I wonder if anyone in graduate school in mathematics had managed to improve considerably his/her speed of problem solving. I had failed to get my PhD pass and obtained only Master Pass on my qualifiers. I have been trying to increase speed of solving problems but alas, achieved only slight improvement. I know that most problems on these exams are manageable and may be few are hard. Problems of similar level of difficulty would take me days or even weeks of solving. Any suggestions and especially real life examples of improving problem solving speed would be highly appreciated.
Success in well-designed qualifying exams in mathematics will depend very little on "problem-solving" speed or talent, but, rather, will depend on whether you've already done problems nearly identical to the problems which appear.
That is, as you observe, it would be prohibitively slow to "solve" many of those problems "in real time". Thus, that is not the expectation. Rather, the "test" is whether examinees have studied sufficiently broadly so as to have seen examples resembling the instances occurring in the given exam.
And, no, it's not about "memorization", either, which tends to be insufficiently flexible to allow easy adaptation to slightly changed situations.
So, really, it's not about "speeding up in problem-solving", but to be able to merely "remember" instead of "solving".
I don't think speed of problem solving is a criteria for graduate school in mathematics. You're doing research and inventing new mathematics, and that usually requires slowing down, not speeding up. If you're in a graduate program where not solving problems quickly leads to failure, I'd say you're not in the right program.
This is not to say that slower is better. If you're unable to solve problems relating to your subject area, that might point to deeper problems. But speed of problem solving should never be a goal (unless you're competing in a math competition like the Putnam)