This is a response to this, but I thought it should have its own post so you can post thoughts and ask questions.
Since this is a long post, let me put the main overall points at the top, those from my personal story that helped me:
Make math something fun, such that you study it not just to do well on competitions (although it is a wonderful side-effect) but because you enjoy it.
Treat a "failure" as motivation rather than responding with some sort of depression. On the same point, when someone else succeeds, use this as motivation rather than responding in a negative way.
Stay calm. Don't put too much pressure on yourself.
Here are some tips about how to improve:
Study one level up. In the US, this means the AMC 10 and AIME for Mathcounts, and Olympiads for AIME, and so on. This is #1 because I feel, although maybe not as important as some of the others, this is the one that is least emphasized in my opinion.
Do practice problems. This is essential. You must know which topics are repeated. You also get a feel for different types of problems, and common mistakes to watch out for. About timing, I personally never really time myself, although this may be because I am fast, so decide what works for you (on Mathcounts sprints I can have 1-20 done in 10-15 minutes, and on AMC 10 I can have the first 20 done and checked in 25-30 minutes).
On competitions, check your work. Even if you still have problems to do, you should always check. Spending a couple minutes to check quite a few problems is usually more profitable than spending those minutes to solve one more hard problem at the end.
Make a strategy. Plan out timing. Plan when (not if!) you will check, and which problems you will attempt. This should not be strict, change as you see fit during the test, but I find this really helps me.
Understand what you do not understand. By this, I mean you should always try to understand any problem you cannot do, and also, make sure you see the motivation behind the method used.
Don't be afraid to skip problems. This is probably the least important, but it is still important. If a #7 stumps you, skip it. Don't think, "This is #7 it must be easy," because different people have different strengths. Move on, and you may figure it out when you come back, with a fresh look.
Teach it. You also learn when you teach concepts to others.
Here is my math story (not to brag, but to show how I got where I am): From first grade, I was above average in math. I enjoyed it, and my mom taught me extra at home. I started competition math with a little known and very simple (relatively) competition called Mathfax. I was very happy in 5th grade when I got a perfect score and first place in the nation, along with a sizable trophy. However, the problems weren't very hard.
In 6th grade, I did no competition preparation at all. No Mathcounts, which I see as a mistake looking back. I did take the AMC 8 and 10, getting a 20 and a 78, respectively, and I didn't really care.
7th grade is the year I really picked it up. I was 2nd place at my school in Mathcounts (which continued, usually behind Ashwin Sah), 5th at chapter, and placed 8th at state. This was a little disappointing, as I wanted to make nationals, but I realized later that I probably wasn't good enough. The biggest event math-wise of my 7th grade year was my "failure" (in my perspective) on the AMC 10. I tried hard to make it to the AIME, and yet, I got 114 on the 10A, and... 114 on the 10B as well (for those who don't know, the cutoff is 120 or so and each question is 6 points). This was very disappointing. Due to this, I was more motivated to study, and I took some AoPS classes (all competition, no topic classes), and did a lot of practice over the summer. I discovered a lot of problems that really interested me, and were very intriguing. This was also a key factor.
When the 8th grade year started, until the AMCs I spent a lot of time in class doing past AIMEs and things like that. I again made my school's Mathcounts team, and this time I got 2nd at chapter, and 1st at Countdown (CD). This was encouraging. On the AMC 8, I got 24, unfortunately never getting the perfect 25. On the AMC 10s, I got 138 and 132. Next was state, where I got 2nd (making nationals!). On the AIME, I managed to get a 10, with 2 silly mistakes. On the USAJMO, I got problems 1 and 2 for a 14, while messed up my proof for 4. At nationals, I got 6th written with a 44 (compare to the 40 I got at state), and 10th after CD, and Oregon got 4th (no trophies or plaques, just missed it :). Although I am usually quite good at CD, I messed up because I was too nervous. My 8th grade year was very successful.
Finally, this year, I got a 144 and 150 (yay!) on the AMC 10s. Sorry about this being so long and (maybe) hard to understand.
Now, you may see two big improvements, both from 7th grade to 8th. One is on the AMCs, with a AMC 10 score jump from 114 to 138, and an AIME jump from 0 to 10 (although this 0 is not entirely accurate). On Mathcounts, I went from 8th at state to 6th (written) at nationals. I credit this to mainly motivation, and as a product of that, studying. I had three motivating factors in 7th grade. First of all, I discovered AIME problems, which I found to be very, frankly, fun and also interesting. Secondly, my "failure" on the AMC 10 really motivated me, and I think this is a good lesson. Treat a "failure" as motivation rather than responding with some sort of depression. Lastly, Ashwin from my school placed 2nd at national MC, and, as I could easily compare myself with him, I could see that a high goal would not be impossible to reach.
Overall, see the top of the post for the main points (everything bolded).
Thanks for reading this long post!
Sorry I haven't been active for quite a while. Mostly this is due to school and the fact that the AMCs have started=more prep... I'll try to become more active again when I can.