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

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

310 questions
Filter by
Sorted by
Tagged with
1 vote
178 views

### 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 ...
71 views

### 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. ...
1 vote
105 views

• 707
1 vote
66 views

1 vote
62 views

### 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 ...
152 views

### 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 ...
1 vote
97 views

### 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-...
• 78
326 views

180 views

1 vote
87 views

### deform equation into sum of squares

I met an equation in a paper: $$\mu^2(\mu^2-rs)=(\mu^2-rs)(a+b)+2ab\mu+a^2s+b^2r$$ we already have known $\mu^2<rs$ and $r>0$.In the paper, the author gives a magical ...
1 vote
62 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 ...