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sqing

#1 h Yesterday at 7:54 AM

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Let $a,b,c\ge \frac{1}{2}$ and $a+b+c=2.$ Prove that
$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 1$ Let $a,b,c\ge \frac{1}{3}$ and $a+b+c=1.$ Prove that
$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 9$

This post has been edited 1 time. Last edited by sqing, Yesterday at 7:57 AM

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41020 posts

sqing

#2 h Yesterday at 9:16 AM

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Let $a,b\ge \frac{1}{2}$ and $a+b=2.$ Prove that
$\left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \ge \frac{20}{9}$ $\left( \frac{1}{a}+\frac{1}{b}+2\right)\left( \frac{1}{a}-\frac{1}{b}+2\right) \ge \frac{28}{9}$ $\left( \frac{1}{a}-\frac{1}{b}+1\right)\left( \frac{1}{b}-\frac{1}{a}+1\right) \ge -\frac{7}{9}$ $\left( \frac{1}{a}+\frac{1}{b}+1\right)\left( \frac{1}{a}-\frac{1}{b}+1\right) \ge -\frac{11}{9}$

This post has been edited 2 times. Last edited by sqing, Yesterday at 9:32 AM

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41020 posts

sqing

#3 h Yesterday at 1:18 PM

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Let $a,b\ge \frac{1}{2}$ and $\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\le \frac{13}{3}.$ Prove that
$a^2+b^2\geq 1$

This post has been edited 1 time. Last edited by sqing, Yesterday at 1:21 PM

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#4 h Yesterday at 1:35 PM

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sqing wrote:

Let $a,b\ge \frac{1}{2}$ and $\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\le \frac{13}{3}.$ Prove that
$a^2+b^2\geq 1$

$\begin{align*} \tfrac 4{a^2}-(\tfrac 1b-1)^2\le \tfrac {13}3\xRightarrow{\hspace{25pt}} a^2\ge \tfrac{4}{\tfrac {13}3+(\tfrac 1b-1)^2}\ge \tfrac 34\xRightarrow{\hspace{25pt}}\ a\geq\tfrac{\sqrt3}2\xRightarrow{\hspace{25pt}} a^2+b^2\ge 1. \end{align*}$

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41020 posts

sqing

#5 h Yesterday at 2:25 PM

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Very good.Thanks.

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sqing

#6 h Yesterday at 2:35 PM

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Let $a,b\ge \frac{1}{2}$ and $\left( \frac{1}{a}-\frac{1}{b}+1\right)\left( \frac{1}{b}-\frac{1}{a}+1\right) \le -\frac{7}{9}.$ Prove that
$a+b\geq 2$ Let $a,b\ge \frac{1}{2}$ and $\left( \frac{1}{a}-\frac{1}{b}+1\right)\left( \frac{1}{b}-\frac{1}{a}+1\right)\le -\frac{16}{9}.$ Prove that
$a+b\geq \frac{7}{2}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 2:44 PM

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