Choosing research topic/area Observation. After reading bios of many successful scholars I have come to the conclusion that bs/ms/phd students, who eventually succeed in their research, work on the same problems as their supervisors (usually continuing older works of the supervisors). Which makes me think that the adviser plays the crucial role of the success of a beginner researcher.

Background. During last two years of studies I tried to do some research. I approached a prof who I considered among the most active researches on the faculty, not very young / not old, who's research was in the field (some branch of cs|applied mathematics) which I was and still am interested in. He proposed me several topics and I picked one. I tried to produce the result and failed. The difficulty was that: first, I lacked some necessary knowledge, there was no actual "problem", rather "try to get some theoretical result, connected to a certain algorithm". It was very frustrating experience because neither I understood why the topic is important (so I had no real passion) nor the prof had anything particular in mind and was not researching that topic so I was unable to get any research skill from collaboration.
Next year I have chosen another topic with same prof (I thought that the topic was to blame) with almost the same result.
So I don't have any successful research experience yet. For this reason instead of applying to a PhD program directly I have decided to do a Research Masters (in Europe). No that I am in the first year of masters I want to do some research.

The question. I am not sure if the problem lies with me or I am just having a bad luck with advisers. I believe that there are geniuses who can do self-motived research on their own and who do not actually need an adviser. Is it the case for other students too? How can I actually have a research experience which fits my level?
I tried to produce the result and failed. 


As I see it, the issue here is that you were expecting the result to be some solution to the problem. Unfortunately, research tends to fail to produce results we expect quite often.

First, there is probably some result. Whether it is publishable is debatable, but I would consider the advance of the research skill a very positive result. As with all skills, it takes practice and you shouldn't expect to get it right (i.e. groundbreaking publishable results) the first (few) time(s).

Second, if you were stuck for a year without adjustment, that is a problem, not with research per se, but probably with a combination of supervision and communication. You need to set smaller milestones and adjust the course of your work accordingly. In order to do that, you must communicate with your adviser about the problems you are facing. His experience and knowledge should be able to nudge you in the right direction. However, do keep in mind that this is research, and a vital part of it is getting moving again after you've been stuck by yourself. Your adviser may help you, but can't and shouldn't do your work for you.