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Questions tagged [uvw]

This is a very useful method for the proof of polynomial inequalities with three variables. Sometimes it works for more variables.

-2
votes
1answer
47 views

Prove $\sum\limits_{cyc}\frac{ab}{b^{\,2}+ c^{\,2}}\geqq \frac{3}{2}$

For $a\geqq b\geqq c> 0$. Prove $$\frac{ab}{b^{\,2}+ c^{\,2}}+ \frac{bc}{c^{\,2}+ a^{\,2}}+ \frac{ca}{a^{\,2}+ b^{\,2}}\geqq \frac{3}{2}$$ I used discriminant to find & want to see a solution ...
1
vote
2answers
49 views

Prove $2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$ [on hold]

For $a,\,b,\,c\geqq 0$ and $b\equiv {\rm mid}\,\{\,a,\,b,\,c\,\}$. Prove $$2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$$ Inspried from $\lceil$ Prove $k=0$ is the only non-...
0
votes
0answers
59 views

Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...
0
votes
0answers
63 views

Prove that $\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$ [closed]

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$$ I tried Holder Inequality (it's only the hint to get you ...
0
votes
3answers
93 views

How to find $f(X)$ such that $\sum\limits_{cyc}a^2-f(X)[abc-(1-a)(1-b)(1-c)]\geqq\frac{3}{4}X^2$ for $abc=(X- a)(X- b)(X- c),0\leqq a,\,b,\,c\leqq X$?

We have $\sum\limits_{cyc}\,a^{\,2}\geqq \frac{3}{4}\,X^{\,2}\tag{HaiDangel29}$ with $abc= (\,X- a\,)(\,X- b\,)(\,X- c\,),\,0\leqq a,\,b,\,c\leqq X$. Here is a hint to get you started from above. For $...
0
votes
1answer
84 views

Prove $\frac{1}{a+ 2\,b}+ \frac{1}{b+ 2\,c}+ \frac{1}{c+ 2\,a}\leqq 1$ with $8\,abc\geqq a+ b+ c+ 5$ and $a,\,b,\,c> 0$

Prove $$\frac{1}{a+ 2\,b}+ \frac{1}{b+ 2\,c}+ \frac{1}{c+ 2\,a}\leqq 1$$ with $8\,abc\geqq a+ b+ c+ 5$ and $a,\,b,\,c> 0$ $$constant= 8$$ is the best $constant$, which was found by me (using ...
0
votes
1answer
98 views

Prove $\prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 $

Prove $$\begin{equation}\begin{split} \prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 \end{split}\end{equation}$$ with $a,\,b,\,c> 0$. $$\begin{equation}\begin{...
2
votes
1answer
89 views

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$ [duplicate]

let $x,y,z,w\in R$,and such $x+y+z+w=2$.show that $$\sum_{cyc}\dfrac{x}{x^2-x+1}\le\dfrac{8}{3}$$ I have only solve when $x,y,z,w>0$, because $$\dfrac{x}{x^2-x+1}\le\dfrac{4}{3}x$$ so $$\sum_{...
1
vote
2answers
93 views

Find maximum of function $A=\sum _{cyc}\frac{1}{a^2+2}$

Let $a,b,c\in R^+$ such that $ab+bc+ca=1$. Find the maximum value of $$A=\frac{1}{a^2+2}+\frac{1}{b^2+2}+\frac{1}{c^2+2}$$ I will prove $A\le \dfrac{9}{7}$ and the equality occurs when $a=b=c=\dfrac{...
0
votes
2answers
94 views

Prove/disprove $\sum_{cyc}a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} \ge 2$ where $a$, $b$, $c > 0$ and $a^2 + b^2 + c^2 = 1$

$a$, $b$ and $c$ are positives such that $a^2 + b^2 + c^2 = 1$. Prove/disprove that $$a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} + b\sqrt{\frac{(ab + 1)(bc + 1)}{ca + 1}} + c\sqrt{\frac{(bc + 1)(ca + 1)}{...
1
vote
1answer
76 views

Prove $F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0$

Given that $1\leqq X\leqq k,$ prove that $$F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0 \tag{29}$$ Origin For $a,\,b,\,c\geqq 0$ and $a+ b+ c= 3$, prove that $(2+a^2)(2+b^2)(2+c^2)+...
0
votes
3answers
124 views

Prove $1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geqq a+ b+ c+ ab+ bc+ ca$ for $a,\,b,\,c\geqq 0$

Given that $a,\,b,\,c$ are $3$ non-negatve numbers, prove $$1+ a^{\,2}+ b^{\,2}+ c^{\,2}+ 4\,abc\geqq a+ b+ c+ ab+ bc+ ca$$ Let $X= a+ b+ c$$,$ we have to prove $$\left ( \frac{1}{X^{\,3}}- \frac{1}{X^...
1
vote
2answers
74 views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Let $a,b,c$ be non-negative real numbers such that $c \geq 1$ and that $a+b+c=2$. Find the minimum and maximum values of $$P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$$ To find the minimum of $P$ ...
0
votes
1answer
110 views

For $a,b,c>0,\,a+b+c=3,$ prove $\sum\limits_{cyc}\,\frac{1}{a}\geqq\left(\frac{1}{2}+\frac{5}{18}\,\sqrt{3}\right)(a^2+b^2+c^2)$

Given $a,\,b,\,c> 0$ such that$:$ $a+ b+ c= 3$$.$ Prove$:$ $$\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}\geqq \left ( \frac{1}{2}+ \frac{5}{18}\,\sqrt{3} \right )(\,a^{\,2}+ b^{\,2}+ c^{\,2}\,)$$ I find $...
1
vote
0answers
178 views

Prove $\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$ for $x,\,y,\,z\in [\,j,\,k\,j\,]$

For $x,\,y,\,z\in [\,j,\,k\,j\,]$ $$\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$$ is true with $j= constant,\,k= constant> 8$ and $k_{\,\max}$...
2
votes
3answers
62 views

Inequalities, but working around an absolute value

So what I want to prove is $$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$ for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$. Now I'm able to ...
2
votes
2answers
76 views

Inequality with a+b+c=1 and $18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3)$

Let $a,b,c$ be reals with $a+b+c=1$. Show that : $$18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3).$$ I have tried to something like this: $$18a^4-24a^3+6a^2-12a+12\geq 0$$ $$18b^4-24b^3+6b^2-12b+...
4
votes
3answers
300 views

$\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ if $a+b+c=ab+bc+ca$?

For $a,b,c>0 $ and $a+b+c=ab+bc+ca$ . Prove or disprove that : $\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ I checked in very many cases. Example :$c=1, a=2,b=\frac{1}{2}...$ then it’s ...
-1
votes
2answers
65 views

Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ [closed]

Given the three real numbers a, b, c are not negative, in which at most some are equal to zero. Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ ...
3
votes
1answer
83 views

Prove $\Sigma_{cyc}(\frac{a}{b-c}-3)^4\ge193$

The inequality is expected original question of this MSE question. The exact statement is "If $a$, $b$ and $c$ are positive real numbers and none of them are equal pairwise, prove the following ...
2
votes
1answer
206 views

Prove $ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $

Show that if $a,b,c > 0$, such that $ab + bc + ca = 1$, then the following inequality holds: $$ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $$ What I ...
12
votes
3answers
292 views

Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$

Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$ If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ ...
0
votes
1answer
74 views

How to prove the given inequality using CS or AM GM HM inequality [closed]

I was solving a problem on triangular inequalities and I have ended up with the following inequality required to be proven $$\sum_{cyc}\frac{x}{2x+y+z}\geq\frac{9\sqrt{3(x+y+z)xyz}}{4(x+y+z)^2},$$ ...
0
votes
2answers
105 views

UVW method application of basic theorem

Find the minimum and maximum value of $x+y+z+xy+yz+xz$ if $x^2+y^2+z^2 = 1$ I converted it to uvw, $3u+3v^2$ is the expression, $(3u)^2-2(3v^2)=1$, is the constraint. now I don't know what to do, ...
1
vote
1answer
183 views

If $x+y+z=3$ then $\sum x\sqrt{x^3+3y} \ge 6$

Let $x,y,z>0$ such that $x+y+z=3$. Prove that $$\sum x\sqrt{x^3+3y} \ge 6$$ This trying doesn't help. With Cauchy Schwarz $(\sum x\sqrt{x^3+3y})^2\geq \sum x^2\sum(x^3+3y) = (x^2 + y^2 + z^2)(x^...
2
votes
2answers
218 views

Prove that: $xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$

Let $x$,$y$ and $z$ are positive and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\leq 3$$ Prove that: $$xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$$ The things I have done so far $$3\geq \sum \limits_{cyc}\...
3
votes
2answers
202 views

Inequality with $abc=2$

Let $a,b,c$ be positive real numbers such that $abc=2$. Prove that $\frac{a+b+c}{4}\geq \sqrt[4]{\frac{a^2+b^2+c^2}{6}}$ Sorry, I don't have an idea for this problem.If I had, I would have showed them ...
1
vote
1answer
174 views

Prove that $\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$.

If $x,y,z \in[0,1/2]$, with $x+y+z=1$, then prove that: $$\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$$ OK so... I've tried to square the expression ...
0
votes
1answer
123 views

How to prove this inequality ??? [closed]

Let $a,b,c$ are positive real numbers such that $a+b+c=3$. Prove that $$\sum\frac{(7a^{3}+3)(b+c)}{7a+3} \geq 6 $$ I try to prove $LHS \geq \sum\frac{9}{5}a+\frac{1}{5}$ but don't succeed
7
votes
1answer
225 views

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

For $x,y,z>0,$ I have to prove that $$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$$ I tried to use $$ \sqrt {\frac {1}{3} \...
0
votes
2answers
115 views

What tools one should use for inequalities?

If $a,b,c>0$ prove that: $$\frac{1}{a+4b+4c}+\frac{1}{4a+b+4c}+\frac{1}{4a+4b+c}\leq \frac{1}{3\sqrt[3]{abc}}.$$ My first try was the following: $$\sum_{cyc}\frac{1}{a+4b+4c}\leq\sum_{cyc}\frac{1}{\...
0
votes
1answer
40 views

Double integral of $(y-2)^2$ over $\{(x,y)\in\mathbb{R}^2\colon x^2 +y^2 \le 6x\}$

Trying to solve this for hours... will appreciate any help $$\int\int_{D}^{} (y-2)^2dxdy$$ $$D=\{(x,y)| x^2+y^2\leq6x\}$$ Tried doing it in a few ways... doesn't seem to work out.
0
votes
1answer
63 views

An inequality with a condition

If $x,y,z>0$ and $\sum_{cyc}^{} x^2 (1-x) \ge 0 $, I have to prove that: $\sum_{cyc}^{} x^2 y^2 (1-xy) \ge 0 $ Then, if $\sum_{cyc}^{} x=3,$ then I have to prove $\sqrt [8] \frac {\sum_{cyc}^{} ...
4
votes
1answer
166 views

A tricky algebraic inequality

This is an old inequality but I haven't seen a satisfactory solution yet and am hoping someone here can provide one. There are a couple of brute force solutions but they provide no insight into the ...
11
votes
1answer
423 views

prove this inequality with $63$

Let $x,y,z,w>0$, and such $x^2+y^2+z^2+w^2=1$. show that $$x+y+z+w+\dfrac{1}{63xyzw}\ge\dfrac{142}{63}\tag{1}$$ I know $$x^2+y^2+z^2+w^2\ge 4\sqrt[4]{x^2y^2z^2w^2}\Longrightarrow xyzw\le \dfrac{...
4
votes
2answers
627 views

Whenever $a+b+c=1$, $\frac{bc+a+1}{a^2+1} + \frac{ac+b+1}{b^2+1} + \frac{ab+c+1}{c^2+1} \le \frac{39}{10}$

Prove that for a + b + c =1 and a,b,c are positive real numbers, then $$\frac{bc+a+1}{a^2+1} + \frac{ac+b+1}{b^2+1} + \frac{ab+c+1}{c^2+1} \le \frac{39}{10}$$ My try: if one term is proven to be $\...
1
vote
1answer
70 views

show this $\prod(a^3+b^3+ab)^2\ge (\sum a^2b)^3( \sum ab^2)^3$

let $a,b,c>0$ show that :$$(a^3+b^3+abc)^2(b^3+c^3+abc)^2(c^3+a^3+abc)^2-(a^2b+b^2c+c^2a)^3\cdot(ab^2+bc^2+ca^2)^3\ge 0$$Maybe there's some kind of identity in there.so I use wolfampha Calculated ...
0
votes
1answer
61 views

Stronger than Nessbit

Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc\neq0$. Prove that: $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+\frac{3\sqrt[3]{a^2b^2c^2}}{2(a^2+b^2+c^2)}\geq2\tag{1}$$ we know ...
1
vote
2answers
137 views

If $x+y+z=2$ prove that $\sum_{cyc}\frac{1}{\sqrt{x^2+y^2}}\ge2+\frac{1}{\sqrt{2}}$

Let $x,y,z$ be non-negative reals whose sum is $2$. Prove that $\frac{1}{\sqrt{x^2+y^2}}+\frac{1}{\sqrt{y^2+z^2}}+\frac{1}{\sqrt{z^2+x^2}}\ge2+\frac{1}{\sqrt{2}}$ I have tried bounding them up (...
2
votes
1answer
85 views

An inequality with condition

I have a new inequality this is the following : Let $x,y,z$ be real strictly positive number such as : $$-2 = - x y z + x + y + z $$ Then we have : $$\sqrt{\frac{1}{x}}+\sqrt{\frac{1}{y}}+\...
1
vote
1answer
97 views

Prove this inequality $Σ_{cyc}\sqrt{\frac{a}{b+3c}}\ge \frac{3}{2}$ with $a;b;c>0$

Let $a,b,c>0$. Prove $$\sqrt{\frac{a}{b+3c}}+\sqrt{\frac{b}{c+3a}}+\sqrt{\frac{c}{a+3b}}\ge \frac{3}{2}$$ $A=\sqrt{\frac{a}{b+3c}}+\sqrt{\frac{b}{c+3a}}+\sqrt{\frac{c}{a+3b}}$ Holder: $A^2\cdot ...
3
votes
3answers
133 views

Prove $\left ( a+ b \right )^{2}\left ( b+ c \right )^{2}\left ( c+ a \right )^{2}\geq 64abc$

Give $a, b, c$ be positive real numbers such that $a+ b+ c= 3$. Prove that: $$\left ( a+ b \right )^{2}\left ( b+ c \right )^{2}\left ( c+ a \right )^{2}\geq 64abc$$ My try We have: $$\left ( a+ b \...
0
votes
1answer
83 views

Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$

Let $a,b,c>0$ such that $abc=1$. Prove the inequality $$\frac{1}{\left(a+1\right)^2}+\frac{1}{\left(b+1\right)^2}+\frac{1}{\left(c+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$$ I proved $\frac{1}{\left(a+1\...
1
vote
1answer
53 views

Find the minimum of a three variate function

We consider the function $$f(x,y,z)=(7(x^2+y^2+z^2)+6(xy+yz+zx))(x^2y^2+y^2z^2+z^2x^2)$$ Find $$m=\min\{f(x,y,z):xyz=1\}$$ Using the AM-GM inequality it is clear that $$m_+=\min\{f(x,y,z):xyz=1,x,y,z&...
1
vote
1answer
97 views

Triangle inequality with the exradii $r_{a}$, $r_{b}$, $r_{c}$, the medians $m_{a}$, $m_{b}$, $m_{c}$

Given a triangle with the exradii $r_{a}$, $r_{b}$, $r_{c}$, the medians $m_{a}$, $m_{b}$, $m_{c}$. Show that $$r_{a}^{2}+ r_{b}^{2}+ r_{c}^{2}\geq 3\sqrt{3}. S+ \left ( m_{a}- m_{b} \right )^{2}+ \...
1
vote
2answers
329 views

Prove $\sum\limits_{cyc}\sqrt[3]{\frac{a^2+bc}{b+c}}\ge\sqrt[3]{9(a+b+c)}$

Let $a,b,c$ be positive real numbers. Show that $$\sum_{cyc}\sqrt[3]{\dfrac{a^2+bc}{b+c}}\ge\sqrt[3]{9(a+b+c)}$$ I have tried C-S and Holder inequalities, without success. How to solve it?
-1
votes
1answer
90 views

Inequality with $ab+bc+ca=3$ [closed]

Let $a$, $b$, $c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc}{\frac{a^2+b^2}{a+b+1}}\geq \frac{6abc+6}{a+b+c+3}$. I have tried using Holder $\sum_{cyc}{\frac{a^2+b^2}{a+b+1}}\geq \frac{2(a+b+c)^2}{\...
2
votes
2answers
86 views

How to solve (in)equality of three variables with trigonometric solutions

I'm working through a set of inequality problems and I'm stuck on the following question: Find all sets of solutions for which $$(a^2+b^2+c^2)^2=3(a^3b+b^3c+c^3a)$$ holds. Note that $a,b,c\in\...
3
votes
3answers
227 views

Finding the maximum of $p^3 + q^3 +r^3 + 4pqr$

$p$,$q$,$r$ are $3$ non-negative real numbers less than or equal to $1.5$ such that $p+q+r = 3$, what will be the maximum of $p^3 + q^3 + r^3 + 4pqr$ ? I tried AM-GM on $p,q,r$ to get the maximum of $...
3
votes
1answer
147 views

maximum and minimum of $\frac{a^4+b^4+c^4}{(a+b+c)^4},$

Finding maximum and minimum of $\displaystyle \frac{a^4+b^4+c^4}{(a+b+c)^4},$ Where $a,b,c>0$ and $(a+b+c)^3=32abc$ $\bf{My Attempt}$ with AM-GM inequality $\displaystyle \frac{a+b+c}{3}\geq (abc)^...