Some areas have additional olympiads at the regional or local level. The jump from short answers to olympiads is a tough one. Here are some tips for students making this transition. Practice, practice, practice. The only way to learn math is by doing.
Proofs are essays.
The better written a proof is, the more likely it is to be understood. Even such mundane things as grammar, spelling and handwriting are worth a bit of attention. Define your terms. If you're going to use a word in a way that might not be commonly understood, define it precisely. Read the masters.