2 var inquality

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sqing

#1 h Yesterday at 4:06 AM

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Let $a , b\geq 0$ and $\frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{3}{2}.$ Show that $a+b+ab\geq1$ Let $a , b\geq 0$ and $\frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{5}{6}.$ Show that $a+b+ab\geq2$

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sqing

#2 h Yesterday at 4:13 AM

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Let $a , b\geq 0$ and $\frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{15+\sqrt 5 }{10}.$ Show that $a+b+a^2 +b^2 \geq 1$ Let $a , b\geq 0$ and $\frac{1}{a^2+1}+\frac{1}{b^2+1}\ge \frac{9 }{5} .$ Show that $a+b+a^2+ab+b^2 \leq1$

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#4 h Yesterday at 12:25 PM

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sqing wrote:

Let $a,b >0 .$ Prove that
$\frac{a+b}{2b}+\frac{2b}{a+b}+1\geq \sqrt{3\left(\frac{a+b}{2b}+\frac{2b}{a+b}+1\right)}$

$\frac{a+b}{2b}+\frac{2b}{a+b}+1\ge \sqrt{3\left( \frac{a+b}{2b}+\frac{2b}{a+b}+1 \right)}\Leftrightarrow \frac{a+b}{2b}+\frac{2b}{a+b}\ge 2$ -it's true

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#5 h Yesterday at 12:26 PM • 1 Y

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sqing wrote:

Let $a,b >0 .$ Prove that
$\frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\geq \sqrt{3\left(\frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\right)}$ Where $k>0.$

$\frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\ge \sqrt{3\left( \frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1 \right)}\Leftrightarrow \frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}\ge 2$

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sqing

#6 h Yesterday at 12:30 PM • 1 Y

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Good.Thanks.
Let $a,b,c>0 .$ Prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\sqrt{3\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)}$

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#7 h Yesterday at 1:29 PM

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Let $a,b>0.$ Prove that $\left( \frac{a+b}{2a}+\frac{2a}{a+b}+2 \right)\left( \frac{a+b}{2b}+\frac{2b}{a+b}+2 \right)\ge 12+2\left( \sqrt[4]{\frac{a}{b}}+\sqrt[4]{\frac{b}{a}} \right)$

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#8 h Yesterday at 1:40 PM

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Let $a,b>0.$ Prove that $\left( \frac{a+b}{2a}+\frac{2a}{a+b}+1 \right)\left( \frac{a+b}{2b}+\frac{2b}{a+b}+1 \right)\ge 5+2\left( \sqrt[4]{\frac{a}{b}}+\sqrt[4]{\frac{b}{a}} \right)$

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