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.
337 questions
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Prove that $\frac{7-4a}{a^{2}+2}+\frac{7-4b}{b^{2}+2}+\frac{7-4c}{c^{2}+2}\ge 3$
Let $a,b,c$ be real variables with $ab+bc+ca+abc=4.$ Prove that:$$\color{black}{\frac{7-4a}{a^{2}+2}+\frac{7-4b}{b^{2}+2}+\frac{7-4c}{c^{2}+2}\ge 3.}$$When does equality hold?
This inequality is ...
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Maximum of $\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}$ for non-negative real numbers satisfying $x+y+z=3$ [duplicate]
Let $x, y, z$ be non-negative real numbers such that $x + y + z = 3$.
Find the maximum value of
$$
\frac{1}{x^2 + 1} + \frac{1}{y^2 + 1} + \frac{1}{z^2 + 1}.
$$
My attempt:
First, I assumed $z = 0$, ...
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$\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3.$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c\ge 0$ with $ab+bc+ca=1.$ Prove that$$\color{black}{\frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}...
- 504
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Creating lemmas to use in proving inequalities involving sums of radicals (square roots, cube roots, etc.)
I am very interested in a motivated systematic approach to generating lemmas such as those invoked by River Li to transform inequalities stated in terms of radicals into inequalities that are radical ...
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If $a,b,c>0$ such that $(a+b+c)^3=125abc$, then show that $ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$
Problem. Let $a,b,c$ be positive real numbers such that $$(a+b+c)^3=125abc$$
Prove that :
$$ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$$
This problem ...
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Math Olympiad Inequality Hard Homogeneous
I found a interesting problem in Aops. It is a hard inequality problem. Here is the problem and my approach(and the link):
https://artofproblemsolving.com/community/c6h3188980p34729637
Let $a,b,c$ be ...
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Prove $\sum \frac{1}{\left(1+\sqrt{\dfrac{a}{bc}}\right)^2} \ge \frac{3}{4}, \forall \ a,b,c>0:a+b+c=3$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c>0: a+b+c=3$ then prove that$$\frac{1}{\left(1+\sqrt{\dfrac{a}{bc}}\right)^2}+\frac{1}{\left(1+\sqrt{\dfrac{b}{ca}}\...
- 957
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How to solve this inequality?
Prove $\dfrac{a}{\sqrt{b^2 + 3c^2}} + \dfrac{b}{\sqrt{c^2 + 3a^2}} + \dfrac{c}{\sqrt{a^2 + 3b^2}} \geq \dfrac{3}{2}$ if $a,b,c > 0$.
I have tried using Cauchy-Schwarz and Holder, but nothing works ...
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Prove $\left(\sum \sqrt{ab}\right)^2+3\sum a\cdot \sqrt{\sum ab} \ge 5\sum ab+\sum a \cdot \sum \sqrt{ab},\forall a,b,c \ge 0$
I'm looking for some ideas to solve the following inequality.
Problem. For all non-negative real variables $a,b,c,$ prove that$$m^2+3pn\ge 5n^2+pm$$holds where$$p=a+b+c;\quad m=\sqrt{ab}+\sqrt{bc}+\...
- 957
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Prove $\sum \frac{1}{4ca+4cb+3}\ge \frac{12}{a+b+c+41}, \forall a,b,c\ge 0: ab+bc+ca+abc=4$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c\ge 0: ab+bc+ca+abc=4$ then prove $$\color{black}{\frac{1}{4ab+4ac+3}+\frac{1}{4bc+4ba+3}+\frac{1}{4ca+4cb+3}\ge \...
- 957
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Prove $\sum \frac{a}{6a^{2}+a+2}\le\frac13$ for reals $a+b+c=3$
I'm looking for some ideas to solve the following inequality.
Problem. For any real numbers $a,b,c$ with $a+b+c=3,$ prove that$$\color{black}{\frac{a}{6a^{2}+a+2}+\frac{b}{6b^{2}+b+2}+\frac{c}{6c^{2}...
- 957
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Prove that: $\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ca}}+\frac{1}{\sqrt{c^2+4ab}}\ge \frac{5}{4}$
Let $a,$ $b$ and $c$ be non-negative numbers such that: $a+b+c+abc=4$ and $ab+bc+ca\neq 0.$ Prove that $$\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ca}}+\frac{1}{\sqrt{c^2+4ab}}\ge \frac{5}{4}.$$
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Prove that $\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1$ for real numbers
How do we prove that for all $a, b, c \in \mathbb{R}$,
$$\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1.$$
I haven't really made much progress in finding a way to tackle this....
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How can I prove this inequality
I'm looking for some ideas to solve the following inequality.
Problem. For any non-negative real numbers $a,b,c$ with $ab+bc+ca+abc=4,$ prove that$$\color{blue}{\sqrt{\frac{a}{bc+8}}+\sqrt{\frac{b}{...
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Prove that $\frac{ab+3c}{5c^2+4}+\frac{bc+3a}{5a^2+4}+\frac{ca+3b}{5b^2+4}\leq\frac{4}{3}$
Let $a$, $b$ and $c$ be real numbers such that $a+b+c=3$. Prove that:
$$\frac{ab+3c}{5c^2+4}+\frac{bc+3a}{5a^2+4}+\frac{ca+3b}{5b^2+4}\leq\frac{4}{3}.$$
This inequality was posted here.
I solved it ...