Sam has 3 numbers $\large a,b,c0$. Sam has stated that the numbers follow the condition:
$\large ab+bc+ac=abc$
Adi has been assigned the task of finding the smallest possible value of $\large a+b+c$. Can you help him by finding this value?

Question

Sam has 3 numbers $\large a,b,c>0$. Sam has stated that the numbers follow the condition:
$\large ab+bc+ac=abc$
Adi has been assigned the task of finding the smallest possible value of $\large a+b+c$. Can you help him by finding this value?

anonymous · Answer

Hint: apply AM-HM

anonymous · Answer

$\large \color{blue}{Solution} $

Given numbers are positive. We can always think of applying AM GM HM inequality. Note that we have

$\large ab+bc+ac=abc$

Now, what we can think here is dividing both sides by abc nd we get

$\large \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

Success. Now we can directly apply AM HM. Nd we obtain

$\large (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c} )\ge9$

Simply giving minimum of $\large a+b+c$  equal to 9