I am comfortable with majority of the medium problems I encounter, but I just can't do hard problems. Some of the more popular ones I can, but most of them I literally have no idea where to even begin, sometimes not even a brute force approach. Sometimes when I look for hints, like what the topic is, I get even more confused, like "wait, you can use BINARY SEARCH to solve the problem?".

People that can solve hard problems, how did you reach that point? Any advice on how someone of average intelligence can do so too?