Questions tagged [sum-of-squares-method]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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_{...
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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: $$...
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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,...
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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 ...
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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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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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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 ...
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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 \...
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6 votes
3 answers
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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})]^...
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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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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 ...
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5 votes
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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}} \...
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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 $\...
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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 ...
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2 votes
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$3\sum_{sym}x^2y^2z+\sum_{sym}x^4y\ge 4\sum_{sym}x^3yz$

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 ...
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2 answers
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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: ...
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1 vote
2 answers
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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 ...
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4 votes
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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 ...
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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^...
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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 ...
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6 votes
1 answer
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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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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 ...
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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
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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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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, \...
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2 votes
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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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0 votes
1 answer
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Simplifying this equation to get sum of squares

$$\frac{1}{n\sum x_{i}^2-{(\sum x_{i}})^2}$$ I have this equation above. I am trying to simplify it such that I can get: $\dfrac{1}{nSS_{x}}$ Where $SS_{x}$ is the sum of squares of $x$. Any pointers ...
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1 vote
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Express $169$ as the sum of $1,2,3,4,5$ non-zero squares

I'm trying to solve the following exercise. Show that $169$ can be expressed as a sum of $1,2,3,4,5$ non-zero squares, and deduce that any $n \ge 169$ is the sum of five non-zero squares. The latter ...
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1 vote
4 answers
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If$x^2+y^2+z^2+t^2=x(y+z+t)$prove $x=y=z=t=0$

If $$x^2+y^2+z^2+t^2=x(y+z+t)$$ Prove $x=y=z=t=0$ I added $x^2$ to both side of the equation: $$x^2+x^2+y^2+z^2+t^2=x(x+y+z+t)$$ Then rewrite it as: $$x^2+(x+y+z+t)^2-2(xy+xz+xt+yz+yt+zt)=x(x+y+z+t)$$...
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28 votes
4 answers
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Proving $abcd+3\geq a+b+c+d$

If $a,b,c,d$ are non negatives and $a^2+b^2+c^2+d^2=3$ prove that $$abcd+3\ge a+b+c+d$$ The inequality is not as simple as it looks.The interesting part is that the equality occurs when $a=0,b=c=d=1$...
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3 votes
4 answers
246 views

Prove the polynomial $x^4+4 x^3+4 x^2-4 x+3$ is positive

Given the following polynomial $$ x^4+4 x^3+4 x^2-4 x+3 $$ I know it is positive, because I looked at the graphics and I found with the help of Mathematica that the following form $$ (x + a)^2 (x + b)...
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1 vote
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If the real zeroes of real polynomial $p(x,y)$ are disjoint points and curves, is $p(x,y)$ a positive sum of squares?

For example, $p(x,y) = x^2(x-1)^2 + y^2(y-1)^2$ has real zeroes in the set $\{(0,0), (0, 1), (1, 0), (1, 1)\}$ and admits a decomposition into a sum of squares. How can I find decompositions like this ...
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1 vote
1 answer
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Inequality $\frac{xy+z}{x+yz}+\frac{yz+x}{y+zx}+\frac{zx+y}{z+xy}-\frac{x+y+z}{3}\leq 1$

For $x,y,z \in [2,\infty)$, prove that $\frac{xy+z}{x+yz}+\frac{yz+x}{y+zx}+\frac{zx+y}{z+xy}-\frac{x+y+z}{3}\leq 1$ I tried to group the terms and prove that $\frac{xy+z}{x+yz} - \frac{y}{3}\leq \...
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  • 153
1 vote
2 answers
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prove thatt $\sum_{cyc} \frac{1}{{(x+y)}^2}\ge 9/4$

prove that $$\sum_{cyc} \frac{1}{{(x+y)}^2}\ge 9/4$$ where $x,y,z$ are positives such that $xy+yz+xz=1$ By Holder;$$\left(\sum_{cyc} \frac{1}{{(x+y)}^2} \right){\left(\sum yz+zx \right)}^2\ge {\sum \...
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1 vote
5 answers
104 views

$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$ Proof

Does anyone know hot to prove this inequality? Having: $a, b, c \gt 0$ $$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$$ I tried with the AM-GM inequality but I couldn't get ...
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1 vote
1 answer
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Prove $3\left(9-5\sqrt{3}\right) \sum \frac{1}{a} \geqslant \sum a^2+\frac32\cdot\frac{\left[(\sqrt3-2)(ab+bc+ca)+abc\right]^2}{abc}$

Let $a,\,b,\,c$ are positive real numbers satisfy $a+b+c=3.$ Prove that $$3\left(9-5\sqrt{3}\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant a^2+b^2+c^2 + \frac32 \cdot \frac{\left[(\...
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2 votes
2 answers
136 views

Prove $5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$

Problem. (?) For $a,b,c$ be non-negative numbers such as $a \geq 2(b+c).$ Prove:$$5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$$ My Solution. We write the ...
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0 votes
3 answers
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Proving a cyclic inequality

Show that $a^4 + b^4 + c^4 \geq a^3b + b^3c + c^3a$ for any postive integers $a, b, c$ I'm not sure how to approach this problem. I've tried assuming that WLOG $a > b > c$ so that it is clear ...
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2 votes
2 answers
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For $a,b,c>0$ proving $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geqslant a + b + c + \frac{4(a - b)^2}{a + b + c}$ [duplicate]

The problem with which I have a problem it's this: For $a,b,c>0$ prove that $$ \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geqslant a + b + c + \frac{4(a - b)^2}{a + b + c} $$ Titu's Lemma ...
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1 vote
1 answer
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How to formulate Augmented Langrangian as an objective function to SOS problem?

I am implementing ADMM algorithm to solve a complex optimization problem with SOS constraints. The x-update of the algorithm looks like this: $$ x^{k+1}= {\arg\min}_{x \in S} \|x-z^{k}+\lambda^{k}\|^...
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  • 137
0 votes
2 answers
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Sum of two squares question

Which of the following statements is true for every value of $n$? A: If $n$ is not a sum of two squares, then neither is 69$n$ B: If $n$ is a sum of two squares, then so is 34$n$ C: If $n$ is not a ...
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2 votes
0 answers
112 views

Expressing a polynomial as a sum of squares in Maple

While sostools in MATLAB would find such a sum of squares decomposition, I am wondering whether a similar package exists for Maple. Example. Express the following polynomial as Sum Of Squares:$$\frac{...
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  • 1,950
3 votes
0 answers
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Prove $\frac{4}{a^2 + 2b^2 + 3c^2 + 10} \le \frac{5a + 3b + c + 7d}{16(a+b+c+d)}$ for positive reals $abcd=1$

Problem: Let $a, b, c, d > 0$ with $abcd = 1$. Prove that $$\frac{4}{a^2 + 2b^2 + 3c^2 + 10} \le \frac{5a + 3b + c + 7d}{16(a+b+c+d)}.$$ Background Information: It is verified by Mathematica. This ...
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0 votes
2 answers
70 views

Explanation to an unmotivated step in an inequality.

Follows the original problem: Let $a, b, c$ be non-negative real numbers. Prove that $$ \frac{a^2}{a^2 + 2\left(a + b\right)^2} + \frac{b^2}{b^2 + 2\left(b + c\right)^2} + \frac{c^2}{c^2 + 2\left(c + ...
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3 votes
1 answer
126 views

A cyclic inequality of degree 10

Suppose that $x,y,z\geq 0$. I would like to prove that $$(x^5+y^5+z^5)^2\geq (x+y+z)(x^3y^6+y^3z^6+z^3x^6).$$ I can prove this inequality using some standard methods. For example, I can let $x=1, y=1+...
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  • 2,708
4 votes
3 answers
109 views

$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ for $a,b,c\in\mathbb{R}^+$ with $abc=1$

Suppose that $a,b,c$ are positive reals such that $abc=1$. Prove that $$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}.$$ Hint: Use Titu's lemma. My approach: I am trying to use Titu'...
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