Questions tagged [uvw]
The uvw method is a very useful method for the proof of polynomial inequalities with three variables. Sometimes it works for more variables as well. This tag should be used for questions that could be tackled with this method, or questions about the method itself.
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Proving $5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.$
For all $a,b,c\ge 0$ prove that$$\color{black}{5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.}$$
I've tried to use AM-GM without success.
Indeed, $$3\cdot RHS=2\sum_{cyc}3a\sqrt{5a^...
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$a+b+c−3≥k(a−b)(b−c)(c−a)$
Given that $a,b,c \ge 0$ satisfy $ab+bc+ca=3$. Find the maximum value of the real number $k$ such that the following inequality is always true:$$a+b+c-3 \ge k(a-b)(b-c)(c-a)$$
"I got $c=0,ab=3 \...
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Proving $\sum_{cyc}\frac{a}{\sqrt{8a+bc}}\le \frac{\sqrt{a+b+c+abc}}{2}$ if $ab+bc+ca=3.$
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=3.$ Prove that$$\color{black}{\frac{a}{\sqrt{8a+bc}}+\frac{b}{\sqrt{8b+ca}}+\frac{c}{\sqrt{8c+ab}}\le \frac{\sqrt{a+b+c+abc}}{2}.}$...
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Proving $\sqrt{31a+b+c} +\sqrt{31b+a+c} +\sqrt{31c+b+a}\le 3\sqrt{3}\cdot\sqrt{a+b+c+8}$? when $a+b+c+abc=4.$
I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it.
Problem. Given non-negative real numbers $a,b,c$ ...
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Find a nice proof for an elegant problem.
Question. For any $a,b,c>0$ then prove that$$\frac{\sqrt{5a^2+4bc}}{bc}+\frac{\sqrt{5b^2+4ca}}{ca}+\frac{\sqrt{5c^2+4ab}}{ab}\le \frac{9}{4}\left(\frac{a^2+b^2+c^2}{abc}+\frac{a+b+c}{ab+bc+ca}\...
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Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.
Remark. I really like inequality problems, but I've found that I have severe difficulties with ones that are very elegant form. I can prove something simple like this
$$\color{black}{\sqrt{3a+bc}+\...
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How to solve $\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}$? [closed]
Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\color{black}{\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}.}$$
Equality holds at $a=b=c=1.$
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Inequality $\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$
Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$$
Equality occurs when $a=b=c=1.$ Also, there's an ...
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Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $
Let $a,b,c\ge 0: ab+bc+ca+abc=4.$ Prove that
$$\color{black}{\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. }$$
Equality holds at $a=b=...
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Prove $\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}$
For all $a,b,c\ge 0: ab+bc+ca>0$ then prove
$$\color{black}{\sqrt{\frac{a}{b^2+bc+c^2}}+\sqrt{\frac{b}{c^2+ca+a^2}}+\sqrt{\frac{c}{a^2+ab+b^2}}\ge 2\sqrt{2}\sqrt{\frac{(ab+bc+ca)}{(a+b)(b+c)(c+a)}}....
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Prove $\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}; a+b+c=3.$ [closed]
Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}.$$
Equality holds at $a=b=c=1$ or $a=b=\...
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Prove $\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}$ if $a^2+b^2+c^2=3.$
Given non-negative real numbers $a,b,c$ satisfying $a^2+b^2+c^2=3.$ Prove that $$\color{black}{\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}.}$$
Here is my ...
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Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$
Let $a,b,c>0: abc=1.$ Prove that: $$2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a). $$
I've tried to use a well-known lemma but the rest is quite complicated for me.
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Prove $\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}$ for $ab+bc+ca=3.$
If $a,b,c\ge 0: ab+bc+ca=3$ then prove $$\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}.$$
I tried to use AM-GM, Cauchy-Schwarz without success.
Notice that $a^...
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If $ab+bc+ca=3,$ prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$
If $a,b,c\ge 0: ab+bc+ca=3,$ then prove $$\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$$
I've tried to use Cauchy-Schwarz and it's $$\frac{1}{a+b}+\...