Olympiad Minimization Problem I've been struggling to find a solution to this problem that I found in the archive of my country's Olympiad questions. 
I'm particularly interested in a solution that doesn't involve the use of calculus since I know that Olympiad questions do not require the knowledge of calculus to solve but I will also like to see one that uses it.
Here's the problem:
Find the minimum value of $\frac{18}{a+b} + \frac{12}{ab} + 8a + 5b$  when $a$ and $b$ are positive real numbers.

For the calculus approach, setting the derivatives with respect to $a$ and $b$ to $0$ yields $$\frac{18}{(a + b)^2} + \frac{12}{a^2 b} = 8$$ and $$\frac{18}{(a + b)^2} 
 + \frac{12}{a b^2} = 5.$$
So 
$$\frac{12}{a^2 b} - \frac{12}{a b^2} = 3,$$
equivalently,
$$4(b - a) = a^2 b^2,$$
for which $(a,b)=(1,2)$ is an obvious solution that happens to satisfy both equations.
The second derivatives are all positive for positive $a$ and $b$.
