Questions tagged [sum-of-squares-method]

Proofs of inequalities by the Sum of Squares method (SOS).

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when could $5^m+5^n$ represented as sum of two squares [closed]

How can we prove that $5^m+5^n$ could be expressed as a sum of two squares if and only if $m-n$ is even with $m,n\in\mathbb{Z}_{>0}$ I was able to prove that any power of $5$ could be expressed as ...
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Prove $\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$

Let $a,b,c>0: abc=1.$ Prove that$$\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$$ I've tried to use equivalent steps but it is quite complicated. ...
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$a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right) \geq 3\left(a^3b+b^3c+c^3a\right)$

Let: $a,b,c >0$. Prove that: $$a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right) \geq 3\left(a^3b+b^3c+c^3a\right)$$ I read an ugly solution: Take $LHS-RHS$, get: $$\sum\left(-ab+ac-5bc\right)\left(a-b\...
Lục Trường Phát's user avatar
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Solving the inequality $\sum_{cyc a, b, c}a(\sqrt[3]{b/a} -b) \leq 2/3$ with constraint $a+b+c = 1$

I was doing some contest math exercises, and stumbled onto this problem Let $a, b, c$ be positive real numbers such that $a + b+ c = 1$. Show that $$a\left(\left(\frac{b}{a}\right)^{1/3} -b\right) + ...
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If $a^2+b^2+c^2=1$ prove $\frac{a^4-a^2}{bc-1}+\frac{b^4-b^2}{ca-1}+\frac{c^4-c^2}{ab-1}\le ab+bc+ca$

Let $a,b,c\ge 0: a^2+b^2+c^2=1.$ Prove that $$\frac{a^4-a^2}{bc-1}+\frac{b^4-b^2}{ca-1}+\frac{c^4-c^2}{ab-1}\le ab+bc+ca$$ Here is just my thought: After clear denominator, it suffices to prove $$\...
Dragon boy's user avatar
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Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$

Let $a,b,c>0 : a+b+c=4.$ Find minimum $$T=\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}.$$ For $a=b=c=\dfrac{4}{3},$ I got a value $\dfrac{15}{4}$ ...
Dragon boy's user avatar
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Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?

Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7.$ Calculus way: Let $A=\sqrt{1-2xy}$, $B=\sqrt{1-2yz}$, $C=\sqrt{1-2zx}$, $f(x,y,z)=A+B+C$ and $g(x,y,z)=x+y+z-1.$ By the ...
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Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$

Let $a,b,c\ge 0: ab+bc+ca>0$ such that $a^2+b^2+c^2=a+b+c.$ Find minimum $$M=\frac{a^2(b+c)}{b^2+bc+c^2}+\frac{b^2(c+a)}{c^2+ca+a^2}+\frac{c^2(a+b)}{a^2+ab+b^2}$$ When $a=b=c=1$ we get minimum is ...
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Prove $\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right)$ when $xy+yz+zx=1$

Given $x,y,z$ be non negative real numbers satisfying $xy+yz+zx=1.$ Prove that $$\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right). $$ My thoughts is proving$$xy\left(2\...
Anonymous's user avatar
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Find minimum and maximum $P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$

Problem. Let $a,b,c$ be real numbers. Find minimum and maximum$$P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$$ I worked on maximum number of days but I did not find ...
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Prove: $\sum\limits_{cyc} \sqrt{9a^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$ with $a,b,c>0.$

Let $a,b,c>0.$ Prove that $$ \sqrt{9a^2+(a+b+c)^2}+\sqrt{9b^2+(a+b+c)^2}+\sqrt{9c^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$$ I see it on Facebook here. I tried Mincopxki, but it doesn't ...
Nguyễn Thái An's user avatar
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Decomposition of nonnegative polynomial on interval into sum of squares

My professor went over the following theorem Consider a univariate polynomial $p(x)$. Then, If $p(x)$ has degree $2d$, then $p(x)$ is nonnegative on $[-1,1]$ if and only if there exists sum-of-...
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Relationship between sum of squares and square of sums in a recursive way

In the proof of Theorem 2 of Sparse projections onto the simplex authors mention the following equality for any $\mathbf{b} \in \mathbb{R}^k$ and $\lambda \in \mathbb{R}$: $$ \begin{aligned} &\...
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Write $7 \cdot 10^{100} + 7$ as a sum of four squares

How do you write $7 \cdot 10^{100} +7$ as a sum of four squares? I know that you can write it as a sum of four squares by the Lagrange's Four Squares Theorem, but I don't know how to write such a big ...
user1052623's user avatar
5 votes
6 answers
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Prove that $(\sum x^2)^3\ge9\sum x^4yz$

Prove that $\displaystyle\left(x^2+y^2+z^2\right)^3\ge9\left(x^4yz+y^4xz+z^4xy\right)$, for $x$, $y$, $z\in\Bbb R_+$. The $pqr$ method doesn't seem possible because the power is too high. $$\iff\left(...
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180 views

How To Prove $a^3b+b^3c+c^3a>a^2b^2+b^2c^2+a^2c^2 $ if $ a > b > c > 0\,$?

How to prove this : $$a^3b+b^3c+c^3a>a^2b^2+b^2c^2+a^2c^2 $$ if we know: $$ a > b > c > 0 $$ My attempt: $$\frac {a^3b+b^3a}{2}>a^2b^2 ...(1)$$ $$\frac {b^3c+c^3b}{2}>c^2b^2...(...
Moustapha_M_I's user avatar
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5 answers
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Showing $\sum_{cyc}\frac{a^2+bc}{b+c}\geq a+b+c$ for positive $a$, $b$, $c$

The following is an inequality which I have trouble solving: $$\frac{a^2+bc}{b+c}+\frac{b^2+ac}{c+a}+\frac{c^2+ab}{a+b}\geq a+b+c$$ ($a, b, c>0$) I tried multiplying LHS and RHS by 2 and then use $$...
TheUnknown1's user avatar
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1 answer
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deform equation into sum of squares

I met an equation in a paper: \begin{equation} \mu^2(\mu^2-rs)=(\mu^2-rs)(a+b)+2ab\mu+a^2s+b^2r \end{equation} we already have known $\mu^2<rs$ and $r>0$.In the paper, the author gives a magical ...
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Smallest positive $k$ such that $\sum_{cyc}\left(\frac{(k-1)(z+x)}{y}-\frac{3}{2}\right)(x-y)^2\geq 0$

Let $x,y,z$ be positive real numbers. Find the smallest positive $k$ such that $$\sum_{cyc}\left(\frac{(k-1)(z+x)}{y}-\frac{3}{2}\right)(x-y)^2\geq 0$$ holds true for $x,y,z$. I proved it for $k=\frac{...
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On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS. Definition 2. If an $n$-variate polynomial ...
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Writing $1-xyz$ as a sum of squares

Can you write $1 - xyz$ in the form $p + q (1 - x^{2}-y^{2}-z^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z]$? For instance, in ...
Abhiroop Sanyal's user avatar
1 vote
3 answers
133 views

Decomposing $\sum_{i = 0}^{2n} x^i$ as a simple sum of squares

As we have $\sum_{i = 0}^{2n} x^i = (x^{2n + 1} - 1) / (x - 1)$, the polynomial is positive. So we know that there is a decomposition as a sum of squares. Is there a closed simple form for such a ...
Lolo's user avatar
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2 answers
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Find parameters $a,b$ such that $x^6-2 x^5+2 x^4+2 x^3-x^2-2 x+1-\left(x^3-x^2+a x+b\right)^2>0$

The probrem is to prove that $$x^6-2 x^5+2 x^4+2 x^3-x^2-2 x+1>0.$$ (the minimum value is about 0.02, tested by wolframalpha.) I use sos(sum of squares) method, my idea is to reduce the degree of ...
lapcal's user avatar
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Sum of Two Squares of a Quartic

I am trying to write a quartic in sum of two squares, can anyone help me with the following polynomial: $$6y_0^{4}+6y_0^{2}y_1^{2}+y_1^4+4y_0y_1^{2}y_2+4y_0^{2}y_2^{2}+ 6y_1^{2}y_2^2+6y_2^{4},$$ one ...
Bill's user avatar
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Prove that: $a+b+c+a^2+b^2+c^2\ge\sum_{cyc}{\sqrt{a(a^3+b+c)}}$

Given $a,b,c$ be non-negative real numbers such that: $ab+bc+ca+2abc=1.$ Prove that: $$a+b+c+a^2+b^2+c^2\ge\sqrt{a(a^3+b+c)}+\sqrt{b(b^3+c+a)}+\sqrt{c(c^3+a+b)}$$ I tried AM-GM for right side: $$\sum_{...
Sickness's user avatar
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3 answers
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How to prove:$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+a+b+c\ge\sum_{cyc}{\sqrt{2(a^2+b^2)}}$

Problem: Let $a,b,c>0. $ Prove that: $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+a+b+c\ge\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}$$ I have seen problem before, and I tried to prove: $$...
Sickness's user avatar
1 vote
1 answer
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Nice problem: Prove that: $ab+bc+ca \ge \sum{\sqrt{a^2+b^2+3}}$

Problem: Let $a,b,c>0:a+b+c=abc.$ Prove that: $$ab+bc+ca\ge \sqrt{a^2+b^2+3}+\sqrt{b^2+c^2+3}+\sqrt{c^2+a^2+3}$$ Please help me give a hint to get a nice proof! My attempts after squaring both side,...
Sickness's user avatar
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Find another sum of squares for $3^{12}-6^6+2^{12}$

I have a question about factorization of number $3^{12}-6^6+2^{12}$. By completing the square one can show that$$3^{12}-6^6+2^{12} = (3^6-2^6)^2+6^6 = 665^2+216^2$$ If we can find another ...
serg_1's user avatar
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2 answers
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How do I prove that $\sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge6(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})$ given $x,y,z > 0$ and $x+y+z=3$?

I tried to apply AM-GM inequality: $$ \dfrac{1}{x^2-xy+y^2} + (x^2-xy+y^2) \ge 2 \implies \dfrac{1}{x^2-xy+y^2} \ge 2 - (x^2-xy+y^2) $$ Then, $$ \sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge 21-2(...
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3 votes
4 answers
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Find the minimum value of $a^8+b^8+c^8+2(a-1)(b-1)(c-1)$

Let $a,b,c$ be the lengths of the three sides of the triangle, $a+b+c=3$. Find the minimum value of $$a^8+b^8+c^8+2(a-1)(b-1)(c-1)$$ My attempts: $\bullet$ The minimum value is $3$, equality holds iff ...
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0 votes
1 answer
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How to write a square of a trigonometric polynomial cosine?

How to write a square of a polynomial of the form $$\left(1 + 2\sum_{k=1}^n a_k \cos k \theta\right)^2$$ with an explicit formula for just the coefficient of $$\cos k\theta$$ in terms of $k$ and the ...
V Bert's user avatar
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7 votes
4 answers
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Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$

Let $a,b,c>0$: Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$ My solution: We have: $\left[\begin{matrix}\frac{1}{x}+\frac{1}{y} \geq \...
Leomessi's user avatar
6 votes
3 answers
278 views

how to prove $\sum_{cyc}(1-a_{1}+a_{1}a_{2})^2\ge\frac{n}{2}$

let $a_{i}\in [0,1]$,prove or disprove $$f_{n}=(1-a_{1}+a_{1}a_{2})^2+(1-a_{2}+a_{2}a_{3})^2+\cdots+(1-a_{n}+a_{n}a_{1})^2\ge\dfrac{n}{2}$$ I can only prove $n=3$. $$f_{3}=\sum_{cyc}[1-a_{1}(1-a_{2})]^...
math110's user avatar
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4 votes
2 answers
152 views

Prove $\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3$ when $x^2+y^2+z^2+(x+y+z)^2\leq 4$.

Let $x,y,z\in \Bbb{R}^+$ such that $x^2+y^2+z^2+(x+y+z)^2\leq 4$. Prove that $$\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3.$$ As there are three fractions in the left side and a single term in the right ...
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1 answer
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show this inequality $\sum_{cyc}\sqrt{a(b+c)}(b^2+c^2-a^2-bc)\ge 0$

let $a,b,c>0$,show this inequality $$\sqrt{a(b+c)}(b^2+c^2-a^2-bc)+\sqrt{b(c+a)}(c^2+a^2-b^2-ca)+\sqrt{c(a+b)}(a^2+b^2-c^2-ab)\ge 0$$ I want use S-O-S methods to solve ,But I can't, see this ...
math110's user avatar
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5 votes
3 answers
244 views

Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $

Show that for positive reals $x,y,z$ the following inequality holds and that the constant cannot be improved $$ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \...
vand's user avatar
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2 votes
1 answer
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Are elements in each stage of the Lasserre hierarchy convex?

The Lasserre hierarchy is a schema for proving multivariate polynomials positive via a sum of squares decomposition. At the first level, a polynomial $p$ is written $$p = \sum_i f_i^2$$ where each $\...
Alex Meiburg's user avatar
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7 votes
6 answers
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Prove that $\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}$.

Let $a,b,c$ be positive real numbers. Prove that $$\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}.$$ I want to prove the inequality with ...
Oshawott's user avatar
  • 3,956
2 votes
2 answers
102 views

$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz$

Prove that $$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz,\quad\forall x,y,z>0,$$ where $\sum_{sym}$ is the symmetric sum notation. Context: I was reading about the Muirhead inequality and ...
Elgar Gumm's user avatar
2 votes
2 answers
117 views

Prove that $a^3b^3 + b^3 + 1 \geq a^2b^2 + ab^3 + b$ (for real, positive $a$,$b$

I was working on an Olympiad-level inequality, which I was able to boil down to the following inequality: Prove that: $a^3b^3 + b^3 + 1 \geq a^2b^2 + ab^3 + b$ I think it's more useful to write it as: ...
Aditya Gupta's user avatar
1 vote
2 answers
104 views

solving $\sqrt{2 x^4-3 x^2+1}+\sqrt{2x^4-x^2}=4 x-3.$

I am trying to solve this equation $$\sqrt{2 x^4-3 x^2+1}+\sqrt{2x^4-x^2}=4 x-3.$$ By using Mathematica, I know that, the equation has unique solution $x=1$. I tried to write the equation in the form ...
minhthien_2016's user avatar
4 votes
3 answers
212 views

Can any polynomial in $\Bbb R [x]$ be written as the difference of two sum-of-squares polynomials in $\Bbb R [x]$?

Let $f \in \Bbb R[x_1,\dots,x_n]$ be an arbitrary polynomial, not necessarily non-negative. Are there always two sum-of-squares (SOS) polynomials $g, h \in \Bbb R[x_1,\dots,x_n]$ such that $f=g-h$? If ...
kxxwz's user avatar
  • 143
2 votes
1 answer
266 views

Choi Lam homogeneous polynomials as sums of squares

I came across two polynomials that Choi and Lam gave in 1976, that are not sum of squares of polynomials, despite being evidently non-negative by AM-GM $$ S(x,y,z) = x^4 y^2 + y^4 z^2 + z^4 x^2 - 3 x^...
Will Jagy's user avatar
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2 votes
1 answer
576 views

Find $g(x,y,z);p(x,y,z)\ge 0$ so that $f(x,y,z):=x\cdot g(x,y,z)+p(x,y,z)$

We have the following fact: (I don't remember where I read it, but there is.) If $f(x)$ is which is non-negative for $x\ge 0,$ then $f(x)=g(x)+x\cdot h(x),$ where $g(x)$ and $h(x)$ are SOS. So I ...
NKellira's user avatar
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6 votes
1 answer
432 views

Prove that there exist four polynomials $p_1,p_2,p_3,p_4$ in $x,y,z$ so that $(x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2$

Prove that there exist four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that $$\left ( x^{2}+ y^{2}+ z^{2} \right )^{3}- 8\left ( z^{3}x^{3}+ x^{3}y^{3}+ y^{3}z^{3} \right )= p_{1}^{2}+ ...
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3 answers
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Trigonometric Equation - can this be solved using complex numbers

How do I solve the following: $\cos (12x) = 5 \sin (3x) + 9 \tan^2( x )+ \cot ^2 (x)$ for $x \in (0,360)$ I tried converting cos and sin term into single angle i.e. into x but the equation becomes ...
Mathronza's user avatar
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2 votes
1 answer
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Question about Sui Zhen Lin's proof for inequality $\sqrt{\frac{a^2}{9b^2-8b+4}}+\sqrt{\frac{4b}{a+4}}\leq 1$ with positive numbers $a,b$ so $a+b=1$

given two positive numbers $a, b$ so that $a+ b= 1$ Sui Zhen Lin ; @szl6208 gave a very beautiful proof for the following inequality $$\sqrt{\frac{a^{2}}{9b^{2}- 8b+ 4}}+ \sqrt{\frac{4b}{a+ 4}}\leq 1$...
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3 votes
2 answers
177 views

To need a way of thinking about Ji_Chen's nice result $(a- c)^{2}+ (b- d)^{2}\geq\frac{7}{9}ab- \frac{7}{20}(c^{2}+ 4d^{2})$

given four real numbers $a, b, c, d$ Ji Chen gave a nice result on.AoPS $$\left ( a- c \right )^{2}+ \left ( b- d \right )^{2}\geq\frac{7}{9}ab- \frac{7}{20}\left ( c^{2}+ 4d^{2} \right )$$ The ...
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2 votes
1 answer
241 views

Rational rank one decomposition of symmetric positive semidefinite integer matrices

Problem: Given an $n\times n$ symmetric positive semidefinite (PSD) $\color{blue}{\textbf{integer}}$ matrix $Q$ with $\mathrm{Rank}(Q) = r$, find $\color{blue}{\textbf{integer}}$ vectors $u_i, i=1, \...
River Li's user avatar
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1 vote
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SOSTOOLS - Express a Matrix polynomials as Polynomial

We know that to express an polynomial as SOS, we can write it as a Matrix SOS Now we take an example, use SOStools in Matlab to get SOS for $$f(a,b,c)=16(a^2+b^2+c^2)^3-9\left[(a^3+3b^2c)^2+(3ac^2+b^3)...
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