Question

If $a, b, c$ are three positive numbers then

• A. $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}<3$
• B. $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 9$
• C. $a+b+c>3(a b c)$
• D. $a^2 b+b^2 c+c^2 a \geq 9 a b c$

Answer 1

Answer: (B)

(1) $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq\left(\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a}\right)^{1 / 3}$
$ \Rightarrow \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq 3$
(2) $AM \geq HM \Rightarrow \frac{ a + b + c }{3} \geq \frac{3}{\frac{1}{ a }+\frac{1}{ b }+\frac{1}{ c }}$
(3) $\frac{a+b+c}{3} \geq(a b c)^{1 / 3}$
(4) $\frac{a^2 b+b^2 c+c^2 a}{3} \geq\left(a^3 b^3 c^3\right)^{1 / 3} $
$\Rightarrow a^2 b+b^2 c+c^2 a \geq 3 a b c$