Does anyone know any competitions for high school students in the writing or musical area?
Are you a musician? How about auditioning for Boston University Tanglewood Institute, or Aspen Music Festival? Not free…but if you are very notable they do have scholarships.
If you take private lessons, ask your teacher about concerto competitions on your instrument.
Do you mean writing songs or scores, or writing prose or poems?
Both. Those two areas just happen to be things that I enjoy doing 
Look in your community for local events. There may also be opportunities through your high school. We have audition opportunities for musicians at the regional and state level. Have you asked your music and English or writing teachers?
Try googling Scholastic. Also Young Arts.
I just googled writing competitions, and there are dozens. Just start looking around. And if a teen just wants to be published, submit stuff to Teen Ink. They publish most things, I beleive.