Does anyone have long-term advice for becoming better at timed problem solving, such as in a test or competition setting? I can dedicate one hour per day 5 days/week to train.
Read the problem carefully. Clearly define your variables. Typically each sentence can be translated to one equation.
Thats not exactly the type of advice I was looking for... Has anyone here been involved in competitions at some point? How did you practice, and did it work?
Practice by... practicing. Do some sample competitions (usually you can get older versions of the test) and see what kind of problems come up. Some you'll know how to do, some you won't. Pick out the ones you don't know how to do, learn how to do them, then find more problems of that type to practice on. It depends on what kind of competition it is, in high school a lot of them were big on geometry so learning obscure theorems involving circles and secants were worthwhile, depending on what kind of test you're taking different subjects will be important to know
Learn algebra tricks. Learn your trig identities. Memorize the Taylor series for common functions. Memorize Gaussian integrals. And practice.