x,y,z>1

For discussing Olympiad Level Algebra (and Inequality) problems
photon
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x,y,z>1

Unread post by photon » Wed Feb 19, 2014 1:17 pm

Prove that if $x,y,z$ $>$ $1$ , and $\displaystyle \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 2$ , then
$\displaystyle \sqrt{x+y+z}\geq\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$

Source - IMO longlist (1992)
Try not to become a man of success but rather to become a man of value.-Albert Einstein

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asif e elahi
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Re: x,y,z>1

Unread post by asif e elahi » Wed Feb 19, 2014 9:39 pm

As $\sum \frac{1}{x}=2$,$\sum \frac{x-1}{x}=1$
By Cauchy Schwarz inequality
$\sum x=\sum x \times \sum \frac{x-1}{x}\geq (\sum \sqrt{x-1})^{2}$
or $\sum x\geq (\sum \sqrt{x-1})^{2}$
or $\sqrt{\sum x}\geq \sum \sqrt{x-1}$
So $\sqrt{x+y+z}\geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$ :mrgreen:

photon
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Re: x,y,z>1

Unread post by photon » Sat Feb 22, 2014 10:38 pm

that was cool 8-)

Let $x=a^2+1 , y=b^2+1 ,z=c^2+1 $, ($a,b,c>0$)
then inequality becomes $\displaystyle \sqrt{a^2+b^2+c^2+3} \geq a+b+c$
$\Rightarrow a^2+b^2+c^2+3 \geq a^2+b^2+c^2+2ab+2bc+2ca$
$\Rightarrow \frac{3}{2}\geq ab+bc+ca $
We have $\sum \frac{1}{x}=2 \Rightarrow \sum \frac{1}{a^2+1}=2$
Let $f(x)=\frac{1}{x^2+1}$ , $f''(x)>0$ , by jenson's inequality ,
$\displaystyle \frac{1}{3} \sum \frac{1}{a^2+1} \le \frac{1}{(\frac{a+b+c}{3})^2+1}$
$\displaystyle \Rightarrow \frac{2}{3} \le \frac{9}{(a+b+c)^2+9}$
$\Rightarrow (a+b+c)^2 \le \frac{9}{2}$
$\Rightarrow \sum a^2 + 2\sum ab \le \frac{9}{2}$ ......(1)
Again , $f(x)=\frac{1}{x}$ , $f''(x)>0$
Again applying Jenson , $\frac {1}{3}[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}] \le \frac{3}{x+y+z}$
$\displaystyle \Rightarrow \frac{2}{3} \le \frac{3}{a^2+b^2+c^2+3}$
$\Rightarrow a^2+b^2+c^2 \le \frac{3}{2}$ .......(2)
adding 1 and 2 ,
$2(\sum a^2 + \sum ab) \le 6$
$\Rightarrow \sum a^2 + \sum ab \le 3$
as , $\sum ab \le \sum a^2 $ - adding last 2 inequalities -
$2\sum ab \le 3 \Rightarrow ab+bc+ca \le \frac{3}{2}$ which satisfies the inequality .
Try not to become a man of success but rather to become a man of value.-Albert Einstein

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