# Toán 9Cho a, b , c > 0 và abc = 1.

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#### c3lttrong.0a1.nhphat

##### Học sinh
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Cho a, b , c > 0 và abc = 1. Chứng minh rằng: $\frac{b+c}{\sqrt{a}} + \frac{c + a}{\sqrt{b}} + \frac{a + b}{\sqrt{c}} \geq \sqrt{a} + \sqrt{b} + \sqrt{c} + 3$
Edgarnguyen248
$\frac{b+c}{\sqrt{a}} + \frac{c + a}{\sqrt{b}} + \frac{a + b}{\sqrt{c}} \geq 2(\sqrt{\dfrac{bc}{a}}+\sqrt{\dfrac{ca}{b}}+\sqrt{\dfrac{ab}{c}}) = (\sqrt{\dfrac{ca}{b}}+\sqrt{\dfrac{ab}{c}})+(\sqrt{\dfrac{ab}{c}}+\sqrt{\dfrac{bc}{a}})+(\sqrt{\dfrac{bc}{a}}+\sqrt{\dfrac{ca}{b}})$$\geq 2(\sqrt{a}+\sqrt{b}+\sqrt{c}) \geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3\sqrt[6]{abc}=\sqrt{a}+\sqrt{b}+\sqrt{c}+3$

Edgarnguyen248

#### c3lttrong.0a1.nhphat

##### Học sinh
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$\frac{b+c}{\sqrt{a}} + \frac{c + a}{\sqrt{b}} + \frac{a + b}{\sqrt{c}} \geq 2(\sqrt{\dfrac{bc}{a}}+\sqrt{\dfrac{ca}{b}}+\sqrt{\dfrac{ab}{c}}) = (\sqrt{\dfrac{ca}{b}}+\sqrt{\dfrac{ab}{c}})+(\sqrt{\dfrac{ab}{c}}+\sqrt{\dfrac{bc}{a}})+(\sqrt{\dfrac{bc}{a}}+\sqrt{\dfrac{ca}{b}})$$\geq 2(\sqrt{a}+\sqrt{b}+\sqrt{c}) \geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3\sqrt[6]{abc}=\sqrt{a}+\sqrt{b}+\sqrt{c}+3$
c3lttrong.0a1.nhphatdấu bằng xảy ra khi a=b=c=1

Edgarnguyen248