Art of Problem Solving - aops.com
This is probably the best free resource for anyone starting or looking to get more practice in competition math. Specific Tabs that are useful include:
https://artofproblemsolving.com/alcumus - Alcumus has problems for most categories of competition math and is great for learning each concept due to the detailed explanations provided.
https://artofproblemsolving.com/community/c13_contest_collections - A collection of hundreds of contests from various countries, ranging from AMC to IMO and beyond.
Contest Sites are useful for more practice problems:
MMATHS - http://www.mmaths.org/archive.html
PUMaC - https://jason-shi-f9dm.squarespace.com/archives#directory
Some AIME Resources:
Pre-Olympiad Math - A collection of hundreds of AIME problems categorized by topic
AIME "Study Guide" - A study guide of sorts with tips and strategies on how to approach various types of AIME level problems.
A few helpful/interesting links about Competition Math in general:
A fun read: