I remember competing in several contests in both elementary and high school and they were invaluable experiences. Some of the skills I learned included public speaking, designing a project on a deadline, following specific instructions for entries and more. In some cases, I also learned how to achieve a goal within a group.
The good news for homeschoolers is that many academic contests are open to participation from homeschoolers.
Here are some links to academic contests for homeschoolers:
HSLDA (Homeschool Legal Defense Association)
National History Day Contest: A Homeschoolers Experience
A to Z Homeschool
Homeschool Math Contests
Homeschool Writing Contests
If you know of any homeschool academic contests, please leave a comment below. Thanks!