Don't feel bad you couldn't complete one problem. This is not a problem that tests knowledge but rather skill. I have a lot of passion for mathematics and spend my free time reading about it. For this particular problem, try to see why it works.
First, put all a=b=c. Then you can get an idea of when the equality conditions holds. Then try proving the same result for two variables. $1/a + 1/b > 1/sqrt(ab)$. Why is the true ?
Now, you see that same inequality in action in rawer terms. Now, you go back to your general question and ask if you can start with $a+b/2 > 1/sqrt(ab)$, and write the same inequality thrice and add it to see if you end up where the question asked. In this case, we do.
In school, I liked mathematics a lot but the teachers weren't good. Also, I didn't have much exposure to things outside the syllabus. But, after entering college I found about exams like the Olympiad. Of course, I was too old to write it but I was just fascinated with questions such as these which require no knowledge of higher mathematics but still test you and are many times insolvable. It tested mathematical skill and not just mathematical knowledge. In school, we learn questions at the back of the chapter and then do them using the methods we learnt. We learn the problems on the context of the method instead of learning the methods in the contexts of the problems.
For example, here is a question. Given a point D inside a triangle ABC, we draw another triangle ADC. Can you prove that AB + BC > AD + DC ?
This is a middle school problem, which becomes very difficult when taken out of its context.
I continue to study problems deeply. As people become expert performers, their problem perception improves along with their performance. What that means is if there were three problems and an expert and a novice were asked to classify them into two groups, the novice would probably use superficial characteristics like geometry, numbers, algebra. The expert would use deeper characteristics like proof by contradiction, induction, analogy, etc.
An expert would have seen this problem and seen it as a problem dealing with analogies rather than inequalities.
What you need is to learn problem solving. Spend some time with some elementary books which will inspire you again. The Hungarian Problem book is great but it's hard to begin with. Here are some fun books to start with.
Mathematical Problem Solving - Alan Schoenfeld (This is more about the psychology of problem solving, than an actual Maths book.)