Want Cauchy and AM-GM-HM

[Uranus Hunter]

ข้อความเดิมเขียนโดยคุณ nooonuii

3. Find all nonnegative real numbers satisfying the following system of inequalities

$a+b+c\geq 1$

$ab+bc+ca\leq \sqrt{3abc}$

From the condition we have
$\sqrt {a+b+c}\sqrt{3abc} \geq ab+bc+ca$
$3abc(a+b+c) \geq (ab+bc+ca)^2$

Then, expand it

$abc(a+b+c) \geq (ab)^2+(bc)^2+(ca)^2$

But by Cauchy ; $xy+yz+zx \leq x^2+y^2+z^2 $
Plugging $x=ab,y=bc,z=ca$

$abc(a+b+c) \leq (ab)^2+(bc)^2+(ca)^2$

So $abc(a+b+c)=(ab)^2+(bc)^2+(ca)^2$

hence,we get $a=b=c=\frac 13$