The American Math Competition is the largest math competition in the United States. Open to anyone and packed full of interesting problems for anyone, it forms the first in a series of tests that, ultimately, chooses the US team for the International Math Olympiad.
Follow the link to sign up! Even if you are unsure if you will be able to compete, filling out the form will ensure that I order enough tests to include you.
Finally, here is a geometry problem from the AMC 10 in 2011 to get you warmed up:
Square EFGH has one vertex on each side of square ABCD. Point E is on line segment AB, with AE = 7*EB. What is the ratio of the area of EFGH to the area of ABCD?