Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,481 questions
2 votes
1 answer
39 views
Construct AM–GM Proofs of Muirhead Inequalities (From Majorization to Explicit Weighted AM–GM Chains)
Motivation I am experimenting with symbolic implementations of algorithms that, given a majorization relation between two exponent vectors, automatically generate Olympiad-style inequality proofs ...
- 6,087
2 votes
0 answers
163 views
Prove or disprove that $a+b+c \geq ab+bc+ca$, for positive $a$, $b$, $c$ satisfying $\sum_{cyc}\frac{1}{a+b+1}\geq1$ [closed]
Prove that for positive $a,b,c$, if $$ \frac{1}{a+b+1} + \frac{1}{b+c+1} + \frac{1}{c+a+1} \geq 1 $$ then $$a+b+c \geq ab+bc+ca$$ My attempt: expanded everything and stuck at $$ \begin{aligned} 2(a+b+...
- 37
4 votes
0 answers
105 views
How to comprehend a step of caculation from Ramanujan's Notebook?
$$\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\...
4 votes
0 answers
144 views
Prove $a^6+b^6+c^6 + \frac14(1 - a^2 - b^2 - c^2)^3 \ge \frac{11}{180}$ subject to $a^2+b^2+c^2 + \frac12(a+b+c)^2 \le 1$
Problem. Let $a, b, c \ge 0$ with $a^2 + b^2 + c^2 + \frac12(a + b + c)^2 \le 1$. Prove that $$a^6+b^6+c^6 + \frac{(1 - a^2 - b^2 - c^2)^3}{4} \ge \frac{11}{180}.$$ Equality case: $a = b = c = \sqrt{\...
- 51.4k
1 vote
0 answers
57 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
6 votes
1 answer
321 views
Finding a degree-$d$ polynomial over $\Bbb C$ that has no multiple roots
Let $h \in {\Bbb C}[x]$ be a univariate polynomial of degree $d > 3$ of the following form: $$ h(x)=x^d+ax^{d-1}+bx^{d-2}+cx^{d-3}+\cdots,$$ where $a, b, c \in {\Bbb C}$ are given. In other words, ...
1 vote
2 answers
201 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
1 vote
2 answers
280 views
Maximum value of $(a^2 + bc)(b^2 + ac)(c^2 + ab)$ when $(a+b)(b+c)(c+a) = 2$ for $a,b,c \in \mathbb{R_{\ge 0}}$
Let $a,b,c$ be non-negative real numbers satisfying $(a+b)(b+c)(c+a) = 2$. Find the maximum value of the expression $$P = (a^2 + bc)(b^2 + ac)(c^2 + ab).$$ Context: I am training to get selected into ...
- 13
10 votes
2 answers
505 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
3 votes
0 answers
99 views
Find Factorization of Symmetric Polynomials in Several Variables
Consider following symmetric polynomial in $a,b,c$: $$ P_1(a,b,c):= (5-a-b-c-2abc)^2 - (1-abc)(a+b+c-3)^2$$ Assume $a,b,c$ are subjected to relation $ab+bc+ac+abc=4$, so we can express $c = \frac{4-ab}...
- 10.1k
6 votes
4 answers
184 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 $...
- 579
-1 votes
1 answer
87 views
$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$
I'm looking for some ideas to solve the following inequality. Problem. Let $a,b,c\ge 0$ with $ab+bc+ca=1.$ Prove that$$\color{black}{\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}...
- 504
0 votes
1 answer
238 views
Creating lemmas to use in proving inequalities involving sums of radicals (square roots, cube roots, etc.)
I am very interested in a motivated systematic approach to generating lemmas such as those invoked by River Li to transform inequalities stated in terms of radicals into inequalities that are radical ...
- 57
5 votes
2 answers
216 views
Prove $ \sum\limits_{\mathrm{cyc}} \frac{bcd}{(a+1)^3} \le 1$ for non-negatives $a+b+c+d=3$
Problem. Let $a,b,c,d$ be non-negative real numbers such that $a+b+c+d=3$. Show that $ \dfrac{bcd}{(a+1)^3}+\dfrac{cda}{(b+1)^3}+\dfrac{dab}{(c+1)^3}+\dfrac{abc}{(d+1)^3} \le 1$ My work. WLOG, We ...
- 111
1 vote
3 answers
212 views
If $a,b,c>0$ such that $(a+b+c)^3=125abc$, then show that $ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$
Problem. Let $a,b,c$ be positive real numbers such that $$(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