# 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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### MOP 2011 inequality

If $a,b,c$ are positive integers prove that $\sqrt{(a^2-ab+b^2)} +\sqrt{(b^2+c^2-bc)} +\sqrt{(a^2+c^2-ac)} +9(abc)^{1/3} \le 4(a+b+c)$ My attempt: I tried to split inequality and prove it bit by ...
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### For $a,b,c \ge 0$ real numbers, prove $\frac{2}{(1+a)^2}+\frac{2}{(1+b)^2}+\frac{2}{(1+c)^2} \geq \frac{9}{3+ab+bc+ca}$

Let be $a,b,c \ge 0$ real numbers. Prove that: $$\frac{2}{(1+a)^2}+\frac{2}{(1+b)^2}+\frac{2}{(1+c)^2} \geq \frac{9}{3+ab+bc+ca}$$ It is question 2 from here : https://artofproblemsolving.com/...
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### Hard inequality for positive numbers

The problem is to prove that for $a,b,c>0$ we have $$\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+\frac{9abc}{4(a^3+b^3+c^3)}\geq \frac{15}{4}.$$ I have tried to use Bergstrom/Engel inequality ...
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### If $x,y,z>0.$Prove: $(x+y+z) \left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right) \geq9\sqrt[]\frac{x^2+y^2+z^2}{xy+yz+zx}$

If $x,y,z>0.$Prove: $$(x+y+z) \left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right)\geq9\sqrt[]\frac{x^2+y^2+z^2}{xy+yz+zx}$$ I was not able to solve this problem instead I could solve similar ...
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### Maximizing $\frac{a^2+6b+1}{a^2+a}$, where $a=p+q+r=pqr$ and $ab=pq+qr+rp$ for positive reals $p$, $q$, $r$

Given $a$, $b$, $p$, $q$, $r \in\mathbb{R_{>0}}$ s.t. $$\begin {cases}\phantom{b}a=p+q+r=pqr \\ab =pq+qr+rp\end{cases}$$ Find the maximum of $$\dfrac{a^2+6b+1}{a^2+a}$$ This question is ...
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### Given $x+y+z=3, x,y,z>0$ how to prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} >= x^2+y^2+z^2$

Given $x+y+z=3, x,y,z>0$ how to prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} >= x^2+y^2+z^2$ ? I tried something basic like $(x+y+z)^2 = x^2+y^2+z^2 + 2xy+2zx+2yz$, so we just need to ...
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### Given three positive numbers $a,b,c$. Prove that $\sum\limits_{cyc}\sqrt{\frac{a+b}{b+1}}\geqq3\sqrt{\frac{4\,abc}{3\,abc+1}}$ .

Ji Chen. Given three positive numbers $a, b, c$. Prove that $$\sum\limits_{cyc}\sqrt{\frac{a+ b}{b+ 1}}\geqq 3\sqrt{\frac{4\,abc}{3\,abc+ 1}}$$ Of course, we've to solve it by $uvw$, before that,...
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### Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ . My own problem is given a solution, and I'm looking forward to seeing a ...
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### How do you come up with great and ingenious questions [closed]

In mathematics how are the professionals and great authors able to come up with such ingenious questions which which are so difficult yet elegant. Like in integral calculus it involve a substitution ...
### Minimize value of the function $a^2+b^2+c^2+2\sqrt{3abc}$
Let $a,b,c$ be the positive real numbers such that $a+b+c=1$. Find Minimize of $$P=a^2+b^2+c^2+2\sqrt{3abc}$$ WA says that $P$ gets only a local minimum. But i think it must be maximum value of $P$. ...