Help me
Cho a,b,c>1 và 1/a+1/b+1/c=2
$\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a+b+c}$
(Olympic Iran 98)
Help me
Cho a,b,c>1 và 1/a+1/b+1/c=2
$\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a+b+c}$
(Olympic Iran 98)
Vì $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$ => $\frac{x-1}{x}+\frac{y-1}{y}+\frac{z-1}{z}=1$.
Áp dụng BĐT Cauchy - Schwarz ta có:
$x+y+z\doteq (x+y+z)(\frac{x-1}{x}+\frac{y-1}{y}+\frac{z-1}{z})\geq (\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1})^{2}$.
=> đpcm.
Dấu bằng "=" xảy ra khi x = y = z = 3/2.
"IF YOU HAVE A DREAM TO CHASE,NOTHING NOTHING CAN STOP YOU"_M10
Đặt $a,\,b,\,c= \frac{x+ y+ z}{y+ z},\,\frac{x+ y+ z}{z+ x},\,\frac{x+ y+ z}{x+ y}$ với $x,\,y,\,z>0$
Ta có $\sum\limits_{cyc}\sqrt{a- 1}= \sum\limits_{cyc}\sqrt{\frac{x}{y+ z}}\leqq \sqrt{\sum\limits_{cyc}x}\,\sqrt{\sum\limits_{cyc}\frac{1}{y+ z}}= \sqrt{\sum\limits_{cyc}a}$
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