How significant are math contests / competitions?

[PetSounds]

Confession: I'm creating this thread because I'm a little disappointed in some contest results I just received. That said, hopefully it leads to an interesting general discussion. Feel free to move it if this is not the correct subforum.

Besides their role in university admissions and scholarships, how much significance should an aspiring mathematician assign to contest results? How should one contrast them with grades?

Obviously, a very high score is a sign of mathematical aptitude and a very low score a sign that improvement is required. What does a middling score (i.e. second quartile or low first quartile) indicate?

Lastly, how are such contests perceived by professional mathematicians?

 

[scottdave]

I was a member of the Math Club in Jr. High and High School. We went to contests. I would say that I usually was in the top half, sometimes in the top quarter. A few times I got a place award. It didn't discourage me. It was always the top people attending, so being in the middle of the top people isn't too bad, I figured. I kind-of enjoyed the challenge. It was good test taking practice under some actual stress, without the pressure of my GPA being affected.

 

[ISamson]

I compete in many competitions, from hacking and penetration to writing, from mathematics competitions to chemistry quizzes. This makes me very "popular" among the teachers, so they come to "respect" me. This helps with grades and further career. Competitions are a very useful add-on to further academical applications, CV and so on. Most of these competitions are held at good universities and schools around the place here, so I also get to show myself to the public and community. This is useful.
This answer might not really apply to specifically mathematics contests, but it is all similar, in general.