Questions tagged [sum-of-squares-method]
Proofs of inequalities by the Sum of Squares method (SOS).
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How to write $\displaystyle \sum\limits_{\text{cyc}}(a-b)(a-c)(a-d)(a-e)$ as the sum of squares?
How to write $\displaystyle \sum\limits_{\text{cyc}}(a-b)(a-c)(a-d)(a-e)$ as a sum of squares?
This problem comes from 1971 IMO problem 1,which is stated as follows. Prove that the following ...
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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:
$$\sum\left(-ab+ac-5bc\right)\left(a-b\...
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Solving the inequality $\sum_{cyc a, b, c}a(\sqrt[3]{b/a} -b) \leq 2/3$ with constraint $a+b+c = 1$
I was doing some contest math exercises, and stumbled onto this problem
Let $a, b, c$ be positive real numbers such that $a + b+ c = 1$. Show that $$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 $$\...
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Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$
Let $a,b,c>0 : a+b+c=4.$ Find minimum $$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}.$$
For $a=b=c=\dfrac{4}{3},$ I got a value $\dfrac{15}{4}$
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Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?
Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7.$
Calculus way:
Let $A=\sqrt{1-2xy}$, $B=\sqrt{1-2yz}$, $C=\sqrt{1-2zx}$, $f(x,y,z)=A+B+C$ and $g(x,y,z)=x+y+z-1.$
By the ...
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Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$
Let $a,b,c\ge 0: ab+bc+ca>0$ such that $a^2+b^2+c^2=a+b+c.$ Find minimum
$$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\...
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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 ...
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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 ...
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