I am looking for a problem that can not be solved by computer but can be solved by human while computer can verify if the answer is correct or not. In fact what is the question in my head is that is there a way to clearly show the smartness in human brain compared to huge computation power of the processors.
We know that there are problems (like the halting problem) that absolutely cannot be solved by a computer because there is no algorithmic solution. But a human cannot solve those problems either.
We also know that there are problems that humans can solve, but which we do not know how to solve with a computer with our current state of knowledge and technology. Problems in image processing ("is there a teddy bear in this picture ?") or natural language processing ("do these two English sentences have equivalent meanings ?") fall into this category. The difficulty here is showing that any of these problems absolutely cannot ever be solved by a computer, given a clever enough program and enough processing power and time - it is not at all clear how you could establish that.
We do not know of any problem that can be solved by a human and absolutely cannot ever be solved by a computer, no matter how powerful.
This is indeed, an open problem. The Church-Turing thesis, in its most popular conception, states that anything that can be computed by a human with a pen and paper can be computed by a Turing Machine. We have not yet been able to prove or disprove this thesis, therefore no one can tell you such a problem.