Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,479 questions
3
votes
0
answers
94
views
How to comprehend a step of caculation from Ramanujan's Notebook?
$$\frac{x^5-a}{x^2-y}=\frac{y^5-b}{y^2-x}=5(xy-1)$$
solve the equation for x,y;a,b are arbitary
here is a problem from Ramnujan's Notebook.The procedure is
let$$x=\alpha+\beta+\gamma=S_1,y=\alpha\...
1
vote
0
answers
54
views
Has the topology or combinatorics of real polynomials parameterized by degree, critical points, and inflection points been studied?
I have been investigating a way to visualize a space of real polynomials by describing each polynomial through three integer features:
Its degree $d$
The number of real critical points (real zeros of ...
- 365
1
vote
2
answers
200
views
Prove that $\frac{7-4a}{a^{2}+2}+\frac{7-4b}{b^{2}+2}+\frac{7-4c}{c^{2}+2}\ge 3$
Let $a,b,c$ be real variables with $ab+bc+ca+abc=4.$ Prove that:$$\color{black}{\frac{7-4a}{a^{2}+2}+\frac{7-4b}{b^{2}+2}+\frac{7-4c}{c^{2}+2}\ge 3.}$$When does equality hold?
This inequality is ...
- 504
10
votes
2
answers
498
views
Proofs to Ji Chen's Lemma (Symmetric Function Theorem)
Symmetric Function Theorem, also commonly known as Ji Chen's Lemma, is a very powerful and beautiful result in algebraic inequalities. I have used it many times to prove Olympiad-level inequalities ...
- 510
6
votes
4
answers
178
views
Let $a,b,c$ and $d$ be distinct reals $:~~a>b>c>d>0;~~abcd=1$, and $a+b+c+d = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$. Show that $ad=bc=1$.
Here is the problem statement:
Let $a,b,c$ and $d$ be distinct real numbers such that $a>b>c>d>0$, $abcd=1$, and
$$a+b+c+d = \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$$
Show that $...
- 575
0
votes
1
answer
227
views
Creating lemmas to use in proving inequalities involving sums of radicals (square roots, cube roots, etc.)
I am very interested in a motivated systematic approach to generating lemmas such as those invoked by River Li to transform inequalities stated in terms of radicals into inequalities that are radical ...
- 57
1
vote
3
answers
211
views
If $a,b,c>0$ such that $(a+b+c)^3=125abc$, then show that $ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$
Problem. Let $a,b,c$ be positive real numbers such that $$(a+b+c)^3=125abc$$
Prove that :
$$ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$$
This problem ...
- 111
7
votes
4
answers
739
views
Cubic polynomial with equal absolute values at $6$ points [duplicate]
Let $P \in {\Bbb R} [x]$ be a cubic polynomial with real coefficients such that $$ |P(1)| = |P(2)| = |P(3)| = |P(5)| = |P(6)| = |P(7)| = 12 $$ Find the value of $\frac19 P(0)$
My approach so far:
...
- 151
1
vote
0
answers
106
views
Prove that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge \frac{Ap^3+Bpq+Cr}{Dp^3+Er}$ for $a, b, c > 0$
Problem (by KaiRain@AoPS).
Let $a, b, c > 0$ and $w \in [-3, -2]$. Prove that
$$\frac{a}{b} +\frac{b}{c}+\frac{c}{a} \ge \frac{Ap^3+Bpq+Cr}{Dp^3+Er} $$
where $p := a + b + c, q := ab + bc + ca, r :=...
- 51.4k
0
votes
1
answer
140
views
Prove $\sum \frac{1}{\left(1+\sqrt{\dfrac{a}{bc}}\right)^2} \ge \frac{3}{4}, \forall \ a,b,c>0:a+b+c=3$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c>0: a+b+c=3$ then prove that$$\frac{1}{\left(1+\sqrt{\dfrac{a}{bc}}\right)^2}+\frac{1}{\left(1+\sqrt{\dfrac{b}{ca}}\...
- 957
2
votes
2
answers
112
views
$\sigma_i(A) = \frac{x^{n+1}-x^{i-m-1}}{x^i-1} \cdot \sigma_{i-1}(A)$
Assume that $\sigma_i(\{x_1,\ldots,x_n\})$ denotes the $i$th symmetric polynomial on $\{x_1,\ldots,x_n\}$ and set $A = \bigcup_{i = 0}^n \{x^i\} \cup \bigcup_{i = 0}^m \{x^{-i}\}$ for some $n,m \in \...
- 909
-1
votes
1
answer
78
views
Symmetric function theorem - example
The book that I am reading says that
The symmetric polynomial $u_1^2+u_2^2+\dots u_n^2$ ,
because it has degree $2$ , is
a linear combination $c_1s_1^2 + c_2s_2$ ,
where $s_1=\sum_{i\leq n} u_i$ and $...
- 517
0
votes
0
answers
28
views
References to complement Macdonald's "Symmetric Functions and Hall Polynomials"?
I am currently reading Macdonald's Symmetric Functions and Hall Polynomials and was interested in if there was another book that covered similar topics, specifically those topics directly mentioned in ...
2
votes
2
answers
221
views
Prove $\sum \sqrt{3x^2+1} \ge \sqrt{2x^2+2y^2+2z^2+30}$ for reals $x+y+z=3$
I'm looking for some ideas to solve the following inequality.
Problem. For any real numbers $x,y,z$ with $x+y+z=3,$ prove that$$\sqrt{3x^2+1}+\sqrt{3y^2+1}+\sqrt{3z^2+1}\ge \sqrt{2x^2+2y^2+2z^2+30}.$...
- 957
36
votes
2
answers
815
views
Is $a_1\cdots a_n + \frac1{a_1\cdots a_n}$ a polynomial in $a_i + \frac1{a_{i+1}}$?
Question
Let $n$ be a positive integer, and let $a_1,\ \dots,\ a_n$ be formal variables. Is
$$ P = a_1\cdots a_n + \frac1{a_1\cdots a_n} $$
a polynomial in the $n$ expressions $a_1 + \frac1{a_2},\ a_2 ...
- 363
3
votes
0
answers
126
views
Prove $\sum\limits_{cyc}\sqrt{\left(a^{2}+b^{2}\right)\left(c+1\right)}\ge 6$ for $a+b+c=3.$
Let $a,$ $b$ and $c$ be non-negative numbers such that $a+b+c=3$. Prove that:
$$\sqrt{\left(a^{2}+b^{2}\right)\left(c+1\right)}+\sqrt{\left(b^{2}+c^{2}\right)\left(a+1\right)}+\sqrt{\left(c^{2}+a^{2}\...
2
votes
2
answers
288
views
Prove that: $\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ca}}+\frac{1}{\sqrt{c^2+4ab}}\ge \frac{5}{4}$
Let $a,$ $b$ and $c$ be non-negative numbers such that: $a+b+c+abc=4$ and $ab+bc+ca\neq 0.$ Prove that $$\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ca}}+\frac{1}{\sqrt{c^2+4ab}}\ge \frac{5}{4}.$$
...
5
votes
1
answer
548
views
Prove that $\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1$ for real numbers
How do we prove that for all $a, b, c \in \mathbb{R}$,
$$\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1.$$
I haven't really made much progress in finding a way to tackle this....
- 920
3
votes
3
answers
237
views
Prove that $\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \ge \frac{3}{1 + \sqrt[3]{abc}}$ for $a\ge b \ge c > 0$ and $abc \ge 1$ and $ac \ge 1$
Problem. Let $a \ge b \ge c > 0$ with $abc \ge 1$ and $ac \ge 1$. Prove that
$$\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \ge \frac{3}{1 + \sqrt[3]{abc}}.$$
When I dealt with this question, I ...
- 51.4k
3
votes
5
answers
227
views
An olympiad-like inequality problem with square roots: $\sum_{c y c} a \sqrt{3 a^2+5(a b+b c+c a)} \geq \sqrt{2}(a+b+c)^2$
The problem seeks to prove
$$
\sum_{\text{cyc}} a \sqrt{3 a^2+5(a b+b c+c a)} \geq \sqrt{2}(a+b+c)^2
$$
when $a, b, c$ are nonnegative reals.
I exerted all my effort, only found that the equal ...
- 796
3
votes
5
answers
293
views
Prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ \ge\ \frac{12}{1+3abc}$ for positives $a + b + c = 3$
Problem. Let $a, b, c > 0$ with $a + b + c = 3$. Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ \ge\ \frac{12}{1+3abc}.$$
Motivation. The original question was closed. I think it is interesting ...
- 51.4k
1
vote
2
answers
120
views
Prove $\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$
Problem. Let $a,b,c$ are positive real numbers. Prove that
$$\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$$
The inequality is equivalent to $$...
user1656827
3
votes
2
answers
140
views
Prove $5(a+b+c)+\frac{9abc}{ab+bc+ca}\ge 2\sum_{cyc}\sqrt{4ab+4ac+bc}$
Problem. Let $a,b,c\ge 0$ and $ab+bc+ca>0.$ Prove that $$5(a+b+c)+\frac{9abc}{ab+bc+ca}\ge 2\left(\sqrt{4ab+4ac+bc}+\sqrt{4bc+4ba+ca}+\sqrt{4ca+4cb+ab}\right).$$
I've tried to use CBS inequality ...
user1656827
1
vote
1
answer
177
views
minimum of a function
Lets define complex-valued polynomials $P_n$ and $\bar{P}_n$
$$P_n=(e^{-i\theta}-x_1)(e^{-i\theta}-x_2)...(e^{-i\theta}-x_n)$$
$$\bar{P}_n=(e^{i\theta}-x_1)(e^{i\theta}-x_2)...(e^{i\theta}-x_n)$$
...
- 11
3
votes
2
answers
328
views
A general USA MO 2003
From a famous inequality
Problem 1. (USA MO 2003) Let $ a$, $ b$, $ c$ be positive real numbers. Prove that $$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \...
- 6,289