation of degree n
Let $K$ be a field and $\{t_1,t_2,\ldots,t_n\}$ be an algebraically independent set over $K$.
For every permutation $\sigma \in S_n$ we have an automorphism of $K(t_1,t_2,\ldots,t_n)$ given by mapping ...
- 243
1
vote
0
answers
16
views
Definition of Macdonald polynomial $P_\lambda^{\mathfrak{g}}$ associated to a Lie algebra $\mathfrak{g}$ (unlike $P_\lambda$)
I want to find the definition for the Macdonald polynomial associated to a Lie algebra $\mathfrak{c}_n$, i.e. $P_\lambda^{\mathfrak{c}_n}(x,t,q)$. This appears in the physics paper https://arxiv.org/...
- 433
8
votes
0
answers
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views
Local max of ratio of elementary symmetric polynomials
Let $e_k(x_1,\cdots, x_n) := \sum_{1\leq j_1 < \cdots < j_k \leq n}x_{j_1}\cdots x_{j_k}$ be elementary symmetric polynomials (see e.g. https://en.wikipedia.org/wiki/...
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0
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$x_i>0, \forall\ 1\leq i\leq n$ is equivalent to $\sigma_1,\cdots,\sigma_n$ are all $>0$, where $\sigma_i$ are the elementary symmetric polynomials. [duplicate]
$x_i>0, \forall\ 1\leq i\leq n$ is equivalent to $\sigma_1,\cdots,\sigma_n$ are all $>0$, where $\sigma_i$ are the elementary symmetric polynomials.
Here $\sigma_1=x_1+\cdots+x_n, \sigma_2=...
- 2,031
0
votes
0
answers
75
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$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$
Problem: Prove that in any triangle ABC, the following inequality is true: $$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$$
Anyone can help me? I tried AM-GM for right ...
- 409
0
votes
0
answers
20
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Mathematica package for computing Macdonald polynomials
0
I want to implement computation of Macdonald polynomials in mathematica. A similar question was raised in another question 5 years ago (Macdonald-Koornwinder polynomials?), but received no clear ...
- 433
3
votes
1
answer
89
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Prove that: $2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$
Problem: For $a,b,c\ge0: ab+bc+ca>0.$ Prove that: $$2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$$
Recently, i have seen a post on AoPS link
My approach: ...
- 409
2
votes
2
answers
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In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$
Problem: Given a,b,c are length of triangle. Prove that: $$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$$
Happy Vietnamese Women's ...
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