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How can I improve my problem-solving ability?

How do you improve your problem solving skills?

Should I learn design patterns or algorithms to improve my logical thinking skills?

What to do when you're faced with a problem that you can't solve quickly?

Are there non-programming related activities akin to solving programming problems?

I am a computer engineering graduate. I have studied programming since three years. I am good in coding and programming. I have been trying to compete in algorithmic competitions on sites such as topcoder,spoj since one and a half year, but I am still unable to solve problems other than too easy problems. I have learned from people that it takes practice to solve such problems. I try to solve those problems but sometimes I am unable to understand and even if I do understand I am unable to think of a good algorithm for solving it. Even if I solve I get Wrong answer and I am unable to figure out what is the problem with my code as it works on samples given on the sites but fails on test cases which they do not provide.

I really want to solve those problems and become good in algorithms. I have read books for learning algorithms like Introduction to algorithms by CLRS,practicing programming questions.

I have gone through some questions but they don't answer this question. I have seen the questions which are said duplicates but those questions focus on overall programming, but I am asking for algorithm related programming, basically for competing in programming which involve solving a problem statement then online judge will automatically evaluate it, such type of programming is quite different from the type of programming these questions discuss.