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Questions tagged [sos]

Inequalities proofs by Sum of Squares method (SOS).

0
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3answers
55 views

Cyclic Olympiad Inequality

Given $a^2+b^2+c^2=1$ Prove $\sum_\text{cyc} \frac{1}{6ab+c^2}-\frac{1}{2+c^2}$ is nonnegative I have tried substituting 1 with $a^2+b^2+c^2$, but nothing is working. I’m trying to reduce it into a ...
1
vote
1answer
69 views

Decomposing bivariate quadratic form into sum of two squares

I would like to decompose $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ into two expressions, each involving only one variable. I'm trying to use a transform like $x_1 = x_+ + x_-$ and $x_2 = x_+ - x_-$ to ...
1
vote
1answer
126 views

Prove $\sum \sqrt{\frac{a^2}{6a^2+5ab+b^2}}\le \frac{\sqrt{3}}{2}$

Let $a,b,c\in R^+$ prove that the inequality $$\sqrt{\frac{a^2}{6a^2+5ab+b^2}}+\sqrt{\frac{b^2}{6b^2+5bc+c^2}}+\sqrt{\frac{c^2}{6c^2+5ca+a^2}}\le \frac{\sqrt{3}}{2}$$ My try:$$\sum\limits_{cyc} \sqrt{...
-2
votes
1answer
109 views

Inequality with $a+b+c=1$, a,b,c positive numbers

If a,b,c are real positive numbers, such as $ a+b+c=1$ prove that $ \left( ab+bc+ca\right) \left( 1+3abc\right) \geq 10abc $. Thank you! First try:$$\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)(...
7
votes
1answer
159 views

Inequality. ${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}} + {{1} \over {\sqrt{b}}} + {{1} \over {\sqrt{c}}}$

Question. If ${{a} \over {1+a}}+{{b} \over {1+b}}+{{c} \over {1+c}}=2$ and $a$, $b$, $c$ are all positive real numbers, prove that $${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}}...
0
votes
1answer
38 views

Inequality triangle Radon substitutions

I have this inequality: $$\sum \frac {a^3}{p-a}\geq 8(2R-r)^2$$ I have tried using Radon substitutions and I get this: $$\sum \frac{(y+x)^3}{x}\geq 8(2R-r)^2$$ I know from Holder that : $$\sum \frac{(...
0
votes
1answer
111 views

For $x$, $y$, $z$ the sides of a triangle, show $\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$

in $\triangle ABC$, let $AB=z,BC=x,AC=y$,show that $$\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$$ by well kown Iran 96 inequality $$(xy+yz+xz)\left(\frac{1}{(...
3
votes
1answer
123 views

If $ab+bc+ca=3$ for non-negative $a$, $b$, $c$, show that $\sum_{cyc}a^2b^2+\sum_{cyc}\frac{12a^2b^2c^2}{(a+b)^2}\ge 12abc$

Problem Let $a,b,c\ge 0$,and such $ab+bc+ca=3$, show that $$\sum_{cyc}a^2b^2+\sum_{cyc}\dfrac{12a^2b^2c^2}{(a+b)^2}\ge 12abc\tag{1}$$ A few hours ago, I asked for an error inequality. Wrong ...
5
votes
1answer
101 views

A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

Let $\triangle ABC$ have sides $BC=a$, $CA=b$, and $AB=c$. Let $m_a$, $m_b$, $m_c$ be the medians to $BC$, $CA$, and $AB$, respectively. Prove that $$(ab+bc+ca)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{...
0
votes
1answer
68 views

Prove that $\frac{3a^3+7b^3}{2a+3b}+\frac{3b^3+7c^3}{2b+3c}+\frac{3c^3+7a^3}{2c+3a}\ge 3\left(a^2+b^2+c^2\right)-\left(ab+bc+ca\right)$

Let $a;b;c>0$. Prove that $$\frac{3a^3+7b^3}{2a+3b}+\frac{3b^3+7c^3}{2b+3c}+\frac{3c^3+7a^3}{2c+3a}\ge 3\left(a^2+b^2+c^2\right)-\left(ab+bc+ca\right)$$ My proof: That inequality is equivalent $...
0
votes
2answers
76 views

Prove that the inequality is true

Prove that forl all positive real numbers $x, y, z$ we have that $ (x^3 + y^3 + z^3)^2 \geq 3(x^2y^4 + y^2z^4 + z^2x^4)$. I tried to apply Cebasev, Muirhead but doesn't work.
2
votes
3answers
173 views

Prove that $(x−2y+z)^2 \geq 4xz−8y$

Let $x,y,z$ be nonnegative real numbers such that $x+z\leq2$ Prove that, and determine when equality holds. $(x−2y+z)^2 \geq 4xz−8y$ Please correct me if my methods are incorrect or would lead ...
0
votes
1answer
519 views

$\;32\displaystyle\left(\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \sum_{c} \frac{x^2+yz}{y+z}+6$

Let $x, y, z > 0$ Prove that $\;32\displaystyle\left(\displaystyle\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \displaystyle\sum_{c} \frac{x^2+yz}{y+z}+6$ My work, $\displaystyle\sum_{c} \frac{x^2+...
1
vote
0answers
45 views

Duality gap in polynomial optimization problem Lasserre relaxation

Consider a polynomial optimization problem of the following type \begin{equation} \begin{array}{cl} \text{maximize} & p(\mathbf{x}) \\ \text{subject to} & \mathbf{x} \in K \\ \end{array} \...
2
votes
1answer
127 views

Prove $\sum\limits_{cyc} \frac{1- x_{1}}{5\,{x_{1}}^{4}- 9\,{x_{1}}^{2}- 450}\geq 0$

With $\sum\limits_{i= 1}^{3}x_{i}= 3,\,\,x_{i}\geq 0$, prove that: $$\sum\limits_{cyc} \frac{1- x_{1}}{5\,{x_{1}}^{4}- 9\,{x_{1}}^{2}- 450}\geq 0$$ Case $x_{i}\geq \frac{2}{\sqrt{5}}$, we have: $$\...
0
votes
1answer
386 views

Prove that $3\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right) \geq 2(x + y + z)\left (\frac{1}{y + z} + \frac{1}{x + z} + \frac{1}{x + y}\right)$

Let $x, y, z > 0$. Prove that: $$3\left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right) \geq 2(x + y + z) \left(\frac{1}{y + z} + \frac{1}{x + z} + \frac{1}{x + y}\right)$$ I tried proving this ...
-2
votes
2answers
64 views

$ a^3+b^3+c^3 \lt bc(b+c) +ca(c+a)+ab(a+b) \lt 2(a^3+b^3+c^3) $ [closed]

Given a, b, c are three positive real numbers such that sum of any two is grater than the third. Prove that $$ a^3+b^3+c^3 \lt bc(b+c) +ca(c+a)+ab(a+b) \lt 2(a^3+b^3+c^3) $$
0
votes
1answer
99 views

Question from a Moscow summer camp

Prove that, given $a,b,c > 0$ and $n$ a positive integer, $$\frac{a^n}{b+c}+\frac{b^n}{a+c}+\frac{c^n}{a+b} \geq \frac{a^{n-1}+b^{n-1}+c^{n-1}}{2}\ .$$ I've tried every rathole for hours on this ...
0
votes
3answers
62 views

If $a_i>0$ and $\sum_{i=1}^n a_i = 1$, then $\sum_{i=1}^n \frac{1}{a_i} \geq n^2$?

If $a_i>0$ and $\sum_{i=1}^n a_i = 1$, is $\sum_{i=1}^n \frac{1}{a_i} \geq n^2$? I'm doing an inequality exercise. If I can confirm that's true, then my proof is done. I wrote down some examples ...
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votes
2answers
70 views

For $a>0,b>0,c>0$, prove $\frac{a(a^2+bc)}{b+c}+\frac{b(b^2+ca)}{bc+a}+\frac{c(c^2+ab)}{a+b}\geq ab+bc+ca$ [closed]

Suppose $a>0,b>0,c>0$, prove $$\frac{a(a^2+bc)}{b+c}+\frac{b(b^2+ca)}{c+a}+\frac{c(c^2+ab)}{a+b}\geq ab+bc+ca.$$ Any help will welcome! This problem is from a competition. I just hope someone ...
10
votes
4answers
173 views

Express $x^4 + y^4 + x^2 + y^2$ as sum of squares of three polynomials in $x,y$ [duplicate]

I don't know any identity that'd help me simplify it. I know of Brahmagupta's identity and tried using but no good. Any hints? Edit: So far I've tried various things, $(x^2+1/2)^2 + (y^2 +1/2)^2 -1/...
2
votes
1answer
86 views

Prove this stronger inequality

Let $a,b,c>0$, and $a+b+c=1$,show that $$\sum_{cyc}\dfrac{a^3+b^2}{b+c}\ge\dfrac{2}{3}+\dfrac{5+\sqrt{2}}{12}\sum_{cyc}(a-b)^2\tag{1}$$ I have prove $$\sum_{cyc}\dfrac{a^3+b^2}{b+c}\ge\dfrac{2}{...
2
votes
0answers
114 views

Find minimum value of $\sum \frac {\sqrt a}{\sqrt b +\sqrt c-\sqrt a}$ [duplicate]

Find the minimum value of $$\sum \frac {\sqrt a}{\sqrt b +\sqrt c-\sqrt a}$$ Where $a, b, c$ represent the sides of a triangle My approach $$\sum \frac {\sqrt a}{\sqrt b +\sqrt c-\sqrt a}$$ $$=\...
3
votes
2answers
85 views

Homogeneous fourth degree inequality : $\sum x_i^2x_j^2 +6x_1x_2x_3x_4 \geq\sum x_ix_jx_k^2$

Let $x_1,x_2,x_3,x_4$ be four real numbers. The inequality $$ \sum_{i,j} x_i^2x_j^2 +6x_1x_2x_3x_4 \geq \sum_{i,j,k} x_ix_jx_k^2 $$ (where the sums are over unordered uples of distinct indices) is ...
1
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2answers
99 views

Prove expression greater than 0

Prove that the following expression $$6 \left( x_1^2 + \dots + x_n^2 \right) - 4 \left( x_1 x_2 + x_2 x_3 + \dots + x_{n-1} x_n \right) + 2 \left(x_1 x_n + x_2 x_{n-1} + \dots + x_{\frac{n+1}{2}−1} ...
1
vote
2answers
285 views

How can one prove that this polynomial is non-negative?

How one can prove the following inequality? $$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ I plotted the graph on Wolfram Alpha and found that the inequality seems to ...
5
votes
1answer
187 views

I have a inequality, I don't know where to start

Show that for $x,y,z > 0$ the inequality is true: $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}+x+y+z \geq \frac{(x+y)^2}{y+z}+\frac{(y+z)^2}{z+x}+\frac{(z+x)^2}{x+y}$ I have tried Holder, but i had ...
1
vote
1answer
66 views

Prove the next cyclic inequality

Let $a,b,c>0$ such that $$a+b+c=1$$ prove that $$cyclic\sum \frac {ab}{\sqrt {c+ab}}\le\frac {1}{2} $$ By symetrie, i proved it by assuming $a=b=c$ but i cannot justify this hypothesis.
4
votes
2answers
141 views

An algebraic inequality involving $\sum_{cyc} \frac1{(a+2b+3c)^2}$

I was reading through the proof of an inequality posted on a different website and the following was mentioned as being easily proven by AM-GM: Let $a,\ b,\ c>0$, then $$\frac{1}{(a+2b+3c)^2} + \...
0
votes
1answer
126 views

prove this inequality $Σ_{cyc}\frac{a^3+abc}{b^2+c^2}\ge a+b+c$

Let $a,b,c>0$ such that $ab+bc+ca>0$. Prove the inequality $$\frac{a^3+abc}{b^2+c^2}+\frac{b^3+abc}{c^2+a^2}+\frac{c^3+abc}{a^2+b^2}\ge a+b+c$$ My try 1: S.O.S: $$​\Leftrightarrow \frac{a^3+...
2
votes
1answer
181 views

Inequality : $\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$

Let $a, b, c$ be positive real number such that $a+b+c = 3$. Prove that $$\displaystyle\sum_{cyc}\frac{\sqrt{2}a^2b}{2a+b} \leq \displaystyle\sum_{cyc} \frac{\sqrt{a^2+b^2}}{2ab+1}$$ My attempt : ...
3
votes
7answers
607 views

Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

I am searching for an elementary proof of the AM-GM inequality in three variables: $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$ The inequality of the geometric mean vs the arithmetic mean of two variables ...
-1
votes
1answer
30 views

notaion in SOS theorem

The theorem is A polynomial $f$ has a degree-$d$ SOS certificate if and only if there exists a positive semidefinite matrix $A$ such that for all $𝑥\in\{ 0,1 \}^𝑛$, $x\in \{ 0,1 \}^n$, $$𝑓(𝑥)=⟨(...
5
votes
2answers
103 views

Prove that, if $a,b,c$ are positive real numbers $\frac {a} {2a+b+c}+ \frac {b} {2b+a+c}+\frac {c} {2c+a+b}<\frac{19}{25}$

Prove that, if $a,b,c$ are positive real numbers: $$\frac {a} {2a+b+c}+ \frac {b} {2b+a+c}+\frac {c} {2c+a+b}<\frac{19}{25}$$ Can the proof be written without the need for high mathematics?
3
votes
2answers
130 views

A problem :$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f})^3$

Hello during a problem I have this to solve : Let $a,b,c,d,e,f$ be real positiv number such that $a\geq b\geq c\geq d\geq e\geq f\geq 1$ then : $$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\frac{1}...
1
vote
0answers
50 views

Convergence of sum of squares relaxations for global polynomial optimization

I have studied and understood the Moment-SOS hierarchy proposed by Lasserre where a sequence of semidefinite programs are required to be solved and a rank condition for the moment matrices is invoked ...
4
votes
0answers
194 views

Sum-of-squares quartic polynomials in three variables

Suppose $f \in \mathbb R [x,y,z]$ is a homogeneous polynomial of degree $4$. Furthermore, suppose we can write $$f(x,y,z)=\sum_i p_i(x,y,z)^2$$ where each $p_i$ is a homogeneous polynomial of degree ...
3
votes
3answers
392 views

Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$ are different positive real numbers. First, I tried to simplify the proof ...
4
votes
2answers
127 views

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Let $x,y,z$ be real numbers all greater than $1$, then prove that $$\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$$ My Attempt: I am trying to ...
2
votes
1answer
144 views

Nessbit's and Triangle inequality

A question given in a book was as follows: Let $a$ $b$ and $c$ be the sides of a triangle. Prove that: $$\frac{3}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$$ My Attempt: The LHS is the ...
2
votes
1answer
93 views

IMO 2006, A4 Problem, Idea behind the proof

IMO 2006, Problem A4, page 13: Prove the inequality: $$ \sum_{i<j} \frac{a_{i}a_{j}}{a_{i}+ a_{j}} \le \frac{n}{2(a_{1} + a_{2} + \cdots + a_{n}) }\sum_{i<j}a_{i}a_{j} $$ for ...
6
votes
2answers
266 views

Turkevicius inequality

Let $a$, $b$, $c$ and $d$ be positive real numbers. Prove that: $$a^4+b^4+c^4+d^4+2abcd \geq a^2b^2+ a^2c^2+ a^2d^2+ b^2c^2+ b^2d^2+ c^2d^2.$$ Source : Inequalities, Theorems, Techniques and ...
7
votes
2answers
745 views

Write $(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$ as a sum of (three) squares of quadratic forms

The quartic form $$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$$ is non-negative for all real $x$, $y$, $z$, as one can check (with some effort). A theorem of Hilbert implies that there exist ...
0
votes
2answers
122 views

How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$

Let $x>0$, $y>0$ and $z>0$. Show that $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$ I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks?
2
votes
1answer
130 views

Inequality with $ab + bc + ca = 1$

As in the title, let $a>0$, $b>0$ and $c > 0$ be such that $ab + bc + ca = 1.$ Define $S = \frac{a+b+c}{\sqrt{bc} + \sqrt{ca} + \sqrt{ab}}.$ Prove that $$S^2 \ge \sum_{\text{cyc}} \frac{1}{(...
1
vote
3answers
92 views

Is it possible to prove that for $x,y,z \in \Bbb{R}$; if $x,y,z >0$ and $xyz=1$, then $(x+y+z)\ge 3$ without using the AM-GM inequality?

I'm asked in an exercise from an algebra textbook to prove that for$\{ x,y,z\} \subset \Bbb{R}$; if $x,y,z >0$ and $xyz=1$, then $x+y+z\ge 3$. Using the arithmetic and geometric mean inequality ...
0
votes
2answers
117 views

Prove this inequality $\sum _{cyc}a^3+4\left(\sum _{cyc}\frac{ab}{a^2+b^2}\right)\ge 9$

Let $a>0$, $b>0$ and $c>0$ such that $abc=1$. Prove that: $$a^3+b^3+c^3+4\left(\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}\right)\ge 9.$$ $$L.H.S=a^3+b^3+c^3+4\left(\frac{ab}{a^...
1
vote
2answers
129 views

If a, b, c>0 show that $\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}$

For positive real numbers $a$, $b$, and $c$ prove that: $$\frac{a^2}{b^2+bc}+\frac{b^2}{c^2+ac}+\frac{c^2}{a^2+ab} \ge \frac{3}{2}.$$ I let $x=\frac{a}{b}$, $y=\frac{b}{c}$, and $z=\frac{c}{a}$. Then ...
4
votes
1answer
256 views

Wanted: Low-dimensional SOS certificate for the AM-GM inequality

Consider the AM-GM inequality in five variables $$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$ Can one write the LHS as a concrete (finite) sum $\,\sum_i h_i\,s_i\,$ with ...
0
votes
1answer
261 views

Inequality : $ \sum_{cyc} \frac{a+b}{\sqrt{a+2c}} \geq 2 \sqrt{a+b+c}$

Let $a$, $b$ and $c$ be positive real numbers. Prove that: $$ \displaystyle\sum_{cyc} \frac{a+b}{\sqrt{a+2c}} \geq 2 \sqrt{a+b+c}.$$ My attempt : By Holder inequality, $ \left(\displaystyle\sum_{...