# Questions tagged [sum-of-squares-method]

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

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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 ...
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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 ... 98 views

### 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 $\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}$.

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

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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### 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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### 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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### 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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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 $$S(x,y,z) = x^4 y^2 + y^4 z^2 + z^4 x^2 - 3 x^... • 135k 2 votes 1 answer 290 views ### Find g(x,y,z);p(x,y,z)\ge 0 so that f(x,y,z):=x\cdot g(x,y,z)+p(x,y,z) We have the following fact: (I don't remember where I read it, but there is.) If f(x) is which is non-negative for x\ge 0, then f(x)=g(x)+x\cdot h(x), where g(x) and h(x) are SOS. So I ... • 1,965 6 votes 1 answer 427 views ### Prove that there exist four polynomials p_1,p_2,p_3,p_4 in x,y,z so that (x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2 Prove that there exist four polynomials p_{1}, p_{2}, p_{3}, p_{4} in x, y, z so that$$\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}+ ... 92 views

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